Le platonisme est l'idée qu'il existe dans l'abstrait (c'est-à-dire non spatial,
objets non temporels) (voir entrée sur
objets abstraits).
Parce que les objets abstraits sont complètement non spatio-temporels, il s'ensuit
qu'ils sont aussi complètement non physiques (ils n'existent pas dans
monde physique et n'est pas fait de choses physiques) et non mental
(ce ne sont pas des cerveaux ou des idées dans la tête; ce ne sont pas des âmes dissidentes,
ou des dieux, ou toute autre chose dans ce sens). De plus, ils sont
immuable et complètement inerte causalement – c'est-à-dire qu'ils ne peuvent pas être
impliqué dans les relations de cause à effet avec les autres
objets.(1)
Tout cela peut être quelque peu déroutant; car avec tout cela
déclarations sur ce que sont les objets abstraits ne pas, ça peut être
pas clair ce qu'ils est. Cependant, nous pouvons clarifier les choses en
Regardez quelques exemples.
Pensez à la phrase «3 is prime». Cette phrase semble être
dire quelque chose sur un objet particulier, à savoir le numéro 3. Tout comme
l'expression «la lune est ronde» en dit long
lune, donc «3 est premier» semble aussi en dire quelque chose
numéro 3. Mais quoi est le numéro 3? Il en existe plusieurs
vues que l'on pourrait soutenir ici, mais le point de vue platonicien est que 3 est
un objet abstrait. À ce point de vue, 3 est une chose réelle et objective
qui, comme la lune, existe indépendamment de nous et de notre pensée
(c'est-à-dire que ce n'est pas seulement une idée dans nos têtes). Mais selon
Platonisme, 3 est différent de la lune en ce qu'il n'est pas physique
objet; il est complètement non physique, non mentalement et causalement inerte, et
il n'existe ni dans l'espace ni dans le temps. On pourrait dire ça métaphoriquement
en disant que dans la vue platonicienne, les nombres existent "en platonicien
le ciel. "Mais nous ne devons pas dériver de cela comme par
Platonisme, les chiffres sont en un endroit; ils ne le font pas aussi
Le concept de lieu est un concept physique et spatial. C'est plus précis
dire que dans une vision platonicienne, les nombres existent (indépendamment de nous
et nos pensées) mais n'existent pas dans l'espace et le temps.
De même, de nombreux philosophes ont une vision platonicienne
Propriétés.
Par exemple, considérez la propriété comme rouge. selon
Vue platonicienne des propriétés, la propriété de la rougeur existe
indépendamment de toute substance rouge. Il y a des boules rouges et des maisons rouges et
chemises rouges, et tout cela se trouve dans le monde physique. Mais les platonistes
A propos des attributs croient qu'en plus de ces choses, les rougeurs
– le bien lui-même – existe aussi, et selon
Platoniciens, cette propriété est un objet abstrait. Objets rouges communs
est dit à exemplifier ou instancier rougeur. Platon
dit qu'ils participe à rougeur, mais cela suggère une
relation causale entre les objets rouges et les rougeurs, et encore une fois,
les platoniciens contemporains le rejetteraient.
Les platoniciens de ce type en disent autant d'autres propriétés comme
Eh bien: en plus de toutes les belles choses, il y a aussi la beauté;
et en plus de tous les tigres, c'est aussi la propriété d'être
un tigre. Même lorsqu'il n'y a aucun cas de propriété
La réalité est que les platoniciens prétendent généralement que la propriété elle-même
existe. Cela ne veut pas dire que les platoniciens se sont engagés à la tâche
que c'est une propriété qui correspond à tous les prédicats de
Langues anglaises. Le fait est simplement que dans des cas typiques, il
sera une propriété. Selon ce type de platonisme,
il y a une propriété d'être un bâtiment de quatre cents étages, même
bien qu'il n'y ait pas de bâtiments de quatre cents étages. Cette
La propriété existe en dehors de l'espace et du temps avec des rougeurs. Le seul
la différence est que dans notre monde physique, une caractéristique se produit
devenir instantané tandis que l'autre ne le fait pas.
En fait, les platoniciens étendent la position ici davantage, pour
Leur point de vue, les caractéristiques ne sont qu'un cas particulier de beaucoup plus large
catégorie, à savoir la catégorie de Unités. C'est facile de
voyez pourquoi on peut considérer une propriété comme une rougeur comme un universel. UNE
La boule rouge assise dans un garage de Buffalo est une chose spéciale. Mais
la rougeur est un exemple de nombreux objets; c'est
quelque chose que tous les objets rouges partagent ou ont en commun. C'est pourquoi
Les platoniciens considèrent la rougeur comme un objet rouge universel et spécifique
– par exemple, des balles à Buffalo ou des voitures à Cleveland – comme
information.
Mais selon ce type de platonisme, les propriétés ne sont pas les seules
chariots; il existe également d'autres types d'universels, notamment
des relations. Par exemple, pensez à la relation à
au Nord de; cette relation est médiée par de nombreuses paires d'objets
(ou plus précisément, de paires d'objets ordonnées, car l'ordre compte
ici – par exemple au Nord de est médiée par
selon le platonisme, la relation au Nord de est un
universel deux places, alors qu'une propriété comme la rougeur en est une
un endroit universel. Il existe également des relations entre trois endroits
(qui sont universels à trois places), des relations initiales, etc.
Un exemple de relation à trois endroits est aller relation,
qui admettent un donateur, un donateur et un donateur – comme dans
"Jane a donné un CD à Tim."
Enfin, certains philosophes le prétendent proposition est
objets abstraits. Une façon de penser une proposition est le sens
d'une phrase. Alternativement, nous pouvons dire qu'une proposition est là
qui s'exprime par une phrase à une occasion particulière d'utilisation.
Cependant, nous pouvons dire que, par exemple, l'expression anglaise «Snow is
blanc et l'expression allemande «Schnee ist weiss»
exprimer la même proposition, à savoir la proposition que la neige est
blanc.
Il existe de nombreuses notions platonistes différentes des propositions. À
Par exemple, Frege (1892, 1919) croyait que les propositions sont composées de
sens de mots (par exemple, de ce point de vue, l'affirmation
La neige est blanche est composée des sens de la «neige» et
'Is White', tandis que Russell à un moment donné (1905,
1910-11) croyait que les propositions sont constituées de propriétés,
relations et objets (par exemple, de ce point de vue, la suggestion que Mars
est rouge est composé de Mars (la planète elle-même) et
propriété de rougeur). D'autres estiment que les propositions ne
ont une structure interne importante. Les différences entre ces
les vues n'auront aucune incidence sur notre objectif. Pour plus de détails, voir la liste
sur
proposition.
(Il peut sembler étrange de dire que les propositions russes sont abstraites
objets. Par exemple, imaginez la proposition russe selon laquelle Mars est rouge.
C'est un genre étrange hybride objet. Il a deux composantes,
à savoir Mars (la planète elle-même) et la propriété de la rougeur. Un des
ces composants (à savoir Mars) sont un béton objet (où un
l'objet concret n'est qu'un objet spatio-temporel). Ainsi, bien que
la rougeur est un objet abstrait, elle n'apparaît pas au Russe
la proposition est totalement non spatiotemporelle. Néanmoins,
les philosophes regroupent généralement ces objets avec l'abstrait
objets. Et ce ne sont pas seulement les propositions russes; remarques similaires
peut être créé sur divers autres types d'objets. Par exemple, pensez,
d'ensembles impurs – par ex. l'ensemble contenant Mars et Jupiter. Cela marche
être un objet hybride d'une sorte aussi, parce que même s'il l'a
des objets concrets en tant que membres, il en reste un ensemble, et sur
vue standard, les ensembles sont des objets abstraits. Si nous voulions être très
exactement, il serait probablement préférable d'avoir un terme différent pour
objets – par exemple «Objet hybride» ou «abstrait impur
object & # 39; – mais encore une fois, ce n'est pas ainsi que les philosophes parlent habituellement;
ils traitent généralement ces choses comme des objets abstraits. Rien de tout cela
jouera cependant beaucoup dans ce qui suit, car cet essai est
presque entièrement préoccupé par ce qu'on peut appeler pur
objets abstraits – c'est-à-dire des objets abstraits qui sont complets
non-temps et espace).
Nombre, propositions et universels (ie propriétés et relations)
ne sont pas les seules choses que les gens ont prises pour être des objets abstraits.
Comme nous le verrons ci-dessous, les gens ont également rejoint les vues platoniciennes
association avec des objets linguistiques (en particulier des phrases), possible
mondes, objets logiques et personnages fictifs (par exemple, Sherlock
Holmes). Et il est important de noter ici que l'on peut être platonicien
à propos de certaines de ces choses sans être platoniste pour les autres
– par exemple, on peut être platoniste sur les nombres et les propositions
mais pas des traits ou des personnages fictifs.
Bien sûr, le platonisme concerne certains de ces types d'objets
controversé. Beaucoup de philosophes ne croient pas aux objets abstraits
toutes les personnes. Les alternatives au platonisme seront discutées dans
section 2,
Mais il convient de noter que l'argument principal en tant que platoniciens
Donner à leur avis, c'est que, selon eux, c'est bon
arguments contre tous les autres points de vue. Autrement dit, les platonistes pensent que nous l'avons
croire aux objets abstraits, car (a) il y a de bonnes raisons
pense que des choses comme les nombres et les universaux existent, et (b)
la seule vision durable de ces choses est qu'elles sont des objets abstraits.
Nous examinerons ces arguments en détail ci-dessous.
Il n'y a pas beaucoup d'alternatives au platonisme. On peut rejeter
l'existence de choses comme les nombres et le héros universel. Ou tu peux
soutiennent qu'il y a des choses comme les nombres et les universaux,
et au lieu de dire que ce sont des objets abstraits, vous pouvez dire que
ce sont des objets mentaux d'une certaine sorte (généralement, l'affirmation est qu'ils
sont des idées dans nos têtes) ou des objets physiques en quelque sorte. Donc,
quatre points de vue principaux ici sont les suivants (et rappelez-vous que
les anti-platoniciens peuvent poursuivre différentes stratégies
différents types d'objets abstraits présumés, avec une vue, disons,
nombres, et une vision différente des propriétés ou des propositions).
- platonisme: Voici la vue décrite dans
section 1. -
Réalisme immanent: Les partisans de ce point de vue sont d'accord
Les platoniciens qu'il existe des choses comme des objets mathématiques
– ou universels, ou toute catégorie de résumé présumé
les objets dont nous parlons – et que ces choses sont
indépendamment de nous et de notre pensée; mais les réalistes immanents diffèrent
Les platoniciens qui soutiennent que ces objets existent dans le monde physique.
Selon le type d'objet discuté – c'est-à-dire
nous parlons d'objets ou de propriétés mathématiques ou ce qui a
vous – les détails de cette vue seront élaborés d'une manière différente.
Dans le contexte des propriétés, la norme est une vue immanente et réaliste
que des propriétés comme la rougeur n'existent que dans le monde physique, je
en particulier dans les éléments rouges réels, comme les parties non patiatiques ou les aspects de
ces choses (cette vue remonte à Aristote; dans le présent
il a parfois été défendu par Armstrong (1978). C'est certainement
une certaine probabilité initiale de cette idée: si vous regardez une rouge
balle, et vous croyez qu'en plus de la balle, sa rougeur existe,
alors il semble un peu étrange de dire (comme le font les platoniciens) que sa rougeur
existe en dehors de l'espace. Après tout, la balle se trouve ici
dans l'espace-temps et nous pouvons voir qu'il est rouge; comme il semble au premier abord
susceptibles de penser que si la rougeur existe, alors elle existe
dans le ballon. Cependant, comme nous le verrons ci-dessous, il existe de graves problèmes
avec cette vue.En ce qui concerne les chiffres, une stratégie consiste à prendre les chiffres
universels de toute nature – par exemple, on peut les prendre pour être
les propriétés des tas d'objets physiques, comme par ex.
le numéro 3 serait un trait pour, disons, un tas de trois livres –
et avoir une vision immanemment réaliste de l'universel. (Ce genre de vue
ont été défendus par Armstrong (1978).) Mais des vues de ce genre ont
pas très influent dans la philosophie des mathématiques. Un de plus
stratégie de premier plan pour prendre le chat vocal pour faire face à la physique
le monde considère qu'il s'agit de véritables piles d'objets physiques,
plutôt que les propriétés des pieux. Ainsi, par exemple, vous pouvez le faire
maintenez que dire que 2 + 3 = 5 est vraiment ne rien dire
sur des unités spécifiques (nombres); c'est plutôt de le dire à tout moment
nous poussons une pile de deux objets avec une pile de trois objets,
nous finirons avec une pile de cinq objets – ou quelque chose
ces lignes. De cette façon, l'arithmétique est juste très générale
sciences naturelles. Une vue de ce type a été développée par Mill (1843) et,
plus récemment, un point de vue similaire a été défendu par Philip Kitcher
(1984). Il convient toutefois de noter que, bien qu’il soit
thèmes physiques qui traversent les vues de Mill et Kitcher, qui
il n'est pas clair qu'aucun d'entre eux ne doit être interprété comme immanent
réaliste. Kitcher est probablement mieux classé comme une sorte d'anti-réaliste
(Je vais en dire un peu plus à ce sujet dans
section 4.1),
et on ne sait pas exactement comment Mill devrait être classé, relativement
à notre taxonomie, car il n'est pas clair comment il réagirait
la question: "Y a-t-il des chiffres, et si oui, de quoi s'agit-il?
le?"Enfin, Penelope Maddy (1990) a également développé une sorte d'immanent
vue réaliste des mathématiques. Se concentre principalement sur la théorie des ensembles, Maddy
soutient que des ensembles d'objets physiques sont placés dans l'espace et le temps,
exactement où se trouvent leurs membres. Mais Maddian dit que ça ne peut pas être
identifié à la matière physique qui constitue leurs membres. Sur
La vue de Maddy, correspond à tous les objets physiques, c'est un gros
ensemble infini (par exemple ensemble contenant l'objet donné, ensemble)
qui contient cet ensemble, etc.) qui sont tous différents les uns des autres
mais ils partagent tous le même cas et le même spatio-temporel
placement. Ainsi, de ce point de vue, c'est plus un ensemble que le physique
les choses qui composent les membres et pour que Maddy puisse aller mieux
interprété comme soutenant une version non standard du platonisme. -
conceptualisme (aussi appelé psychologisme et
mentalisme, selon le type d'objets ci-dessous
discussion): C'est l'opinion que les chiffres existent – ou
propriétés ou propositions, ou autre chose – mais ils ne le font pas
exister indépendamment de nous; ce sont plutôt des objets mentaux; dans
en particulier, l'affirmation est généralement qu'ils sont quelque chose que les idées dans
nos tetes. Comme nous le verrons ci-dessous, ce point de vue pose de sérieux problèmes et
peu de gens le soutiennent. Néanmoins, il a connu des périodes de
popularité dans l'histoire de la philosophie. On pense très souvent
Locke avait une vision conceptualiste de l'universel et avant
au XXe siècle, c'était l'image standard des concepts et
proposition. Dans la philosophie des mathématiques, les vues psychologiques
étaient populaires à la fin du XIXe siècle (le plus remarquable
porte-parole est le premier Husserl (1891) et même dans la première partie
du XXe siècle avec l'arrivée d'un psychologue
l'intuitionisme (Brouwer 1912 et 1948, et Heyting 1956). Enfin Noam
Chomsky (1965) a rejoint une vision mentaliste des phrases et d'autres choses
objets linguistiques, et il a été suivi ici par d'autres, la plupart
en particulier Fodor (1975, 1987).Il convient également de noter ici que l’on peut soutenir que l’existence de
les nombres (ou propositions ou autre) sont en fonction de nous
êtres humains sans soutenir une vision psychologique de la question
dispositifs. Car on peut combiner cette affirmation avec l'idée que
les objets en question sont des objets abstraits. En d'autres termes, vous pouvez
affirmation – et certains avoir a affirmé – que les chiffres (ou
propositions ou autre) sont des objets abstraits dépendant de l’esprit, c.-à-d.
les objets qui existent en dehors de l'esprit, et en dehors de l'espace et du temps,
mais cela ne vient que des activités humaines
les êtres. Liston (2003-2004), Cole (2009) et Bueno (2009) se joignent
vues de ce type général par rapport aux objets mathématiques;
Schiffer (2003, chapitre 2), Soames (2014) et King (2014) se joignent
des vues comme celle-ci des suggestions; et Salmon (1998) et Thomasson
(1999) soutient des vues telles que celle des objets de fiction. -
scission (aussi appelé anti-réalisme): C'est
voir qu'il n'y a pas de choses telles que les nombres ou universel ou
quel que soit le type d'objet abstrait allégué dont il est question. Donc,
par exemple, un nominaliste sur les caractéristiques le dirait pendant qu'il était là
sont des choses comme des boules rouges et des maisons rouges, il n'y a rien appelé
la propriété des rougeurs, au-delà des boules rouges et des maisons rouges.
Et un nominaliste sur les chiffres dirait que s'il y a de telles
des choses comme des tas de trois pierres, et peut-être "3 idées"
trouvé dans la tête des gens, il n'y a rien de tel que le nombre 3. Comme
Nous voyons ci-dessous, il existe de nombreuses versions différentes de chaque
sorte de nominalisme, mais pour l'instant nous n'avons besoin de rien de plus
cette formulation générale de la vue. (Quelquefois
Le «nominalisme» est utilisé pour indiquer qu'il n'y en a pas
des choses comme des objets abstraits; sur cette utilisation,
«Nominalisme» est synonyme de
L'anti-platonisme et des vues comme le réalisme immanent comptent comme
versions du nominalisme. Contrairement à l'usage utilisé dans ce
essai, «nominalisme» est essentiellement synonyme de
«Anti-réalisme», et des vues comme le réalisme immanent
ne compte pas ici comme versions du nominalisme.)
À première vuecela peut ressembler au nominalisme ou à l'anti-réalisme,
est plus loin du point de vue platonicien que le réalisme immanent et
Le conceptualisme est pour la simple raison que les deux derniers apparaissent
admettre qu'il existe des choses telles que les nombres (ou universels ou
peu importe). Cependant, il est important de noter que les nominalistes conviennent
avec les platoniciens à un moment important en tant que réalistes immanents et
les conceptualistes rejettent; en particulier les nominalistes (en accord avec
platoniciens) soutient la thèse suivante:
(S) S'il y avait des choses comme des nombres (ou universels ou autre
sorte d’objets abstraits présumés dont nous parlons), ils
être des objets abstraits; c'est-à-dire qu'ils ne seraient pas spatio-temporels,
non physique et non mental.
C’est un point extrêmement important, car il s’avère
sont quelques arguments très convaincants (dont nous discuterons) en faveur de
(S). En conséquence, il y a très peu de partisans du réalisme immanent et
le conceptualisme, en particulier en ce qui concerne les objets mathématiques et
proposition. Il existe un large consensus sur les nombres et
propositions seraient s'il y avait de telles choses (à savoir abstrait
objets) mais très peu de consensus s’il existe de telles
des choses. Ainsi, aujourd'hui, la question controversée est pure
ontologiques: y a-t-il des choses comme des objets abstraits (par ex.
objets mathématiques, propositions, etc.)?
Il convient de noter que, s'il n'y a que quatre points de vue traditionnels
ici (Platonisme, Réalisme immanent, Conceptualisme et
nominalisme) un cinquième point de vue mérite d'être mentionné, à savoir
Meinongianism (voir Meinong (1904)). Dans cette vue, chaque
désignation singulière – par ex. «Clinton», «3»,
et 'Sherlock Holmes' – choisir un objet qui a
une sorte d'être (il subsisteou est, auberge
un peu de sens) mais seuls certains de ces objets ont leur pleine existence.
Selon le méinongianisme, les platoniciens prennent des peines pour agir
objets abstraits – des phrases comme «3 is prime» et
«Le rouge est une couleur» – exprimer des vérités sur des objets comme
n'existe pas.
Le méinongianisme est presque universellement rejeté par les philosophes.
L'argument standard contre lui (voir, par exemple, Quine (1948), p. 3 et
Lewis (1990)) est qu'il ne donne pas une vision claire
différent du platonisme et ne crée que l'illusion d'un autre
voir en changeant le sens du terme «existe». L'idée
voici que la signification par défaut de & # 39; existe & # 39; tout
un objet qui n'a aucun être existe, et donc selon la norme
utilisation, le méinongianisme implique que les nombres et les universaux existent; mais
cette vision ne prend évidemment pas de telles choses pour exister dans l'espace-temps
Ainsi, l'argument conclut, le méinongianisme implique que les nombres et
les universaux sont des objets abstraits – tout comme le platonisme.
Cependant, il convient de noter que si le méinongianisme a été largement
rejeté, il a quelques défenseurs plus contemporains, le plus notable,
Routley (1980), Parsons (1980) et Priest (2003, 2005).
Ce sont deux principaux arguments pour le platonisme. Le premier, qui
remontant à Platon, est un argument pour l'existence de propriétés et
seules les relations; C'est Argument un sur plusieurs. le
d'autres sont également présents dans un certain sens dans les œuvres de Platon (au moins
sur certaines lectures de ces œuvres), mais la première formulation moderne,
et certainement le premier clair formulation, a été donnée par Frege
(1884, 1892, 1893-1903, 1919); J'appellerais ça singulier
le concept d'argument, et contrairement à de nombreux autres, il peut être utilisé dans
connexion avec tous les différents objets abstraits, à savoir.
nombres, propriétés, propositions, etc. Dans cette section,
nous allons discuter de l'argument One Over Many, et dans la section suivante,
nous allons discuter de l'argument singulier.
L'argument One Over Many peut être formulé comme suit:
J'ai devant moi trois objets rouges (disons une balle, un chapeau et un
Rose). Ces objets sont similaires. Voilà pourquoi ils l'ont
quelque chose en commun. Ce qu'ils ont en commun est clairement une propriété,
à savoir, rougeur; donc la rougeur existe.
Nous pouvons considérer cet argument comme la fin de la meilleure explication.
C’est un fait qui nécessite des explications, à savoir celui des trois
les objets se ressemblent. L'explication est qu'ils possèdent tous
une seule caractéristique, à savoir la rougeur. Ainsi les platoniciens se disputent s'il y a
il n'y a pas d'autre explication à ce fait (c'est-à-dire le fait de l'égalité)
c'est aussi bon que leur explication (c'est-à-dire celle qui fait appel
propriétés), alors nous avons le droit de croire aux propriétés.
Notez que comme l'argument a été énoncé ici, ce n'est pas un
argument pour une vision platonicienne des propriétés; c'est un argument pour
thèse qu'il existe des propriétés, mais pas pour la thèse
les propriétés sont des objets abstraits. Ainsi, pour utiliser cet argument
pour motiver le platonisme, il faut le compléter avec quelques
raison de penser que les propriétés en question ne pouvaient être
des idées dans nos têtes ou des propriétés immanentes qui sont spéciales
objets physiques. Il y a un certain nombre d'arguments que l'on peut utiliser
ici, et dans
section 4.3,
nous en discuterons certains. Mais il n'est pas nécessaire de poursuivre
ici, car il y a de bonnes raisons de croire qu'il domine de nombreux
l'argument ne réussit pas de toute façon – c'est-à-dire qu'il ne donne pas
une bonne raison de croire aux propriétés de quelque sorte. En d'autres termes,
L'argument One Over Many ne réfute pas le nominalisme
Propriétés.
Avant de poursuivre, il convient de souligner que sur de nombreux
L'argument décrit ci-dessus peut être simplifié. Comme Michael Devitt (1980)
souligne l'appel à l'égalité, ou à nombreuses des choses
Avoir une propriété donnée est un hareng rouge. Dans le traditionnel
formulation, les nominalistes sont mis au défi d'expliquer ce qui suit
Réalité: la balle est rouge et le chapeau est rouge. Mais si les nominalistes peuvent
expliquer que la balle est rouge, probablement
il suffit de répéter la même explication sur le chapeau,
et ils veulent expliquer le fait que les deux choses sont rouges.
Ainsi, le véritable défi pour le nominaliste est d'expliquer facilement
des faits prédictifs, tels que le fait que la balle est rouge, sans
fait appel à des propriétés telles que la rougeur. Plus généralement, ils doivent le faire
montrer comment on peut expliquer la vérité dans les phrases du formulaire
"une est FSans faire appel à une propriété
de
Fness.(2)
(Vous pouvez également penser à l'argument comme ne demandant pas d'explication
de Mars, par exemple, étant rouge, mais plutôt une déclaration de ce
c'est le monde qui rend la phrase «Mars est
rouge & # 39; vrai. Voir Peacock (2009) à cet égard.)
C'est une réponse nominaliste très familière à One Over Many
argument. Le cœur de la réponse est capturé par ce qui suit
Commentaire de Quine (1948, p. 10):
Que les maisons et les roses et les couchers de soleil soient tous rouges peuvent être pris
comme ultime et irréductible, et on peut supposer que … (le platoniste)
n'est pas mieux, dans le cas d'un réel pouvoir explicatif, pour tout le monde
entités occultes qu'il constitue sous des noms tels que
'Rougeur'.
Il y a deux idées différentes ici. Le premier est que les nominalistes peuvent
Répondez-y à plusieurs en faisant appel à des faits irréductibles, ou
brute faits. La seconde est que les platoniciens ne sont pas meilleurs
que ces nominalistes de fait brut quand il s'agit d'un réel pouvoir explicatif.
Maintenant, Quine n'a pas beaucoup parlé de ces deux idées, mais les deux idées
a été développé par Devitt (1980, 2010), dont nous suivons l'exposition
ici.
Le défi pour les nominalistes est de fournir une explication des faits
une certaine espèce, à savoir les faits prédictifs exprimés par les théorèmes de
la forme & # 39;une est F& # 39;, Par exemple, le fait qu'un
la balle donnée est rouge. Maintenant, quand nous sommes mis au défi d'en donner un
explication d'un fait, ou d'un fait allégué, nous avons un certain nombre d'options. le
La réponse la plus évidente est simplement de donner l'explication souhaitée.
Mais nous pouvons également affirmer que le fait allégué n'est pas vraiment un fait
toutes les personnes. Ou, troisièmement, nous pouvons affirmer que le fait réel est un
brute fait – c'est-à-dire un fait qui n'en a pas
explication. Maintenant, dans le cas présent, les nominalistes ne peuvent pas l'exiger
tous les faits prédictifs sont des faits bruts car il est clair que nous
pouvez expliquer au moins quelques faits de ce genre. Par exemple,
Il semble que le fait qu'une balle donnée soit rouge s'explique beaucoup
simplement en disant qu'il est rouge car il réfléchit la lumière dans tel et tel
de telle manière et qu'il réfléchit la lumière de cette façon à cause de la surface
est structuré de telle ou telle manière. Les nominalistes ne devraient donc pas exiger
que tous les faits prédictifs sont des faits bruts. Mais comme le souligne Devitt,
C'est une manière plus subtile de faire appel à la Bruteness ici, et si les Chinois
les nominalistes en font usage, ils peuvent bloquer One Over Many
argument.
La réponse de Quine-Devitt à One Over Many commence par la revendication
que nous pouvons expliquer que la balle est rouge, sans
fait appel à la propriété de la rougeur, juste en utilisant ce que
les scientifiques expliquent ce fait. Maintenant, en soi, ce
l'explication ne satisfera pas les défenseurs de l'argument One Over Many.
Si nous expliquons le fait que la balle est rouge en soulignant que son
La surface est structurée d'une certaine manière, puis préconise une
Ci-dessus De nombreux arguments diraient que nous venons de reculer le problème d'un
étape, parce que les nominalistes doivent maintenant l'expliquer
la surface du ballon est structurée de la manière indiquée et ils le veulent
pour ce faire sans faire appel à la propriété d'être structuré en
la voie donnée. Plus généralement, le point est le suivant: bien sûr, il est vrai
que si les nominalistes sont invités à rendre compte du fait que quelqu'un s'oppose
une est F, sans faire appel à la propriété
Fness, ils peuvent le faire en soulignant que (i) une
est g et (ii) tous Fs est gs (c'est
le genre d'explication qu'ils obtiendront s'ils empruntent leurs explications
des scientifiques); mais de telles explications ne font que ramener le problème
étape, car ils nous ont laissé la tâche d'expliquer le fait
cette une est get si nous voulons défendre le nominalisme,
Nous devons le faire sans faire appel à la propriété pour
gness.
C'est là que l'appel de la brutalité vient. Les nominalistes peuvent dire
que (a) nous pouvons continuer à fournir des explications du type ci-dessus (c.-à-d.
explications de type 'une est F parce que c'est
g, & # 39; Ou parce que les pièces sont gs, Hs,
et jes, ou quoi que ce soit) aussi longtemps que nous le pouvons, et (b) lorsque
aucune explication de ce type ne peut être donnée, aucune explication
être donné. L'idée ici est qu'à ce stade, nous serons arrivés
par des faits de base qui ne permettent pas d'explications – par exemple
faits sur la nature physique de base des particules élémentaires. Quand
Lorsque nous arrivons à des faits comme celui-ci, nous dirons: «Il n'y a aucune raison
pourquoi ces particules sont de cette façon; ils ont juste est. "
Cela nous donne un moyen de comprendre comment les nominalistes peuvent utiliser de manière plausible
un appel à la brutalité pour répondre à l'argument One Over Many. Mais
L'appel à la brutalité n'est que la moitié de la citation Quinean
au dessus de. Qu'en est-il de l'autre moitié, c'est-à-dire la partie concernant le platoniste
ne pas être mieux que les nominalistes brute-réelle quand il s'agit de réel
pouvoir explicatif? Pour apprécier cette affirmation, supposons que nous
est arrivé à un fait de bas niveau que les nominalistes chinois abordent
être un fait brisé (par exemple le fait que les particules physiques de quelqu'un
des espèces spéciales – disons, les gluons – sont g). Défenseurs
de ce qui précède, beaucoup diront que leur point de vue est supérieur au Quinean
nominalisme, car ils peuvent donner une explication du fait
des questions. Lorsqu'ils annoncent cela, les personnes intéressées
la question de savoir pourquoi les gluons sont g, et qui avait été
déçu d'entendre des scientifiques et des Chinois que c'est tout simplement
un fait cassé, peut être très tendu et écouter attentivement ce que
les partisans de ce qui précède doivent dire. Ils disent ceci:
La colle est g parce qu'ils possèdent la propriété
gness.
Cela ne semble pas très utile. L'affirmation que les gluons possèdent
gsemble ne faire guère plus que de nous dire que les gluons ont
une nature qui fait la façon dont ils sont g, et c'est comme ça
il semble qu'aucune explication réelle n'ait été donnée. Après tout, ceux qui
avait été intéressé à savoir pourquoi les gluons sont g ne serait pas
très content de cette prétendue "explication". Ainsi, pour
en utilisant les mots de Quine, il semble que les partisans de celui-ci sur beaucoup sont
"Pas mieux, dans le cas d'un réel pouvoir explicatif" que
les nominalistes de fait brut sont.
Les nominalistes peuvent essayer de pousser l'argument un peu plus loin ici,
réclamant le verdict
(P) Gluons est propriétaire de la propriété gness
n'est qu'un phrase de jugement
(N) Les gluons sont g.
Dans cette vue, (P) équivalent tonne). Autrement dit, il se dresse
même chose. Et aucune des phrases n'implique selon ce point de vue
l'existence de gness. Nous pouvons appeler cela un
Phrase-nominalisme vues de phrases comme (P). Mais
les nominalistes n'ont pas besoin de se rallier à ce point de vue. Ils peuvent également
fictionnaliste vues de phrases comme (P). Dans cette vue, (P)
et (N) ne dit pas strictement la même chose, car (P)
parler de la propriété get (N) pas. Selon
car cette vision fictive est (P) strictement fausse, car elle
parler de la propriété get selon
nominalisme, il n'y a rien appelé gness. En bref, (P) est
strictement faux sur ce point de vue pour la même raison que,
par exemple, «Tooth Fairy is Fine» n'est pas vrai. Mais tandis que (P)
n'est pas littéral vrai sur ce point de vue, il est
"À toutes fins pratiques vrai", ou quelque chose comme ça,
car ensemble, il peut être utilisé pour dire ce que (N) dit littéralement.
Cette idée est souvent capturée en disant que (P) n'est qu'un le moyen de
parleou façon de perles. (Remarquerez que
Le différend entre le fictionnisme et le paraphrase du nominalisme est le meilleur
compris comme un simple différend empirique sur
sémantique en langage ordinaire de phrases telles que (P); La question est
à propos de ces énoncés disent littéralement les mêmes choses que
phrases similaires à celles de (N).)
Quel que soit le point de vue adopté par les nominalistes ici, ils peuvent y répondre
Over Over argument – c'est-à-dire l'affirmation que nous pouvons expliquer (N)
en soutenant (P) – de la même manière, à savoir en le signalant
comme explication de (N), (P) est complètement non informatif. Même si
les nominalistes soutiennent une vision fictive selon laquelle (P) n'est pas
correspondant à (N), ils peuvent encore dire que l'explication ci-dessus est
pas informatif, car il dit vraiment que les gluons sont g
parce qu'ils ont une nature qui les rend tels qu'ils sont
g.
Etter å ha gjort poenget at den platonistiske forklaringen av (N) er
uninformative, the nominalist's next move is to appeal to Ockham's
razor to argue that we shouldn't believe in Gness (or at
least that we shouldn't believe in Gness for any reason that
has anything to do with the need to explain things like (N)). Ockham's
razor is a principle that tells us that we should believe in objects
of a given kind only if they play a genuine explanatory role. This
principle suggests that if Gness does not play a genuine role
in an explanation of the fact that gluons are G, then we
shouldn't believe in Gness — or, again, we shouldn't
believe in it for any reason having to do with the need to explain the
fact that gluons are G.
The Quinean response to the One Over Many argument is often couched in
terms of a criterion of ontological commitment. A criterion
of ontological commitment is a principle that tells us when we are
committed to believing in objects of a certain kind in virtue of
having assented to certain sentences. What the above response to the
One Over Many suggests is that we are ontologically committed not by
predicates like ‘is red’ and ‘is a rock’, but
by singular terms. (A singular term is just a denoting
phrase, i.e., an expression that purports to refer to a specific
object, e.g., proper names like ‘Mars’ and
‘Clinton’, certain uses of pronouns like
‘she’, and on some views, definite descriptions like
‘the oldest U.S. senator’.) More specifically, the idea
here seems to be this: if you think that a sentence of the form
‘a is F’ is true, then you have to
accept the existence of the object a, but you do not have to
accept the existence of a property of Fness; for instance, if
you think that ‘The ball is red’ is true, then you have to
believe in the ball, but you do not have to believe in redness; or if
you think that ‘Fido is a dog’ is true, then you have to
believe in Fido but not in the property of doghood.
Three points need to be made here. First, the above criterion needs to
be generalized so that it covers the use of singular terms in other
kinds of sentences, e.g., sentences of the form ‘a is
R-related to b’. Second, on the standard view,
we are ontologically committed not just by singular terms but also by
existential statements — e.g., by sentences like ‘There
are some Fs’, ‘There is at least one
F’, and so on (in first-order logic, such sentences are
symbolized as ‘(∃x)Fx’, and the
‘∃’ is called an existential quantifier).
The standard view here is that if you think that a sentence like this
is true, then you are committed to believing in the existence of some
Fs (or at least one F) but you do not have to
believe in Fness; for instance, if we assent to ‘There
are some dogs’, then we are committed to believing in the
existence of some dogs, but we are not thereby committed to believing
in the existence of the property of doghood. (Quine actually thought
that we are committed bare by existential claims and not by
singular terms; but this is not a widely held view.) Third and
finally, it is usually held that we are ontologically committed by
singular terms and existential expressions (or existential
quantifiers) only when they appear in sentences that we think are
literally true and only when we think the singular term or
existential quantifier in question can't be paraphrased away.
We can see what's meant by this by returning to the sentence
(R) The ball possesses the property of redness.
In this sentence, the expression ‘the property of redness’
seems to be a singular term — it seems to denote the
property of redness; thus, using the above criterion of ontological
commitment, if we think (R) is true, then it would seem, we are
committed to believing in the property of redness. But there are two
different responses that nominalists can make to this. First, they can
endorse paraphrase nominalism (defined a few paragraphs back) with
respect to (R). If they do this, they will claim that (R) doesn't
really carry an ontological commitment to the property of redness,
because it is really just equivalent to the sentence ‘The ball
is red’. This idea is often expressed by saying that in (R), the
singular term ‘the property of redness’ can be
paraphrased away — which is just to say that (R) can be
paraphrased by (or is equivalent to) a sentence that doesn't contain
the singular term ‘the property of redness’ (namely,
‘The ball is red’). The second view that nominalists can
endorse with respect to (R) is fictionalism. In other words, they can
admit that (R) does commit to the existence of the property of
redness, but they can maintain that because of this (and because there
are no such things as properties), (R) is, strictly speaking, untrue,
even if it is “for-all-practical-purposes true,” or some
such thing.
Having said all of this, we can summarize by saying that the standard
view of ontological commitment is as follows:
Criterion of Ontological Commitment: We are ontologically
committed by the singular terms (that can't be paraphrased away) in
the (simple) sentences that we take to be literally true; and we are
ontologically committed by the existential quantifiers (that can't be
paraphrased away) in the (existential) sentences that we take to be
literally true; but we are not committed by the predicates in such
sentences. Thus, for instance, if we believe that a sentence of the
form ‘a is F’ is literally true, and if
we think that it cannot be paraphrased into some other sentence that
avoids reference to a, then we are committed to believing in
the object a but not the property of Fness; and,
likewise, if we assent to a sentence of the form ‘a is
R-related to b’, then we are committed to
believing in the objects a et b but not the
relation R; and if we assent to a sentence of the form
‘There is an F’ then we are committed to
believing in an object that is F but we are not committed to
the property of
Fness.(3)
The One Over Many argument is now widely considered to be a bad
argument. Ironically, though, the above criterion of ontological
commitment — which Quinean nominalists appeal to in responding
to the One Over Many argument — is one of the central premises
in what is now thought to be the best argument for platonism. We will
call this argument the singular term argument, although one
might just as well call it the
sucker-punch-on-the-Quinean-nominalist argument, for as we
will see, the strategy is to accept the above criterion of ontological
commitment and turn it against the Quinean nominalist.
The general argument strategy here has roots in the work of Plato, but
its first clear formulation was given by Frege (1884, 1892,
1893–1903, and 1919). We begin with a general formulation of the
argument:
- If a simple sentence (i.e., a sentence of the form
‘a is F’, or ‘a is
R-related to b’, or…) is literally
true, then the objects that its singular terms denote exist.
(Likewise, if an existential sentence is literally true, then there
exist objects of the relevant kinds; e.g., if ‘There is an
F’ is true, then there exist some Fs.) - There are literally true simple sentences containing singular
terms that refer to things that could only be abstract objects.
(Likewise, there are literally true existential statements whose
existential quantifiers range over things that could only be abstract
objects.) Therefore, - Abstract objects exist.
Premise (1) follows straightaway from the criterion of ontological
commitment that we discussed in the last section. Again, this is
widely accepted among contemporary philosophers, and for good reason
— if you think that a sentence of the form ‘a is
F’ is literally true and that it cannot be paraphrased
into some other sentence, then it's hard to see how you can deny that
there is such a thing as the object a. Therefore, since (3)
follows trivially from (1) and (2), the central question we have to
answer, in order to evaluate the above argument, is whether (2) is
true. (Below, I will discuss the possibility of denying (1), but for
now I want to focus on (2).) In any event, in order to motivate (2),
platonists need to provide some examples; that is, they have to
produce some sentences and argue that (i) they contain singular terms
that can only be taken as referring to abstract objects (and that
can't be paraphrased away) and (ii) they are literally true.
Platonists maintain that there are many different kinds of such
sentences. In what follows, we will consider versions of this argument
that attempt to establish the existence of mathematical objects (e.g.,
numbers), propositions, properties, relations, sentence types,
possible worlds, logical objects, and fictional objects.
4.1 Mathematical Objects
Platonists about mathematical objects claim that the theorems of our
mathematical theories — sentences like ‘3 is prime’
(a theorem of arithmetic) and ‘There are infinitely many
transfinite cardinal numbers’ (a theorem of set theory) —
are literally true and that the only plausible view of such sentences
is that they are about abstract objects (i.e., that their singular
terms denote abstract objects and their existential quantifiers range
over abstract objects). This general stance toward mathematics goes
back to Plato, but the first clear statement of the argument in this
form was given by Frege (1884); other advocates include Quine (see his
1948 and 1951, though he doesn't explicitly state the argument there),
Gödel (1964), Parsons (1965, 1971, 1994), Putnam (1971), Steiner
(1975), Resnik (1981, 1997), Zalta (1983, 1999), Wright (1983),
Burgess (1983), Hale (1987), Shapiro (1989, 1997), the early Maddy
(1990),(4)
Katz (1998), Colyvan (2001), McEvoy (2005, 2012), and Marcus
(2015).
Let's begin our discussion of the platonists' argument here by
considering their reasons for thinking that we have to take sentences
like ‘3 is prime’ to be about abstract objects rather than
mental or physical objects of some kind. And let's start with a
discussion of the psychologistic view that mathematics is about mental
objects.
Frege (1884, introduction and section 27; 1893–1903,
introduction; 1894; and 1919) gave several compelling arguments
against psychologism. First, it seems that psychologism is incapable
of accounting for the truth of sentences that are about all
natural numbers, because there are infinitely many natural numbers and
clearly, there could not be infinitely many number-ideas in human
minds. Second, psychologism seems to entail that sentences about very
large numbers (in particular, numbers that no one has ever had a
thought about) are not true; for if none of us has ever had a thought
about some very large number, then psychologism entails that there is
no such number and, hence, that no sentence about that number could be
true. Third, psychologism turns mathematics into a branch of
psychology, and it makes mathematical truths contingent upon
psychological truths, so that, for instance, if we all died, ‘4
is greater than 2’ would suddenly become untrue. But this seems
wrong: it seems that mathematics is true independently of us; that is,
it seems that the question of whether 4 is greater than 2 has nothing
at all to do with the question of how many humans are alive. Fourth
and finally, psychologism suggests that the proper methodology for
mathematics is that of empirical psychology; that is, it seems that if
psychologism were true, then the proper way to discover whether, say,
there is a prime number between 10,000,000 and 10,000,020, would be to
do an empirical study of humans and ascertain whether there is, in
fact, an idea of such a number in one of our heads; but of course,
this is not the proper methodology for mathematics. As Frege says
(1884, section 27), “Weird and wonderful…are the results
of taking seriously the suggestion that number is an idea.”
Platonists do not deny that we have ideas of mathematical objects.
What they deny is that our mathematical sentences are about
these ideas. Thus, the dispute between platonism and psychologism is
primarily a semantic one. Advocates of psychologism agree with
platonists that in the sentence ‘3 is prime’,
‘3’ functions as a singular term (i.e., as a denoting
expression). But they disagree about the referent of this expression.
They think that ‘3’ refers to a mental object, in
particular, an idea in our heads. It is this semantic thesis
that platonists reject and that the above Fregean arguments are
supposed to refute. More specifically, they're supposed to show that
the psychologistic semantics of mathematical discourse is not correct
because it has consequences that fly in the face of the actual usage
of mathematical language.
Another argument for the superiority of the platonist semantics of
mathematical discourse over the psychologist semantics is based on the
fact that in ordinary usage, one way to say that something doesn't
exist is to say that “it exists only in your
head”.(5)
To say that mathematical objects exist only in our heads, it seems,
is just to say that they don't exist. For to say that they (as opposed
to our ideas of them) exist is to say that they exist independently of
us and our thinking. Quine put this point in a very compelling way in
connection with a conceptualistic view of mythical objects like
Pegasus. He writes (1948, p. 2):
McX (who maintains that Pegasus exists and is an idea in our heads)
never confuses the Parthenon with the Parthenon-idea. The Parthenon is
physical; the Parthenon-idea is mental…We cannot easily imagine
two things more unlike…But when we shift from the Parthenon to
Pegasus, the confusion sets in — for no other reason than that
McX would sooner be deceived by the crudest and most flagrant
counterfeit than grant the nonbeing of Pegasus.
The same argument can be run against the psychologistic conflation of
3-ideas with 3: you might doubt that there really is such a thing as
the number 3, existing objectively and independently of us, but you
should not for that reason claim that your idea of 3 is 3,
for that is just a confusion — it is like saying that your idea
of Pegasus is Pegasus, or that your idea of the Parthenon is
the Parthenon.
Let us move on now to immanent-realist, or physicalist, views of
mathematics, and let us concentrate first on views like Mill's (1843,
book II, chapters 5 and 6), i.e., views that maintain that sentences
about numbers are really just general claims about bunches of ordinary
objects. On this view, the sentence ‘2 + 1 = 3’, for
instance, isn't really about specific objects (the numbers 1, 2, and
3). Rather, it says that whenever we add one object to a pile of two
objects, we will get a pile of three objects. Now, in order to account
for contemporary mathematics in this way, a contemporary Millian would
have to take set theory to be about ordinary objects as well. This,
however, is untenable. One argument here is that set theory could not
be about bunches of ordinary objects, or piles of physical stuff,
because corresponding to every physical pile, there are many, many
sets. Corresponding to a ball, for instance, is the set containing the
ball, the set containing its molecules, the set containing its atoms,
and so on. (And we know that these are different sets, because they
have different members, and it follows from set theory that if set
UNE and set B have different members, then UNE
is not identical to B.) Indeed, the principles of set theory
entail that corresponding to every physical object, there is a huge
infinity of sets. Corresponding to our ball, for instance, there is
the set containing the ball, the set containing that set, the set
containing that set, and so on; and there is the set
containing the ball and the set containing the set containing the
ball; and so on and on and on. Clearly, these sets are not just piles
of physical stuff, because (a) there are infinitely many of them
(again, this follows from the principles of set theory) and (b) all of
these infinitely many sets share the same physical base. Thus, it
seems that claims about sets are not claims about bunches of ordinary
objects, or even generalized claims about such bunches. They are
claims about sets, which are objects of a different kind.
Another problem with physicalistic views along the lines of Mill's is
that they seem incapable of accounting for the sheer size of the
infinities involved in set theory. Standard set theory entails not
just that there are infinitely large sets, but that there are
infinitely many sizes of infinity, which get larger and larger with no
end, and that there actually exist sets of all of these different
sizes of infinity. There is simply no plausible way to take this
theory to be about physical stuff.
(These arguments do not refute the kind of immanent realism defended
by the early Maddy (1990). On Maddy's view, sets of physical objects
are located in spacetime, right where their members are. Thus, if you
have two eggs in your hand, then you also have the set containing
those eggs in your hand. Maddy's view has no problem accounting for
the massive infinities in mathematics, for on her view, corresponding
to every physical object, there is a huge infinity of sets that exist
in space and time, right where the given physical object exists. Given
this, it should be clear that while Maddy's view says that sets exist
in spacetime, it cannot be thought of as saying that sets are physical
objects. So Maddy's view is not a physicalist view in the sense that's
relevant here (as was pointed out above, it is better thought of as a
non-standard version of platonism). Thus, in the present context
(i.e., the context in which platonists are trying to undermine views
that take mathematical objects to be physical objects), platonists do
not need to argue against Maddy's view. Of course, in the end, if they
want to motivate the standard version of platonism, then they'll have
to give reasons for preferring their view to Maddy's non-standard
version of platonism, and this might prove hard to do. For some
arguments against Maddy's early view, see, e.g., Lavine (1992),
Dieterle and Shapiro (1993), Balaguer (1998a), Milne (1994), Riskin
(1994), Carson (1996), and the later Maddy (1997).)
If arguments like the ones we've been discussing here are cogent, then
sentences like ‘3 is prime’ are not about physical or
mental objects, and therefore, psychologism and immanent realism are
not tenable views of mathematics. But that is not the end of the
singular term argument for the existence of abstract mathematical
objects, for we still need to consider nominalistic views of sentences
like ‘3 is prime’. In order for platonists to establish
their view, they need to refute these nominalistic views as well as
psychologistic and physicalistic views. And it should be noted that
this is the hard part. There is a good deal of agreement among
philosophers of mathematics that psychologism and immanent realism are
untenable; that is, most philosophers of mathematics are either
platonists or nominalists; but there is very little agreement as to
whether platonism or nominalism is correct.
How can nominalists proceed in developing an account of sentences like
‘3 is prime’? One strategy is to reject premise (1) and
the standard criterion of ontological commitment discussed above. le
most obvious way to do this is to endorse the following view: (a)
simple mathematical sentences like ‘3 is prime’ should be
read as being of the form ‘a is F& # 39; and as
being about abstract objects (e.g., ‘3’ should be taken as
denoting the number 3, which could only be an abstract object, and
‘3 is prime’ should be taken as being a claim about that
object); and (b) abstract objects — in particular, mathematical
objects like the number 3 — don’t exist (and the claim
here is that they don’t have any sort of being whatsoever, so
this is not a Meinongian view); but (c) sentences like ‘3 is
prime’ are still literally true. Thus, on this view, a claim
about an object a can be true even if that object
doesn’t exist at all. Let’s call this view
thin-truth-ism. Views of this general kind have been endorsed
by Azzouni (1994, 2004), Salmon (1998), and Bueno (2005).
Thin-truth-ists endorse a similar view of existence claims. For
instance, on their view, the sentence ‘There are infinitely many
prime numbers’ is literally true, even though there are no such
things as numbers. This might look like a contradiction, but it's not,
because according to thin-truth-ism, existential expressions (or
quantifiers) like 'there is' are ambiguous.
Most philosophers find this view extremely hard to believe. Indeed, a
lot of philosophers would say that it is simply confused, or
incoherent. But, in fact, thin-truth-ism is not incoherent. A better
way to formulate the problem with the view is as follows: in giving up
on the standard criterion of ontological commitment, thin-truth-ists
seem to be using ‘true’ in a non-standard way. Most of us
would say that if there is no such thing as the number 3, and if
‘3 is prime’ is read at face value (i.e., as being about
the number 3), then it follows trivially that ‘3 is prime’
could not be true. Or more generally, most of us would say that if
there is no such thing as the object a, then sentences of the
form ‘a is F’ cannot be literally true.
This, of course, is just to say that most of us accept the standard
criterion of ontological commitment discussed above, but the point
here is that this criterion seems to be built into the standard
meaning of words like ‘true’. Indeed, this explains why
the standard criterion of ontological commitment is so widely
accepted.
Another worry one might raise about thin-truth-ism is that it only
differs from fictionalism in a merely verbal way. La
‘thin-true’ express the kind of truth that thin-truth-ists
have in mind, and let ‘thick-true’ express the kind of
truth that everyone else in the debate has in mind (i.e., platonists,
fictionalists, paraphrase nominalists, and so on). Given this,
fictionalists and thin-truth-ists will both endorse all of the
following claims: (a) platonists are right that ‘3 is
prime’ is a claim about the number 3; and (b) there is no such
thing as the number 3; so (c) ‘3 is prime’ isn’t
thick-true; but despite this, (d) ‘3 is prime’ is
thin-true. Now, of course, thin-truth-ists and fictionalists will
disagree about whether ‘3 is prime’ is true, but
this will collapse into a disagreement about whether thin-truth or
thick-truth is real truth, and this is just a disagreement
about what the word ‘true’ means in ordinary folk English,
and it’s hard to see why an empirical question about how the
folk happen to use some word is relevant to the debate about the
existence of mathematical objects.
In any event, if we reject thin-truth-ism — i.e., if we accept
premise (1) and the standard criterion of ontological commitment
— then there are two general strategies that nominalists can
adopt in giving a view of mathematical sentences like ‘3 is
prime’. First, they can endorse a paraphrase view, and second,
they can endorse a fictionalist view. Those who endorse paraphrase
views claim that while sentences like ‘3 is prime’ are
true, they should not be read as platonists read them, because we can
paraphrase these sentences with other sentences that do not commit us
to the existence of abstract objects. One view of this sort, known as
if-thenism, holds that ‘3 is prime’ can be
paraphrased by ‘If there were numbers, then 3 would be
prime’ (for an early version of this sort of view, see the early
Hilbert (1899 and his letters to Frege in Frege (1980)); for later
versions, see Putnam (1967a and 1967b) and Hellman (1989)). A second
version of the paraphrase strategy, which we can call
metamathematical formalism (see Curry (1951)), is that
‘3 is prime’ can be paraphrased by “‘3 is
prime’ follows from the axioms of
arithmetic”.(6)
A third version, developed by Chihara (1990), is that mathematical
sentences that seem to make claims about what mathematical objects
exist — e.g., ‘There is a prime number between 2 and
4’ — can be paraphrased into sentences about what it's
possible for us to do (in particular, what it's possible for us to
write down). Others to endorse paraphrase views include Hofweber
(2005), Rayo (2008), Moltmann (2013), and Yi (2002).
One problem with the various paraphrase views (not to put too fine a
point on it) is that none of the paraphrases seems very good. That is,
the paraphrases seem to misrepresent what we actually mean when we say
things like ‘3 is prime’ (and by ‘we’, I mean
both mathematicians and ordinary folk). What we mean, it seems, is
that 3 is prime — not that if there were numbers, then 3 would
be prime, or that the sentence ‘3 is prime’ follows from
the axioms of arithmetic, or any such thing. And notice how the
situation here differs from cases where we do seem to have good
paraphrases. For instance, one might try to claim that if we endorse
the sentence
(A1) The average accountant has two children,
then we are ontologically committed to the existence of the average
accountant; but it is plausible to suppose that, in fact, we are not
so committed, because (A1) can be paraphrased by the sentence
(A2) On average, accountants have two children.
Moreover, it seems plausible to maintain that this is a good
paraphrase of (A1), because it seems clear that when people say things
like (A1), what they really mean are things like (A2). But in
the present case, this seems wrong: it does not seem plausible to
suppose that when people say ‘3 is prime’, what they
really mean is ‘If there were numbers, then 3 would be
prime’. Again, it seems that what we mean here is, very simply,
that 3 is prime. In short, when people say things like ‘3 is
prime’, they do not usually have any intention to be saying
anything other than what these sentences seem to say; et
because of this, it seems that the platonist's face-value semantics
for mathematical discourse is correct.
Some paraphrase nominalists (e.g., Chihara 1990, 2004) maintain that
it doesn't matter what we really mean, that paraphrase
nominalists aren't committed to the thesis that their paraphrases
capture the real ordinary-language meanings of our mathematical
sentences. But this is false. If paraphrase nominalists admit that
platonists are right about the ordinary-language meanings of
mathematical sentences like ‘3 is prime’, then their view
will collapse into a fictionalist view, according to which sentences
like ‘3 is prime’ are not literally true. For since
paraphrase nominalists don't believe in the existence of mathematical
objects, if they admit that ordinary utterances of ‘3 is
prime’ are best interpreted as being about mathematical objects,
or purporting to be about such objects, then they will have to admit
that such sentences are literally untrue, as fictionalists maintain.
Thus, if the paraphrase nominalist view is going to be a genuine
alternative to fictionalism, it has to involve the thesis that the
paraphrases that nominalists are offering capture the real meanings of
ordinary mathematical sentences.
On the other hand, paraphrase nominalists might try to argue that they
are right about the ordinary-language meanings of sentences
like ‘3 is prime’, even though this is not obvious or
transparent to ordinary speakers. This stance, however, would be
extremely controversial and difficult to motivate.
One paraphrase view that has become somewhat popular recently holds
that sentences that seem to be about numbers are best read as being
about plurals. For instance, we might read ‘2 + 2 = 4’ as
really saying something like this: two and two are four (or
two objects and two (more) objects are four objects, or some
such thing). Views of this general kind have been endorsed or defended
by, e.g., Yi (2002, forthcoming), Hofweber (2005), and Moltmann
(2013). This view fits better with ordinary usage than some of the
other paraphrase-nominalist views, and for sentences like ‘2 + 2
= 4’ they can seem plausible. But when we switch to sentences
like ‘3 is prime’ — and, even worse, ‘There
are infinitely many primes’ — they can start to seem
cumbersome and less plausible.
(For a good in-depth discussion and critique of some of the
paraphrase-nominalist views, see Burgess and Rosen (1997).)
Let's move on now to a discussion of fictionalism, which is the last
option for nominalists. Unlike paraphrase nominalists, fictionalists
admit that the platonist's face-value semantics for mathematical
discourse is correct; but because fictionalists don't believe in
abstract objects, they think that mathematical sentences like ‘3
is prime’ are not true. In other words, fictionalists maintain
that (a) platonists are right that sentences like ‘3 is
prime’ do purport to be about abstract objects, but (b) there
are no such things as abstract objects, and so (c) these sentences
— and, indeed, our mathematical theories — are untrue.
Thus, on this view, just as Alice in Wonderland is not true
because there are no such things as talking rabbits, hookah-smoking
caterpillars, and so on, so too our mathematical theories are not true
because there are no such things as numbers and sets and so
on.(7).
(Fictionalism has been developed by Field (1980, 1989, 1998),
Balaguer (1998a, 2009), Rosen (2001), and Leng (2005a, 2005b, 2010).
One might also interpret Melia (2000), Yablo (2002a, 2002b, 2005), and
Bueno (2009) as fictionalists. Finally, Hoffman (2004) endorses a kind
of fictionalism, but her view is very different from the one under
discussion here; for a bit more on her view, see the entry on
fictionalism in the philosophy of mathematics.)
There are a few different ways that platonists might try to argue
against fictionalism. The most famous and widely discussed argument
against fictionalism is the Quine-Putnam indispensability
argument (see Quine (1948, 1951), Putnam (1971, 2012), and
Colyvan (2001)). This argument (or at any rate, one version of it)
proceeds as follows: it cannot be that mathematics is untrue, as
fictionalists suggest, because (a) mathematics is an indispensable
part of our physical theories (e.g., quantum mechanics, general
relativity theory, evolutionary theory, and so on) and so (b) if we
want to maintain that our physical theories are true (and surely we do
— we don't want our disbelief in abstract objects to force us to
be anti-realists about natural science), then we have to maintain that
our mathematical theories are true.
Fictionalists have developed two different responses to the
Quine-Putnam argument. The first, developed by Field (1980) and
Balaguer (1998a), is based on the claim that mathematics is, in fact,
not indispensable to empirical science — i.e., that our
empirical theories can be nominalized, or reformulated in a
way that avoids reference to abstract objects. The second response,
developed by Balaguer (1998a), Rosen (2001), Yablo (2005), Bueno
(2009), Leng (2010), and perhaps Melia (2000), is to grant the
indispensability of mathematics to empirical science and to simply
account for the relevant applications from a fictionalist point of
view. (A counterresponse to this second response has been given by
Colyvan (2002) and Baker (2005, 2009), who argue that fictionalists
can’t account for the explanatory role that mathematics
plays in science; responses to the explanatory version of the
indispensability argument have been given by Melia (2002), Leng
(2005b), Bangu (2008), and Daly and Langford (2009).)
There is no consensus on whether the fictionalist responses to the
Quine-Putnam argument are successful. But even if they are, there are
other objections that platonists might raise against fictionalism. For
instance, one might argue that fictionalists cannot account for the
objectivity of mathematics (for responses to this, see Field
(1980, 1989, 1998) and Balaguer (2009)). Or, second, one might argue
that fictionalism is not a nominalistically acceptable view because
formulations of it invariably involve tacit reference to various kinds
of abstract objects, such as sentence types, or stories, or possible
worlds (for responses to this, see Field (1989), Balaguer (1998a), and
Rosen (2001)). For other objections to fictionalism, see, e.g.,
Malament (1982), Shapiro (1983a), Resnik (1985), Chihara (1990,
chapter 8, section 5), Horwich (1991), O’Leary-Hawthorne (1997),
Burgess and Rosen (1997), Katz (1998), Thomas (2000, 2002), Stanley
(2002), Bueno (2003), Szabo (2003), Hoffman (2004), and Burgess
(2004). For responses to these objections, see the various
fictionalist works cited above, as well as Daly (2008) and Liggins
(2010). And for a discussion of all these objections, as well as
fictionalist responses to them, see the entry on
fictionalism in the philosophy of mathematics.)
In the end, it's not obvious whether platonists can successfully
refute fictionalism, and more generally, it's not obvious whether the
version of the singular term argument rehearsed in this subsection
provides a good reason for believing in abstract mathematical
objects.
4.2 Propositions
We turn now to a version of the singular term argument aimed at
establishing the existence of propositions. Once again, the most
important figure in the development of this argument is Frege (1892,
1919). Other relevant figures (who wouldn't all endorse an argument
like the one sketched below) include Russell (1905, 1910–1911),
Church (1950, 1954), Quine (1956), Kaplan (1968–69, 1989),
Kripke (1972, 1979), Schiffer (1977, 1987, 1994), Perry (1979), Evans
(1981), Peacocke (1981), Barwise and Perry (1983), Bealer (1982,
1993), Zalta (1983, 1988), Katz (1986), Salmon (1986), Soames (1987,
2014), Forbes (1987), Crimmins and Perry (1989), Richard (1990),
Crimmins (1998), Recanati (1993, 2000), King (1995, 2014), Braun
(1998), and Saul (1999).
The relevant sentences here are belief ascriptions, i.e., sentences
like ‘Clinton believes that snow is white’ and
‘Emily believes that Santa Claus is fat’. The first point
to note about these sentences is that they involve
‘that’-clauses, where a ‘that’-clause is
simply the word ‘that’ added to the front of a complete
sentence — e.g., ‘that snow is white’. The second
point to be made is that ‘that’-clauses, in English, are
singular terms. A common way to illustrate this point — see,
e.g., Bealer (1982 and 1993) and Schiffer (1994) — is to appeal
to arguments like the following:
I. Clinton believes that snow is white.
Therefore, Clinton believes something (namely, that snow is
white).
This argument seems to be valid, and platonists claim that the best
and only tenable explanation of this fact involves a commitment to the
idea that the ‘that’-clause in this argument, i.e.,
‘that snow is white’, is a singular term.
But if ‘that’-clauses are singular terms, what sorts of
objects do they refer to? Well, it might seem that they refer to
facts, or states of affairs. For instance, it might seem that
‘that snow is white’ refers to the fact that snow is
white. This, however, is a mistake (at least in connection with the
‘that’-clauses that appear in belief reports). For since
beliefs can be false, it follows that the ‘that’-clauses
in our belief reports refer to things that can be false. E.g., if
Sammy is seven years old, then the sentence ‘Sammy believes that
snow is powdered sugar’ could be true; but if this sentence is
true, then (by our criterion of ontological commitment) its
‘that’-clause refers to a real object; but then it cannot
refer to a fact, because (obviously) there is no such thing as the
fact that snow is powdered sugar.
These considerations suggest that the referents of the
‘that’-clauses that appear in belief ascriptions are
things that can be true or false. But if this is right, then it seems
that the objects of belief must be either sentences or propositions.
The standard platonist view is that they are propositions. Before we
consider their arguments for this claim, we need to say a few words
about the different kinds of sentential views that one might
endorse.
To begin with, we need to distinguish between sentence types
and sentence tokens. To appreciate the difference, consider
the following indented sentences:
Cats are cute.
Cats are cute.
We have here two different tokens of a single sentence type. Thus, a
token is an actual physical thing, located at a specific place in
spacetime; it is a pile of ink on a page (structured in an appropriate
way), or a sound wave, or a collection of pixels on a computer screen,
or something of this sort. A type, on the other hand, can be tokened
numerous times but is not identical with any single token. Thus, a
sentence type is an abstract object. And so if we are looking for an
anti-platonist view of what ‘that’-clauses refer to, or
what belief reports are about, we cannot say that they're about
sentence types; we have to say they're about sentence tokens.
A second distinction that needs to be drawn here is between sentence
tokens that are external, or public, and sentence tokens that
are internal, or private. Examples of external sentence
tokens were given in the last paragraph — piles of ink, sound
waves, and so on. An internal sentence token, on the other hand,
exists inside a particular person's head. There is a wide-spread view
— due mainly to Jerry Fodor (1975 and 1987) but adopted by many
others, e.g., Stich (1983) — that we are able to perform
cognitive tasks (e.g., think, remember information, and have beliefs)
only because we are capable of storing information in our heads in a
neural language (often called mentalese, or the language
of thought). In connection with beliefs, the idea here is that to
believe that, say, snow is white, is to have a neural sentence stored
in your head (in a belief way, as opposed to a desire way, or some
other way) that means in mentalese that snow is white.
This gives us two different anti-platonist alternatives to the view
that belief reports involve references to propositions. First, there
is the conceptualistic (or mentalistic) view that belief reports
involve references to sentences in our heads, or mentalese sentence
tokens. And, second, there is the physicalistic view that belief
reports involve references to external sentence tokens, i.e., to piles
of ink, and so on (versions of this view have been endorsed by Carnap
(1947), Davidson (1967), and Leeds (1979)).
There are a number of arguments that suggest that ordinary belief
reports cannot be taken to be about (internal or external) sentences
and that we have to take them to be about propositions. We will
rehearse one such argument here, an argument that goes back at least
to Church (1950). Suppose that Boris and Jerry both live in cold
climates and are very familiar with snow. Thus, they both believe that
snow is white. But Boris lives in Russia and speaks only Russian,
whereas Jerry lives in Minnesota and speaks only English. Now,
consider the following argument:
II. Boris believes that snow is white.
Jerry believes that snow is white.
Therefore, there's at least one thing that Boris and Jerry both
believe, namely, that snow is white.
This argument seems clearly valid; but this seems to rule out the idea
that the belief reports here are about sentence tokens. For (a) in
order to account for the validity of the argument, we have to take the
two ‘that’-clauses to refer to the same thing, and (b)
there is no sentence token that they could both refer to. First of
all, they couldn't refer to any external sentence token (or, for that
matter, any sentence type associated with any natural language),
because (i) if the first ‘that’-clause refers to such a
sentence, it would presumably be a Russian sentence, since Boris
speaks only
Russian;(8)
and (ii) if the second ‘that’-clause refers to such a
sentence, it would presumably be an English sentence, since Jerry
speaks only English; and so (iii) the two ‘that’-clauses
cannot both refer to the same external sentence token (or
natural-language sentence type). And second, they cannot refer to any
mentalese sentence token, because (i) if the first
‘that’-clause refers to such a sentence, it would
presumably be in Boris's head; and (ii) if the second
‘that’-clause refers to such a sentence, it would
presumably be in Jerry's head; and so (iii) the two
‘that’-clauses cannot both refer to the same mentalese
sentence token. Therefore, it seems to follow that the
‘that’-clauses in the above argument do not refer to
sentence tokens of any kind. And since these are ordinary belief
ascriptions, it follows that, in general, the
‘that’-clauses that appear in ordinary belief ascriptions
do not refer to sentence tokens.
Now as it's formulated here, this argument doesn't rule out the view
that ‘that’-clauses refer to mentalese sentence types, but
the argument can be extended to rule out that view as well (e.g., one
might do this by talking not of an American and a Russian but of two
creatures with different internal languages of thought). I won't run
through the details of this here, since, as we've seen,
anti-platonists can't claim that ‘that’-clauses refer to
types anyway, because types are abstract objects. But if we assume
that that version of the argument is cogent as well, then it follows
that ‘that’-clauses don't refer to sentences of any kind
at all and, hence, that they must refer to
propositions.(9)
Now, notice that the issue so far has been purely semantic. What the
above argument suggests is that regardless of whether there are any
such things as propositions, our ‘that’-clauses are best
interpreted as purporting to refer to such objects. Platonists then
claim that if this is correct, then there must be such things as
propositions, because, clearly, many of our belief ascriptions are
true. For instance, ‘Clinton believes that snow is white’
is true; thus, if the above analysis of ‘that’-clauses is
correct, and if our criterion of ontological commitment is correct, it
follows that there is such a thing as the proposition that snow is
white.
This version of the singular term argument might seem even more
powerful than the mathematical-object version of the singular term
argument sketched in
section 4.1,
because in this case, it doesn't seem that there is as much room for
paraphrase nominalism. We saw in
section 4.1
that there are a number of programs for paraphrasing the statements
of mathematics, but there are no obvious strategies for paraphrasing
ordinary belief ascriptions. One might think this could be done by
taking sentences of the form ‘S believes that
p’ to mean ‘If there existed propositions, then
S would believe that p’; but this sort of view
is even less plausible here than it is in the mathematical case. It is
just wildly implausible to suppose that when common folk say things
like ‘Clinton believes that his presidency was
successful’, they mean to be making hypothetical claims about
what the person in question would believe in some alternate
situation.
However, while paraphrase nominalism seems hopeless in the case of
propositions, Balaguer (1998b) has argued that fictionalistic
nominalism carries over very well to the case of propositions. plus
specifically, fictionalists can say that ‘Clinton believes that
snow is white’ is strictly speaking not true (because its
‘that’-clause is supposed to refer to a proposition, and
there are no such things as propositions) but that we can still use it
to say something essentially accurate about Clinton's belief state,
because there are facts about Clinton that make it the case that if
there existed propositions, then it would be true that he believes
that snow is white.
4.3 Properties and Relations
One way to argue for a platonistic view of properties and relations is
first to use the argument of
section 4.2
to argue for a platonistic view of propositions, and then to claim
that this argument already contains an argument for properties and
relations, because properties and relations are components of
propositions. If we adopt a Russellian view of propositions, this is
straightforward, because it is built into the Russellian view that
propositions are composed of objects, properties, and relations (see
section 1).
If we adopt a Fregean view of propositions, however, the situation is
different. On the Fregean view, propositions are composed of senses,
which we can think of as meanings, or concepts. Now, Frege himself did
not use the word ‘concept’ to talk about these things
— he used the German ‘sinn’, which is usually
translated as ‘sense’ — but we can use the word
‘concept’ here. On this way of talking, we can say that on
a Fregean view, if we have good reason to countenance the existence
of, e.g., the proposition that roses are red, then we also have good
reason to countenance the existence of the concept red. Now,
some Fregeans might want to say that the property of redness just
is the concept of redness, and if so, then they could
maintain with Russellians that if there exist propositions, then there
also exist properties and relations. But most Fregeans would want to
deny that properties are concepts, so in order to motivate a
platonistic view of properties and relations, they would need an
entirely different argument. And since we have already found that the
One Over Many argument for properties and relations is not cogent,
this other argument would presumably be a version of the singular term
argument, one that was aimed specifically at establishing the
existence of properties and relations.
The most obvious way to formulate an independent property-and-relation
version of the singular term argument would be to appeal to sentences
like
(P1) Mars possesses the property of redness
et
(R1) San Francisco stands in the north-of relation to Los Angeles.
In order to make out a version of the singular term argument here,
platonists would need to begin by arguing that these sentences commit
us to the existence of the property of redness and the north-of
relation, respectively, because (a) they have singular terms that
denote those things, and (b) they are true. In order to motivate claim
(a), platonists would have to refute the paraphrase-nominalist claim
that sentences like (P1) and (R1) are equivalent to sentences like
‘Mars is red’ and ‘San Francisco is north of Los
Angeles’ and that none of these sentences entails the existence
of properties or relations. And in order to motivate claim (b),
platonists would have to refute the fictionalist view that sentences
like (P1) and (R1) are untrue because they do entail the existence of
properties and relations and because there are no such things as
properties or relations (of course, fictionalists maintain that while
(P1) and (R1) aren't literally true, they can still be used
colloquially to say things that about the world that are essentially
accurate — see
section 3).(dix)
If platonists managed to establish the existence of properties and
relations in this way, they would still need to argue that such things
could only be abstract objects. That is, they would have to argue that
properties and relations are not ideas (as conceptualists claim) or
universals inhering in physical things (as immanent realists
claim).
The arguments listed above against conceptualistic or psychologistic
views of numbers also tell against conceptualism about properties and
relations. For instance, as Russell (1912, chapter IX) points out,
property claims and relational claims seem to be objective; e.g., the
fact that Mount Everest is taller than Mont Blanc is a fact that holds
independently of us; but conceptualism about universals entails that
if we all died, it would no longer be true that Mount Everest bears
the taller than relation to Mont Blanc, because that relation would no
longer exist. And, second, conceptualism seems simply to get the
semantics of our property discourse wrong, for it seems to confuse
properties with our ideas of them. The English sentence ‘Red is
a color’ does not seem to be about anybody's idea of redness; it
seems to be about redness, the actual color, which, it seems, is
something objective.
There are also some very famous arguments against the immanent realist
view of properties and relations. First, it is not clear that it is
coherent to say that there is such a thing as redness and that this
en thing exists in many different objects at the same time.
Second, it is not clear what it is for an object to possess a property
on the immanent realist view. Most immanent realists would not say
that property possession is a full-blown relation, for this
would just be another universal, and it is commonly thought that if
immanent realists adopted this view, it would lead to an unacceptable
infinite regress. (If we're told that an object a possesses
Fness iff a stands in the possession relation to
Fness, then one might ask, “What is it for an object
and a property to stand in the possession relation to one
another?”, and so on. For more on this, see the entry on
properties.)
In light of this, many immanent realists maintain that when an object
a possesses a property Fness, a et
Fness are “linked together” in some
non-relational way, e.g., a way that is more intimate, or primitive,
than ordinary relational connections. But it is not clear what this
really amounts to. (Immanent realists might respond that platonists
also have a problem here — i.e., that platonists also have to
provide an account of the relation, or “connection”,
between objects and properties. But some platonists might argue that
the problem isn't as bad for them because platonistic properties are
causally inert, and so they are not responsible in any way
for objects having the natures they have, and they do not play any
important role in our explanations of why objects have the
natures they have. For instance, if a is F,
Fness is not responsible in any way for a having the
nature that it has. Thus, platonists might claim that a is
simply an eksempel of Fness and that there is no more
to their relation than
that.(11)
Immanent realists, however, think that ordinary physical objects are
the way they are because they possess the properties they do.
Thus, they seem committed to the thesis that there is some sort of
physically substantial connection, or link, between objects and their
properties, and it is not at all clear what this could be. There has
been a lot of philosophy dedicated to this problem, but there is no
consensus on how (or whether) it can be solved.)
It is worth noting that platonists who argue for properties and
relations in conjunction with propositions — i.e., by first
arguing for propositions and then claiming that properties and
relations are components of propositions — will have an easier
time arguing that properties and relations couldn't exist in our minds
(as conceptualists say) or in things (as immanent realists say). DANS
connection with conceptualism, platonists of this sort could claim
that the argument given in
section 4.2
for thinking that ‘that’-clauses don't refer to mentalese
sentence tokens suggests that propositions (which are the referents of
‘that’-clauses) could not be made up of properties that
exist in our heads. And in connection with immanent realism,
platonists of this sort could argue that propositions couldn't be
composed of immanent-realist properties, because people can believe
propositions that are composed of properties that are not instantiated
in the physical world. For instance, it seems that sentences like
‘Johnny believes that there is a four-hundred-story building in
Sally's backyard’ can be true, and so according to the above
platonist argument for propositions, there must be such a thing as the
proposition that there is a four-hundred-story building in Sally's
backyard. But if propositions have properties as components, then this
proposition has as a component the property of being a
four-hundred-story building. But if properties exist only in physical
things, as immanent realists suggest, then there is no such thing as
the property of being a four-hundred-story building, since presumably,
nothing in the universe has this property. Thus, the conclusion here
is that if propositions have properties as components, then the
properties in question have to be transcendent, platonist properties,
not immanent
properties.(12)
4.4 Sentence Types
Linguistics is a branch of science that tells us things about
sentences. For instance, it says things like
(A) ‘The cat is on the mat’ is a well-formed sentence of
English,
et
(B) ‘Visiting relatives can be boring’ is structurally
ambiguous.
The quoted sentences that appear in (A) and (B) are singular terms;
e.g., “‘The cat is on the mat’” refers to the
sentence ‘The cat is on the mat’, and (A) says of this
sentence that it has a certain property, namely, that of being a
well-formed English sentence. Thus, if sentences like (A) are true
— and it certainly seems that they are — then they commit
us to believing in the existence of sentences. Now, one might hold a
physicalistic view here according to which linguistics is about actual
(external) sentence tokens, e.g., piles of ink and verbal sound waves.
(This view was popular in the early part of the 20th
century — see, e.g., Bloomfield (1933), Harris (1954), and Quine
(1953).) Or alternatively, one might hold a conceptualistic view,
maintaining that linguistics is essentially a branch of psychology;
the main proponent of this view is Noam Chomsky (1965, chapter 1), who
thinks of a grammar for a natural language as being about an ideal
speaker-hearer's knowledge of the given language, but see also Sapir
(1921), Stich (1972), and Fodor (1981). But there are reasons for
thinking that neither the physicalist nor the conceptualist approach
is tenable and that the only plausible way to interpret linguistic
theory is as being about sentence types, which of course, are abstract
objects (proponents of the platonistic view include Katz (1981),
Soames (1985), and Langendoen and Postal (1985)). Katz constructs
arguments here that are very similar to the ones we considered above,
in connection with mathematical objects
(section 4.1).
One argument here is that linguistic theory seems to have
consequences that are (a) true and (b) about sentences that have never
been tokened (internally or externally), e.g., sentences like
‘Green Elvises slithered unwittingly toward Arizona's favorite
toaster’. (Of course, now that I've written this sentence down,
it has been tokened, but it seems likely that before I wrote it down,
it had never been tokened.) Standard linguistic theory entails that
many sentences that have never been tokened (internally or externally)
are well-formed English sentences. Thus, if we want to claim that our
linguistic theories are true, then we have to accept these
consequences, or theorems, of linguistic theory. But these theorems
are clearly not true of any sentence tokens (because the sentences in
question have never been tokened) and so, it is argued, they must be
true of sentence types.
4.5 Possible Worlds
It is a very widely held view among contemporary philosophers that we
need to appeal to entities known as possible worlds in order
to account for various phenomena. There are dozens of phenomena that
philosophers have thought should be explained in terms of possible
worlds, but to name just one, it is often argued that semantic theory
is best carried out in terms of possible worlds. Consider, for
example, the attempt to state the truth conditions of sentences of the
form ‘It is necessary that S’ and ‘It is
possible that S’ (where S is any sentence). Il
is widely believed that the best theory here is that a sentence of the
form ‘It is necessary that S’ is true if and only
hvis S is true in all possible worlds, and a sentence of the
form ‘It is possible that S’ is true if and only
hvis S is true in at least one possible world. Now, if we add
to this theory the premise that at least one sentence of the form
‘It is possible that S’ is true — and this
seems undeniable — then we are led to the result that possible
worlds exist.
Now, as was the case with numbers, properties, and sentences, not
everyone who endorses possible worlds thinks that they are abstract
objects; indeed, one leading proponent of the use of possible worlds
in philosophy and semantics — namely, David Lewis (1986) —
maintains that possible worlds are of the same kind as the actual
world, and so he takes them to be concrete objects. However, most
philosophers who endorse possible worlds take them to be abstract
objects (see, e.g., Plantinga (1974, 1976), Adams (1974), Chisholm
(1976), and Pollock (1984)). It is important to note, however, that
possible worlds are very often not taken to constitute a new kind of
abstract object. For instance, it is very popular to maintain that a
possible world is just a set of propositions. (To see how a set of
propositions could serve as a possible world, notice that if you
believed in full-blown possible worlds — worlds that are just
like the actual world in kind — then you would say that
corresponding to each of these worlds, there is a set of propositions
that completely and accurately describes the given world, or is true
of that world. Many philosophers who don't believe in full-blown
possible worlds maintain that these sets of propositions are good
enough — i.e., that we can take them to be possible
worlds.) Or alternatively, one might think of a possible world as a
state of affairs, or as a way things could be. In so
doing, one might think of these as constituting a new kind of abstract
object, or one might think of them as properties — giant,
complex properties that the entire universe may or may not possess.
For instance, one might say that the actual universe possesses the
property of being such that snow is white and grass is green and San
Francisco is north of Los Angeles, and so on.
In any event, if possible worlds are indeed abstract objects, and if
the above argument for the existence of possible worlds is cogent,
then this would give us another argument for platonism.
4.6 Logical Objects
Frege (1884, 1893–1903) appealed to sentences like the
following:
(D) The number of Fs is identical to the number of
Gs if and only if there is a one-to-one correspondence
between the Fs and the Gs.(E) The direction of line a is identical to the direction of
linje b if and only if a is parallel to
b.(F) The shape of figure a is identical to the shape of figure
b if and only if a is geometrically similar
to b.
On Frege's view, principles like these are true, and so they commit us
to the existence of numbers, lines, and shapes. Now, of course, we
have already gone through a platonistic argument — indeed, a
Fregean argument — for the existence of numbers. Moreover, the
standard platonist view is that the argument for the existence of
mathematical objects is entirely general, covering all branches of
mathematics, including geometry, so that on this view, we already have
reason to believe in lines and shapes, as well as numbers. But it is
worth noting that in contrast to most contemporary platonists, Frege
thought of numbers, lines, and shapes as logical objects, because on
his view, these things can be identified with extensions of
concepts. What is the extension of a concept? Well, simplifying a
bit, it is just the set of things falling under the given concept.
Thus, for instance, the extension of the concept white is
just the set of white
things.(13)
And so the idea here is that since logic is centrally concerned with
predicates and their corresponding concepts, and since extensions are
tied to concepts, we can think of extensions as logical objects. Thus,
since Frege thinks that numbers, lines, and shapes can be identified
with extensions, on his view, we can think of these things as logical
objects.
Frege's definitions of numbers, lines, and shapes in terms of
extensions can be formulated as follows: (i) the number of Fs
is the extension of the concept equinumerous with F (that is,
it is the set of all concepts that have exactly as many objects
falling under them as does F); and (ii) the direction of line
a is the extension of the concept parallel to a; et
(iii) the shape of figure a is the extension of the concept
geometrically similar to a. A similar approach can be used to
define other kinds of logical objects. For instance, the truth value
of the proposition p can be identified with the extension of
the concept equivalent to p (i.e., the concept true if
and only if p is true).
It should be noted that contemporary neo-Fregeans reject the
identification of directions and shapes and so on with extensions of
concepts. They hold instead that directions and shapes are sui
generis abstract objects.
For contemporary work on this issue, see, e.g., Wright (1983), Boolos
(1986–87), and Anderson and Zalta (2004).
4.7 Fictional Objects
Finally, a number of philosophers (see, most notably, van Inwagen
(1977), Wolterstorff (1980), and Zalta (1983, 1988)) think that
fictional objects, or fictional characters, are best thought of as
abstract objects. (Salmon (1998) and Thomasson (1999) also take
fictional objects to be abstract, but their views are a bit different;
they maintain that abstract fictional objects are created by humans.)
To see why one might be drawn to this view, consider the following
sentence:
(G) Sherlock Holmes is a detective.
Now, if this sentence actually appeared in one of the Holmes stories
by Arthur Conan Doyle, then that token of it would not be true —
it would be a bit of fiction. But if you were telling a child about
these stories, and the child asked, “What does Holmes do for a
living?”, and you answered by uttering (G), then it seems
plausible to suppose that what you have said is true. But if it is
true, then it seems that its singular term, ‘Sherlock
Holmes’, must refer to something. What it refers to, according
to the view in question, is an abstract object, in particular, a
fictional character. In short, present-day utterances of (G) are true
statements about a fictional character; but if Doyle had put (G) into
one his stories, it would not have been true, and its singular term
would not have referred to anything.
There is a worry about this view that can be put in the following way:
if there is such a thing as Sherlock Holmes, then it has arms and
legs; but if Sherlock Holmes is an abstract object, as this view
supposes, then it does not have arms and legs (because abstract
objects are non-physical); therefore, it cannot be the case that
Sherlock Holmes exists and is an abstract object, for this leads to
contradiction. Various solutions to this problem have been proposed.
For instance, Zalta argues that in addition to exemplifying
certain properties, abstract objects also encode properties.
The fictional character Sherlock Holmes encodes the properties of
being a detective, being male, being English, having arms and legs,
and so on. But it does not exemplify any of these properties. Il
exemplifies the properties of being abstract, being a fictional
character, having been thought of first by Arthur Conan Doyle, and so
on. Zalta maintains that in English, the copula ‘is’
— as in ‘a is F’ — is
ambiguous; it can be read as ascribing either property exemplification
or property encoding. When we say ‘Sherlock Holmes is a
detective’, we are saying that Holmes encodes the
property of being a detective; and when we say ‘Sherlock Holmes
is a fictional character’, we are saying that Holmes
exemplifies the property of being a fictional character. (It
should be noted that Zalta employs the device of encoding with respect
to all abstract objects — mathematical objects, logical objects,
and so on — not just fictional objects. Also, Zalta points out
that his theory of encoding is based on a similar theory developed by
Ernst Mally (1912).)
Those who endorse a platonistic view of fictional objects maintain
that there is no good paraphrase of sentences like (G), but one might
question this. For instance, one might maintain that (G) can be
paraphrased by a sentence like this:
‘Sherlock Holmes is a detective’ is
true-in-the-Holmes-stories.
If we read (G) in this way, then it is not about Sherlock Holmes at
all; rather, it is about the Sherlock Holmes stories. Thus, in order
to believe (G), so interpreted, one would have to believe in the
existence of these stories. Now, one might try to take an
anti-platonistic view of the nature of stories, but there are problems
with such views, and so we might end up with a platonistic view here
anyway — a view that takes sentences like (G) to be about
stories and stories to be abstract objects of some sort, e.g., ordered
sets of
propositions.(14)
Which of these platonistic views is superior can be settled by
determining which (if either) captures the correct interpretation of
sentences like (G) — i.e., by determining whether ordinary
people who utter sentences like (G) are best interpreted as talking
about stories or fictional characters.
It should be noted that some people who take fictional characters to
be abstract objects (e.g., Thomasson 1999) would actually agree with
the idea that (G) should be read in the above way — i.e., as a
claim about the Sherlock Holmes stories and not about Sherlock Holmes
himself. Thomasson’s main argument for believing in fictional
characters is based not on sentences like (G) but rather on sentences
like the following:
(H) Some 19th century heroines are better developed than any 18th
century heroines.
It’s hard to see how to paraphrase this as being about a story,
or even a bunch of stories. But, of course, one could still endorse a
fictionalist (i.e., an error-theoretic) view of sentences like (H). DANS
other words, one could admit that (H) is a claim about fictional
characters and then one could claim that since there are no such
things as fictional characters, (H) is simply not true, although of
course it might be true-in-the-story-of-fictional-characters,
where this just means that it would have been true if there had been a
realm of fictional characters of the sort that platonists believe in.
(Brock (2002) endorses a fictionalist view of fictional characters
that's similar in spirit to the view alluded to here.)
Over the years, anti-platonist philosophers have presented a number of
arguments against platonism. One of these arguments stands out as the
strongest, namely, the epistemological argument. This argument goes
all the way back to Plato, but it has received renewed interest since
1973, when Paul Benacerraf presented a version of the argument. Most
of the work on this problem has taken place in the philosophy of
mathematics, in connection with the platonistic view of mathematical
objects like numbers. We will therefore discuss the argument in this
context, but all of the issues and arguments can be reproduced in
connection with other kinds of abstract objects. The argument can be
put in the following way:
- Human beings exist entirely within spacetime.
- If there exist any abstract mathematical objects, then they do not
exist in spacetime. Therefore, it seems very plausible that: - If there exist any abstract mathematical objects, then human
beings could not attain knowledge of them. Therefore, - If mathematical platonism is correct, then human beings could not
attain mathematical knowledge. - Human beings have mathematical knowledge. Therefore,
- Mathematical platonism is not correct.
The argument for (3) is everything here. If it can be established,
then so can (6), because (3) trivially entails (4), (5) is beyond
doubt, and (4) and (5) trivially entail (6). Now, (1) and (2) do not
strictly entail (3), and so there is room for platonists to maneuver
here — and as we'll see, this is precisely how most platonists
have responded. However, it is important to notice that (1) and (2)
provide a strong prima facie motivation for (3), because they
seem to imply that mathematical objects (if there are such things) are
totally inaccessible to us, i.e., that information cannot pass from
mathematical objects to human beings. But this gives rise to a
prima facie worry (which may or may not be answerable) about
whether human beings could acquire knowledge of mathematical objects.
Thus, we should think of this argument not as refuting
platonism but as issuing a challenge to platonists. The challenge is
simply to explain how human beings could acquire knowledge of abstract
mathematical objects.
There are three ways for platonists to respond. First, they can argue
that (1) is false and that the human mind is capable of somehow
forging contact with abstract mathematical objects and thereby
acquiring information about such objects. This strategy has been
pursued by Plato in The Meno et The Phaedo, and by
Gödel (1964). Plato's idea is that our immaterial souls acquired
knowledge of abstract objects before we were born and that
mathematical learning is really just a process of coming to remember
what we knew before we were born. On Gödel's version of the view,
we acquire knowledge of abstract objects in much the same way that we
acquire knowledge of concrete physical objects; more specifically,
just as we acquire information about physical objects via the faculty
of sense perception, so we acquire information about abstract objects
by means of a faculty of mathematical intuition. Now, other
philosophers have endorsed the idea that we possess a faculty of
mathematical intuition, but Gödel's version of this view —
and he seems to be alone in this — involves the idea that the
mind is non-physical in some sense and that we are capable of forging
contact with, and acquiring information from, non-physical
mathematical
objects.(15)
This view has been almost universally rejected. One problem is that
denying (1) doesn't seem to help. The idea of an immaterial mind
receiving information from an abstract object seems just as mysterious
and confused as the idea of a physical brain receiving information
from an abstract object.
The second strategy that platonists can pursue in responding to the
epistemological argument is to argue that (2) is false and that human
beings can acquire information about mathematical objects via normal
perceptual means. The early Maddy (1990) pursued this idea in
connection with set theory, claiming that sets of physical objects can
be taken to exist in spacetime and, hence, that we can perceive them.
For instance, on her view, if there are two books on a table, then the
set containing these books exists on the table, in the same place that
the books exist, and we can see the set and acquire information about
it in this way. This view has been subjected to much criticism,
including arguments from the later Maddy (1997). Others to attack the
view include Lavine (1992), Dieterle and Shapiro (1993), Balaguer
(1998a), Milne (1994), Riskin (1994), and Carson (1996).
It may be objected that according to the definitions we've been using,
views like Maddy's are not versions of platonism at all, because they
do not take mathematical objects to exist outside of spacetime.
Nonetheless, there is some rationale for thinking of Maddy's view as a
sort of non-traditional platonism. For since Maddy's view entails that
there is an infinity of sets associated with every ordinary physical
object, all sharing the same spatiotemporal location and the same
physical matter, she has to allow that these sets differ from one
another in some sort of non-physical way and, hence, that there is
something about these sets that is non-physical, or perhaps abstract,
in some sense of these terms. Now, of course, the question of whether
Maddy's view counts as a version of platonism is purely
terminological; but whatever we say about this, the view is still
worth considering in the present context, because it is widely thought
of as one of the available responses to the epistemological argument
against platonism, and indeed, that is the spirit in which Maddy
originally presented the view.
The third and final strategy that platonists can pursue is to accept
(1) and (2) and explain why (3) is nonetheless false. This strategy is
different from the first two in that it doesn't involve the
postulation of an information-transferring contact between human
beings and abstract objects. The idea here is to grant that human
beings do not have such contact with abstract objects and to explain
how they can nonetheless acquire knowledge of such objects. This has
been the most popular strategy among contemporary platonists. Its
advocates include Quine (1951, section 6), Steiner (1975, chapter 4),
Parsons (1980, 1994), Katz (1981, 1998), Resnik (1982, 1997), Wright
(1983), Lewis (1986, section 2.4), Hale (1987), Shapiro (1989, 1997),
Burgess (1990), Balaguer (1995, 1998a), Linsky and Zalta (1995),
Burgess and Rosen (1997), and Linnebo (2006). There are several
different versions of this view; we will look very briefly at the most
prominent of them.
One version of the third strategy, implicit in the writings of Quine
(1951, section 6) and developed by Steiner (1975, chapter four,
especially section IV) and Resnik (1997, chapter 7), is to argue that
we have good reason to believe that our mathematical theories are
true, even though we don't have any contact with mathematical objects,
because (a) these theories are embedded in our empirical theories, and
(b) these empirical theories (including their mathematical parts) have
been confirmed by empirical evidence, and so (c) we have empirical
evidence for believing that our mathematical theories are true and,
hence, that abstract mathematical objects exist. Notice that this view
involves the controversial thesis that confirmation is
holistic, i.e., that entire theories are confirmed by pieces of
evidence that seem to confirm only parts of theories. One might doubt
that confirmation is holistic in this way (see, e.g., Sober (1993),
Maddy (1992), and Balaguer (1998a)). Moreover, even if one grants that
confirmation is holistic, one might worry that this view leaves
unexplained the fact that mathematicians are capable of acquiring
knowledge of their theories before these theories are applied in
empirical science.
A second version of the third strategy, developed by Katz (1981, 1998)
and Lewis (1986, section 2.4), is to argue that we can know that our
mathematical theories are true, without any sort of
information-transferring contact with mathematical objects, because
these theories are necessarily true. The reason we need
information-transferring contact with ordinary physical objects in
order to know what they're like is that these objects could have been
different. For instance, we have to look at fire engines in order to
know that they're red, because they could have been blue. But we don't
need any contact with the number 4 in order to know that it is the sum
of 3 and 1, because it is necessarily the sum of 3 and 1. (For
criticisms of this view, see Field (1989, pp. 233–38) and
Balaguer (1998a, chapter 2, section 6.4).)
A third version of the third strategy has been developed by Resnik
(1997) and Shapiro (1997). Both of these philosophers endorse
(platonistic) structuralism, a view that holds that our
mathematical theories provide true descriptions of mathematical
structures, which, according to this view, are abstract. Moreover,
Resnik and Shapiro both claim that human beings can acquire knowledge
of mathematical structures (without coming into any sort of
information-transferring contact with such things) by simply
constructing mathematical axiom systems; for, they argue, axiom
systems provide implicit definitions of structures. Une
problem with this view, however, is that it does not explain how we
could know which of the various axiom systems that we might formulate
actually pick out structures that exist in the mathematical realm.
A fourth and final version of the third strategy, developed
independently (and somewhat differently) by Balaguer (1995, 1998a) and
Linsky & Zalta (1995), is based on the adoption of a particular
version of platonism called plenitudinous platonism (Balaguer
also calls it full-blooded platonism, or FBP, and Linsky and
Zalta call it principled platonism). Balaguer defines
plenitudinous platonism (somewhat roughly) as the view that there
exist mathematical objects of all possible kinds, or the view that all
the mathematical objects that possibly could exist actually do exist.
But, in general, Balaguer would define a different plenitude principle
for every different kind of abstract object. Linsky & Zalta
develop plenitudinous platonism by proposing a distinctive plenitude
principle for each of three basic domains of abstracta: abstract
individuals, relations (properties and propositions), and contingently
nonconcrete individuals (1995, 554). For example, on their view, the
plenitude principle for abstract individuals asserts (again, somewhat
roughly) that every possible description of an object characterizes an
abstract object that encodes — and, thus, in an important sense,
has — the properties expressed in the description.
Balaguer and Linsky & Zalta then argue that if platonists endorse
plenitudinous platonism, they can solve the epistemological problem
with platonism without positing any sort of information-transferring
contact between human beings and abstract objects. Balaguer's version
of the argument proceeds as follows. Since plenitudinous platonism, or
FBP, says that there are mathematical objects of all possible kinds,
it follows that if FBP is true, then every purely mathematical theory
that could possibly be true (i.e., that's internally consistent)
accurately describes some collection of actually existing mathematical
objects. Thus, it follows from FBP that in order to attain knowledge
of abstract mathematical objects, all we have to do is come up with an
internally consistent purely mathematical theory (and know that it is
consistent). But it seems clear that (i) we humans are
capable of formulating internally consistent mathematical theories
(and of knowing that they are internally consistent), and (ii) being
able to do this does not require us to have any sort of
information-transferring contact with the abstract objects that the
theories in question are about. Thus, if this is right, then the
epistemological problem with platonism has been solved.
One might object here that in order for humans to acquire knowledge of
abstract objects in this way, they would first need to know that
plenitudinous platonism is true. Linsky & Zalta respond to this by
arguing that plenitudinous platonism (or in their lingo, principled
platonism) is knowable a priori because it is required for
our understanding of any possible scientific theory: it alone is
capable of accounting for the mathematics that could be used in
empirical science no matter what the physical world was like.
Balaguer's response, on the other hand, is based on the claim that to
demand that platonists explain how humans could know that FBP is true
is exactly analogous to demanding that external-world realists (i.e.,
those who believe that there is a real physical world, existing
independently of us and our thinking) explain how human beings could
know that there is an external world of a kind that gives rise to
accurate sense perceptions. Thus, Balaguer argues that while there may
be some sort of Cartesian-style skeptical argument against FBP here
(analogous to skeptical arguments against external-world realism), the
argument in (1)–(6) is supposed to be a different kind of
argument, and in order to respond to that argument, FBP-ists do not
have to explain how humans could know that FBP is true.
Un solide de polyèdre doit avoir toutes les faces planes ( par exemple, des solides de Platon, des prismes et des pyramides ), tandis qu’un solide non polyèdre a au moins une de ses surfaces qui n’est pas plate ( par exemple, barillet, sphère ou tube ). n Régulier sous-entend que tous les angles sont de la même mesure, toutes les faces sont de formes congruentes ou équivalentes dans tous les aspects, et tous les bords sont de la même taille. n 3D sous-entend que la forme a la largeur, la profondeur et la hauteur. n Un polygone est une forme verrouillée dans une est plane avec au minimum cinq bords droits. n Un duel est un solide de Platon qui s’adapte à l’intérieur d’un autre solide de Platon et se connecte au point médian de chaque face. n

















