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Which philosophers best opposed Plato's Theory of Forms?

I know Aristotle did but what others have also?

Is there a consensus on the best/strongest refutation of the Theory of Forms?

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    There is rarely (never?) consensus on anything in philosophy. Objections to the theory of forms, including most of Aristotle's objections, can be found already in Plato's own dialogues, e.g. in Parmenides. Plato obviously did not find them fatal, and since Platonism survived to our days there is obviously no "refutation" of it. More recently it was criticized by Russell, among others. – Conifold Jun 26 '17 at 23:27
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Nominalism is a school of thought that rejects platonism and as such it supplies arguments against Plato's Forms. Nominalists believe that there are no abstract objects (in this case the abstract objects being Plato's Forms). One of the most ubiquitous nominalist arguments against Plato's Forms is what is called the epistemological argument. In essence the epistemological argument against metaphysical platonism starts by raising this question:

How do we have knowledge of the forms?

Plato's Forms are supposedly abstract objects that do not exist in the same sort of space and time that we ourselves do. It then becomes very apparent to ask "okay, then how do we have knowledge of the forms?" This line of reasoning has been explored by many philosophers (including Plato himself) and is considered by some to be the strongest argument against platonism. The argument has been articulated in "Mathematical Truth" by Paul Benacerraf and "Realism, Mathematics, and Modality" by Hartry Field.

From the Stanford Encyclopedia of Philosohpy's article on Platonism in Metaphysics:

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:

(1) Human beings exist entirely within spacetime.

(2) If there exist any abstract mathematical objects, then they do not exist in spacetime. Therefore, it seems very plausible that:

(3) If there exist any abstract mathematical objects, then human beings could not attain knowledge of them. Therefore,

(4) If mathematical platonism is correct, then human beings could not attain mathematical knowledge.

(5) Human beings have mathematical knowledge. Therefore,

(6) 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.

The epistemological argument is centered around the idea that human beings cannot have any sort of knowledge of abstract objects that exist outside of our spacetime. This is similar to (if not outright the same as) the argument against mind-body dualism referred to as the the problem of interaction. The problem of interaction is exactly what is going on with our premise (3), how can something abstract and unphysical interact with something physical? How can the platonic forms have a causal impact on human knowledge if they have no way to interact with us?

In the Phaedo, Plato gives his original argument for his forms by stating that the soul is immortal and also an immaterial object that exists in the Realm of Forms when it is not inside of a living body. He also makes the argument that learning, gaining knowledge, is actually remembrance; he argues that the soul has eternal knowledge of all forms but suffers from amnesia. So, if the soul already has knowledge of forms there is no need to be worried about whether or not it can gain knowledge from an abstract realm that humans cannot access. This argument might seem to skip the epistemological argument, but unfortunately it goes directly back into the mind-body problem. If the soul is immaterial and is what possesses knowledge, how does it causally communicate that knowledge to the brain? This is another example of how the problem of interaction and the epistemological argument are related in this context.

Counter arguments against the epistemological argument have been recently given by John Burgess and Gideon Rosen in two essays: "Nominalism Reconsidered, part of The Oxford Handbook of Philosophy of Mathematics and Logic, and A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics. Their arguments center mainly on the idea that "nominalist epistemological premises typically turn out to be either too weak or too strong." They go on the say that even if a nominalist were to find a "just right" formulation of the argument, they would still fall into a trap. From "Nominalism Reconsidered":

The development of theories in epistemology is a quasi-inductive affair. Typically one begins with what one takes to be clear cases of justified belief and clear cases of the opposite, and looks for a formula that covers the data and rings true on reflection. A theory developed in this spirit can then be used to correct pretheoretical verdicts at the margins. But it would seem that it cannot be used to undermine a vast and significant class of normally uncontroversial verdicts about justification.

If the theory really has the implications the nominalist wants, implying that our apparent mathematical knowledge is error or delusion, the anti-nominalist can simply claim that the novel theory, however meritorious in other respects, stands refuted by the counterexample of mathematical knowledge. It would be a gross mistake to repudiate the central claims of Mesozoic paleontology or Byzantine historiography on the basis of a theory of justification that had been developed and tested on examples drawn exclusively from, say, particle physics. The nominalist who wields a theory of justification developed by reflection on cases of empirical belief as a club against the mathematicians can with considerable plausibility be charged with a similar mistake.

Their argument centers on how we methodologically go about structuring theories of knowledge. What may be the correct way towards epistemic analysis in one domain may not be the same in another. Most of their argumentation is centered around the structure of the epistemological argument, criticizing and analyzing the way in which we take part in epistemology, as opposed to addressing exactly how a human can gain knowledge from an abstract, nonphysical realm. In this sense they aren't rejecting (3) as being flat out false, they are stating that it does not provide a full account of what is going on.

These are just a few examples of literature on this topic. For further reading see the bibliography of the SEP article here.

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    I haven't got the time to read the entire answer which seems well-written, but The school of thought seems off on in the first sentence. There's several schools of thought including hylomorphism and conceptualism that reject Platonism. Plato even seems to raise unanswered objections in later dialogues. – virmaior Jun 27 '17 at 2:37
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    @virmaior sure, that was definitely oversimplified, I added that into the answer after I ended up talking about Burgess and Rosen's criticism of the argument and didn't introduce it in a thorough way. I will say though, I think traditionally the discussion about universals is usually started off by comparing and contrasting platonism and nominalism. But I agree that statement could use more nuance, I'll go through and do edits tomorrow after I sleep. – Not_Here Jun 27 '17 at 3:07
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    The problem of universals is usually described as the dispute between nominalists and realists, and the dominant kind at the time were Aristotelian realists rather than Platonists. Ironically, Burgess is also closer to Aristotelian realism than to Platonism:"nobody nowadays, except perhaps a stray numerologist or two, would imagine that mathematical objects act on us through some mysterious sixth sense of ESP... Nonetheless, as Maddy has skillfully argued... we do, in a sense, have causal contact with certain abstracta". But Benacerraf's dilemma was indeed directed at Platonism. – Conifold Jun 27 '17 at 4:01
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    Anti-realism without qualifications usually means anti-realism across the board, including about singulars, nominalism is essentially anti-realism about universals only. But contrasting it to Platonism is a good opposition and does highlight the objections/responses nicely. So I like your post, and it deserves more than +1, but that is all I could do :) – Conifold Jun 27 '17 at 22:37
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    Sure I definitely get that, thank you for your feedback! – Not_Here Jun 28 '17 at 4:49

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