I have a long interest in the 'problem of universals' and a conviction that some form of either Platonic or Aristotelian realism in respect of universals is correct. This extends to what is called 'Platonic realism' in mathematics, i.e. that mathematics describes objects (where the term 'object' signifies for example the natural numbers) which are real independently of the apprehension of any individual. (Contemporary Platonists include Kurt Godel and Roger Penrose.)

Platonic realism is the philosophical position that universals or abstract objects (including number) exist objectively and outside of human minds.

Aristotelian realism holds that universals are incorporeal and universal, but only exist only where they are instantiated; they exist only in things, not in some purportedly ethereal domain or Platonic heaven. For this reason it is sometimes called 'immanent realism'.

I have long entertained the idea that the debate about the reality of universals can be resolved by understanding that universals have a different kind of existence to particulars.

For example, the view of representative nominalists was that only names (or, more generally, words) are universal, "for the things named are every one of them singular and individual" (Hobbes, Leviathan, Ch. 4). According to Ockham universals are terms or signs standing for or referring to individual objects and sets of objects, but they cannot themselves exist. For what exists must be individual, and a universal cannot be that; the mistake of supposing that it could was the fatal contradiction of Platonic realism.

Now, I say that the supposed contradiction in the Platonic/Aristotelian view can be resolved if we understand that universals do not exist in the same way as particulars. Their existence (and whether 'existence' is the correct word in this context is part of the issue) is purely intelligible - but universals are the same for all who think. This is why, for instance, there is universal agreement about fundamental logical laws and arithmetical rules. Regular abstractions such as these provide the basis for language, abstract thought, mathematics, and even science itself.

About one of the only places where I've encountered acknowledgement of this is in Bertrand Russell's discussion in the Problems of Philosophy - The World of Universals, where he says:

We shall find it convenient only to speak of things existing when they are in time, that is to say, when we can point to some time at which they exist (not excluding the possibility of their existing at all times). Thus thoughts and feelings, minds and physical objects exist. But universals do not exist in this sense; we shall say that they subsist or have being, where 'being' is opposed to 'existence' as being timeless.


I believe this confession on Russell's part points to something of extraordinary importance that has been mostly lost to modern philosophy, but which was fundamental to the classical tradition of philosophy.

This is that 'what exists' is, as Russell says, those things which are locatable in time (and presumably space), which I would paraphrase as 'the phenomenal domain' or 'sensable objects'. Due to the overwhelming influence of empiricism in modern culture, this is generally presumed to be the only real domain, whereas the domain of numbers, universal ideas, and the like, is nowadays subjectivised as the products of the mind (hence, 'neural output'). But Russell says they 'subsist' or 'have being' albeit of a different order to the phenomenal. I think the correct term for the kind of being they have is actually 'noumenal' - not in the Kantian sense of being unknowable, but in the original meaning of 'objects of pure intellect' (or nous). So here, we see a restatement of the fundamental distinction between 'phenomenal' and 'noumenal' (or the distinction of reality and appearance associated with classical metaphysics.)

So the question is: (1) do others agree that a valid distinction can be made between the nature of the existence of phenomenal objects, and the nature of the existence of intelligible objects such as numbers and universals? and (2) that this is a distinction that has generally fallen from view in modern philosophy?

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    We hope so. You can kick a stone; you cannot kick the idea of stone, the concept stone, the stoneness. Dec 1, 2022 at 6:37
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    See also Beingless Objects — Russell versus Meinong and Alexius Meinong in general. Dec 1, 2022 at 6:50
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    See also Nicholas Griffin and Dale Jacquette (eds.), Russell vs. Meinong: The Legacy of "On Denoting" and Dale Jacquette, Alexius Meinong, The Shepherd of Non-Being and Francesco Berto, Existence as a Real Property: The Ontology of Meinongianism. Dec 1, 2022 at 6:53
  • Thanks! I was rather hoping to avoid discussions of Meinong. The specific implication of this debate that interests me is the nature of the reality of mathematical entities, being, as they are, so fundamental to science. Also the argument that modern empiricism (which is descended from medieval nominalism) has lost sight of the concept of degrees of reality, and consequently of the distinctions that can be made between existence, reality, and being, which are nowadays often regarded as synonyms.
    – Wayfarer
    Dec 1, 2022 at 7:20
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    But there isn’t universal agreement on logical laws. Yet I don’t think that lack of agreement necessarily weakens the idea that the armchair/intellectual/a priori/analytical philosophy side does produce its own knowledge. I think the distinction between analytic/continental or analytic/empirical is still the main distinction around. Still neither side has exclusive rights to numbers and universals. I take your question more as, if you do or don’t want universals to exist at the same ontology as particulars, find the camp/method that espouses such. They both have practitioners.
    – J Kusin
    Dec 1, 2022 at 17:23

3 Answers 3


What you are leaving out of this discussion is how to do epistemology. What is the method by which one should establish what is real, and what is not? Or establish degrees of real? You appear to believe in a process of DECLARATION. Which -- is not actually a reason or justification for anyone who disagrees to accept your declarations. Your attack thru name-calling rhetoric (calling those who disagree as suffering from "poverty" of their thinking) -- is fallacious, and the only attempt at justification I see.

Empiricists operate off inferential indirect realism. This is the methodology of Locke and Popper. Models which are highly useful, and pass falsification tests, are assumed to represent reality. There is only one criterion in this process, and if abstract objects pass this process, they would not be a "different kind" or lesser "real" than material objects. If you want two degrees of "real" then you will need to develop a coherent epistemology that can support your two criteria.

Karl Popper, who is the leading author of our current version of how to do empiricism, considered that both consciousness, and abstractions, were definitively shown to be "real" per indirect realism. He DID accept that there are differences between things with space and time, only time, and neither space nor time properties. He inferred that our universe has three categories of types of real things, which he called Worlds 1, 2, and 3, following Frege. Here is a paper where he spelled out these three worlds: https://www.scribd.com/doc/7187000/Karl-Popper-Tanner-Lectures

Note that for Popper, and most contemporary philosophy, this issue is discussed as a question of "abstract objects" rather than "universals". Something like the image of my lover's face in my mind's eye -- is a singular distinct abstract object, but is not a universal, but offers the same challenge to materialism as the universal concept of facial images (or the even more general "images") does. Therefore the discussion has shifted from "universals" to all abstractions.

  • "Your attack thru name-calling rhetoric (calling those who disagree as suffering from "poverty" of their thinking) -- is fallacious, and the only attempt at justification I see." To whom is this addressed, and why? I myself have neither attacked anyone, nor mentioned 'poverty'. I'm aware of Popper's 'third realm' theory. See also Tyler Burge, Frege and Knowing the Third Realm on Frege's implicit Platonism.
    – Wayfarer
    Dec 8, 2022 at 20:31
  • @Wayfarer -- I reviewed your question and skimmed all dialog before posting, to make sure any reply I provided would be on target. I did not see an epistemology; what I think you are using is "this feels appropriate to me", which is an intuitionist justification. The dismissal I found in your last comment: "the metaphysical poverty of today's empiricist philosophy". If you think we need a different epistemology than Popperian Indirect Realism, suggest one. If you are satisfied with empiricism's epistemology, show how it can support multiple levels of "real". Don't dismiss/disparage.
    – Dcleve
    Dec 8, 2022 at 23:46
  • Hey thanks for the explanation although I thought your initial response harsh. As for 'the metaphysical poverty of today's empiricist philosophy', I believe that is justified by the lengths that empiricist philosophers have gone to to dismiss mathematical realism see e.g. iep.utm.edu/indimath and the refusal to acknowledge anything like innate rationality on basically dogmatic grounds.
    – Wayfarer
    Dec 9, 2022 at 2:02
  • e.g. the statements from the IEP article cited above 'Mathematical objects are not the kinds of things that we can see or touch, or smell, taste or hear. If we can not learn about mathematical objects by using our senses, a serious worry arises about how we can justify our mathematical beliefs. ...the rationalist’s claims appear incompatible with an understanding of human beings as physical creatures whose capacities for learning are exhausted by our physical bodies.' And perhaps that's because we're simply not! The reality of abstract objects challenges physicalism.
    – Wayfarer
    Dec 9, 2022 at 2:09
  • @Wayfarer Quine and Popper were the two most important empiricists of the 20th century, both were platonists relative to abstract objects, and your IEP link goes to an empirical argument for platonism. I don't understand your accusation of poverty in empirical philosophy, which you associate with an implied claim that empiricism is wedded to physicalism. Your own link shows that is not the case.
    – Dcleve
    Dec 9, 2022 at 4:21

The short answer is yes.

I think this is an example of a class of problem that arises because human thought and language is imprecise and riddled with unrecognised ambiguity.

When you ask whether a type of thing is real, the answer depends on what you mean by real.

Certain things are real in the sense of being physical- they can be located in spacetime and are subject to the laws of physics. A car, a tree, a photon, a picture of Mickey Mouse, the Andromeda galaxy- they are real in that physical sense. Note that the laws of physics are not real in that sense.

Certain things are real in the sense of being mental- you as an individual can experience them, but they do not have the same concrete properties as physical objects. Pain, taste, softness, the idea of a tree, the idea of Mickey Mouse- all of these can be experienced, and they might have physical causes or counterparts, but they are not physical. Colour, for example, is a mental sensation. Things that are mental exist in the mind alone.

Then you have a category of reality which is neither physical nor mental, to do with relationships between things that may be physical or mental. For example, the laws of physics. Ohm's law, for instance, is a relationship between certain measurable attributes of certain types of physical objects. Ohm's law isn't physical, in the sense I defined earlier, and nor is it mental, in that the relationship between voltage, current and resistance is there (IMO) regardless of whether minds exist to contemplate it.

Asking a question such as 'is mathematics real?' to me seems both pointless and misguided. It seems to me as if you are asking whether you can take a vague, misused and highly ambiguous word such as real and extend the scope of its ambiguity even further. It is as if you are asking can we confuse ourselves even more by trying to force even more things into this bulging bag of confused concepts we call reality.

Here are some things you can say about mathematics:

  1. It can be figured out by the human human mind.
  2. The way in which it is conventionally presented (eg the symbols we use) has been invented by human minds.
  3. Branches of it describe the relationships between physically really objects
  4. Those relationships would exist regardless of whether human minds figured them out and regardless of whether humans had invented symbols to represent them.
  5. Mathematics is not locatable in spacetime, nor is it subject to the laws of physics.
  6. If all humans died tomorrow, the relationships expressed in mathematics would hold true and could be rediscovered by some other form of intelligence.

Given that, the properties of mathematics mean that it is not real in the physical or mental senses I defined above. If you insist on trying to decide whether or not to think of it as real, then you must define some other category of reality to place it within.

  • Reasonable analysis but the question of the ontological status of number is by no means trivial. See the SEP article Platonism in Philosophy of Mathematics plato.stanford.edu/entries/platonism-mathematics/#PhiSigMatPla . And figuring out what precisely are the difference in meaning between 'to exist' 'to be' and 'real' comprises a vast bulk of the subject of philosophy itself. Regarding your last comment, a third possibility is that reality is neither physical nor mental, but that they're both aspects of a larger whole.
    – Wayfarer
    Dec 8, 2022 at 20:42
  • 'Figuring out the precise difference between'... etc- my point is that such an endeavour is utterly misguided. It assumes words have precise meanings, which is a naive notion. Words mean whatever people take them to mean. Dec 8, 2022 at 22:18
  • OK, so relativism and subjectivism. Whatever floats your boat. ''When I use a word,' Humpty Dumpty said in rather a scornful tone, 'it means just what I choose it to mean — neither more nor less.'
    – Wayfarer
    Dec 8, 2022 at 22:26

You are confused because you are unable to distinguish between an idea and implementation of the idea.

Are you a superman? Just because you (and I) can think about it dont mean you are it. See an idea can exist without any implementation of the idea.

There, solved. Ideas of Shapes can exist without those Shapes themselves (physically) existing.

  • You don't exhibit an understanding of the distinction between the uses of idealism in philosophy and ordinary language. A common newbie error.
    – Wayfarer
    Jan 8, 2023 at 22:12
  • Me newbie or not (how would you know?) idealism don't make a thing real. A thing remain what it is - an idea remain an idea, implemented or not.
    – Atif
    Jan 9, 2023 at 4:16

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