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In mathematical philosophy, one asks the question "do mathematical objects really exist"?

This is then followed by "yes" or "no" answers, but does the question even make sense? Is it even meaningful to talk about the existence of an idea? Of a concept? Of a equation?

So basically, that's my question. When philosophers talk about whether mathematics is real or not, what definition of 'real' are they using? What definition of 'exist' do they use to judge whether mathematical objects exist or not?

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    If the person asking the question can't respond to "what do you mean, really exist?" with a good coherent answer, than the original question is meaningless. And if they give you a good coherent answer, it will probably contain the answer to the question within it. Feb 24, 2016 at 17:39
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    This seems very closely related to philosophy.stackexchange.com/questions/8512/are-numbers-real, can you edit the question to distinguish it? Feb 24, 2016 at 22:34
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    This might be of interest to you, even if it does not answer the question. en.wikipedia.org/wiki/Mu_(negative)#.22Unasking.22_the_question
    – JeD
    Mar 28, 2016 at 0:08
  • I've encountered a term like "sufficiently cheap form of existence" that's useful here, but I can never find a source for it.
    – J.G.
    Apr 28, 2021 at 15:38
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    Most people use mind-independence as a criterion for existence of mathematical objects. The real question (appears to) come down to mind-independence vs. psychologism. If neo-logicism is successful, then it can be shown that mathematics has a mind-independent ontology. May 28, 2021 at 14:29

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You are, in a sense, begging the question against (a part of) those who ask themselves the question whether mathematical objects really exist. That is, because you already come equipped with a certain theory of what mathematical objects are. According to you, mathematical objects are ideas, concepts and equations.

A big part of the discussion about the existence of mathematical objects concerns the question what mathematical objects could be. Can we have mathematics as we practice it today if mathematical objects are concrete? Can we have knowledge of mathematics when mathematical objects are abstract? If so, how?

I suggest that you read the related SEP entry on the philosophy of mathematics.

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  • Can all mathematical objects be classified in a singular class of abstract objects? I am more inclined to believe in the existence of the "finite set", as opposed to the distinctly more notional "infinite set". May 30, 2021 at 17:27
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You've spotted something good by separating out "real" and "exist"!

The philosopher Willard Van Orman Quine suggested that the second of these questions is decided by our use of classical, first order predicate logic. When we ask what it means for an entity to exist, what it means for us to be "Ontologically Committed" to something, we look at what, in logically rigorous terms, we mean, and we ask about what objects our theory needs to include in order to make sense of the idea of a Variable having the Values it needs to make the sentences we say true.

Let's take an example in number theory: "there is a prime number p such that p+2 and p+6 are also prime numbers". It's clear to us what this means (for example, p=5 and p=11 are known solutions), even if we don't intuitively know how many things there might be that satisfy it!

In order to make sense of it, we seem to think about p as a variable ranging over a domain of things - the natural numbers. For Quine, this is sufficient to say that we are Ontologically Committed to (at least some) natural numbers as being in our class of stuff in the world. Now, does this mean we have to think of numbers as any particular stuff? Not in itself - we might want to think of numbers as just bundles of patterns realized by physical systems, or component elements of human formal symbol practices, or spatiotemporally isolated abstract entities on a higher plane of existence - but we still want to find some way to validate our talk of prime numbers existing, and that's what it means to be Ontologically Committed.

This does prompt a question of how we flesh out our explanation of whether mathematical objects "really" exist. What lies at the bottom of this chain of ontological reasoning, if there is even such a bottom? This is the question of Realism, and this field is much more complex than a simple answer here might start to pick at, but to keep going with Quine's approach, Quine was a Scientific Realist - the stuff that is real is whatever is needed to ground our best scientific practice.

Interestingly, seeing Mathematics as a science in its own right, he concluded that there was no question of needing to reduce mathematical objects to anything else - if it's central to our good scientific practice, then if you want to know what mathematical objects really exist, you just need to ask a mathematician!

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The problem is that the only way mathematicians can rigorously define and reason about mathematical objects is through finite chains of reasoning based on some finite set of axiom schema. If a mathematical object "really exists" there may be no satisfactory way to say exactly what that means.

Yet it ought to mean that the mathematical object has fixed properties, independent of humans' abilities to determine what those properties are.

For example, it has been proven that using the Zermelo-Frankel axioms plus the Axiom of Choice, it can be neither proved nor disproved that the set of real numbers has a subset whose cardinality is strictly in between that of the integers and that of the real numbers. Many non-platonists will insist this question is just a matter of which axioms you might prefer to use. Many platonists would instead say that since the real numbers has a well-defined existence, such a subset either really exists or really does not (even if we can perhaps never know the answer).

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Rovelli, a well-known European theoretical physicist remarked it's hard to understand how reality exists. Whilst relativity revolutionised our understanding of space and time, the quantum revolutionised our understanding of ontology, that is how things that exist, actually exist as they are.

Given this, we should not be too surprised if numbers and hence ideas exist in some sense. Whilst Plato declared there to be a realm of ideas, he was hazy about how they relate to our actually existing world.

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A language is a tool of communication. They are crafted by human beings called linguists. Each tool has s special purpose and so do languages. Languages have their domain of usage: from ordinary talk of everyday objects of consciousness (food, household utensils, people, cars, etc.) up to the sophisticated languages used by an intelligent class of men (high-class scientist and engineers, psychologists, philosophers, theologists, etc.) to describe objects of consciousness of their domain of thought.

Mental concepts are verbal constructs (constructs of thoughts) that describe and give meaning to objects of consciousness. But mental concepts are not reality.

"The map is not the territory." ― Alfred Korzybski

Today, the majority of people suffer from mental identifications. People think names are real and choose to live in a verbal construct of thoughts for which they believe to be their real self. Real self is conscious. Aware.

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  • "A language is a tool of communication." This is incomplete. The language is not a theoretical primitive. Information is. How do you have a language without information and relationships? "Mental concepts are not reality." Really? That depends on whether information transfer is unidirectional or bidirectional. If information is coming into a mind (bidirectional) and that is represented by mental patterns, the internal information processing of the mind is not the genesis of that information or structure. Hence, "The map is not the territory" is not a satisfying answer to this question. May 28, 2021 at 14:30
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    Interestingly you have not used the word mathematics or the term "mathematical objects" even once in your answer. May 30, 2021 at 17:08
  • And mathematics appears to have a universality other languages do not, which is exactly the issue to be addressed
    – CriglCragl
    Sep 25, 2021 at 13:33

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