The Greeks postulated that the world we observe may be just an illusion and Kant based some of his philosophy on that very idea. From that idea, came the idea that mathematical truths are more certain than empirical truths, but how can we know that mathematics cannot be an illusion created by the mind or is negatively affected by this illusion?

  • Descartes in Meditations actually fears this very possibility. I'm not sure precisely which meditation it is in ..... But he mentions that the evil demon might be confusing even his arithmetic. – Squirtle Feb 24 at 4:32
  • Greeks postulated no such thing, it is a much later Cartesian idea, and Kant did not share it. That mathematical truths are more certain than empirical truths had nothing to do with it either, at least for Kant. Where are you getting this from? And, by your own reasoning, "how can we know" has a trivial answer - we do not need to know, we can just postulate it. – Conifold Feb 24 at 8:12
  • @Conifold Yes the Greeks did – Math Bob Feb 26 at 16:14
  • @MathBob Please provide a reference. – Conifold Feb 26 at 21:01
  • The Eleatics come close. I the multplicity of things is an illusion, then so is most of our experience. – user9166 Feb 26 at 21:34

Kant supposes that certain ideas are "a priori, but synthetic". They reference the outside world, and we experience them only in response to it, but they come from within us. They arise from the context we lay upon our experience, instead of from the experience itself. We are more certain of them not because they are somehow any truer than other things, but because we assume them automatically by virtue of being ourselves. They are still phenomena, but they are part of our structure of understanding. We find them very difficult to question.

The structure of space and time are among those things. Kant describes those as 'forms of our intuition' rather than facts or basic ideas. Brouwer later elaborated the idea that mathematics is entirely based upon our perceptions of space and time, and therefore is entirely interior to human thought, and not part of external reality. It is a creative activity of the human mind, giving structure to things, and not a part of external reality, learned by exposure.

By that theory. whatever violates our deepest intuitions about space and time, we can not really comprehend. We can only allude to it, or form abstruse models that adapt our real understanding in artificial ways that provide them leverage over what we cannot really grasp. So we are not certain that these intuitions are true, only that to the degree they fail to be true, we have no chance of understanding the things to which they do not apply. Some humility regarding our own ability to grasp deeper truth is in order.

That is very different from saying they are really more certainly true of reality.


I’d argue that minds are real, and the ideas that they have a reality to them. This is the angle taken by Kant where he postulated that the basic categories of explanation articulated by Aristotle were in fact categories of this kind, including ideas of modality, of number, of quality and of space and temporality.

This does not mean that minds cannot entertain bad or false ideas, when they most obviously can.


There are three questions here. I will pass over the first question (whether any philosopher made the claim) out of ignorance.

The second question I crafted from the original post: How can we know that mathematics cannot be an illusion created by the mind?

My answer to this question depends on the definitions of the terms, in context. I can offer an answer of sorts, but some people may necessarily object to my definitions. Some may object to my paraphrasing of the question above. I am new to the site and welcome your criticism.

I personally consider "mathematics", in the context of this question, to be any one in a family of languages with strict syntax and consistent semantic rules. The language has strict enough syntax that any grammatical statement is unambiguous. The rules are consistent in the sense that rules don't contradict each other and no valid statement will ever contradict a rule; in elementary mathematics, "one times seven equals one" is always illegal because it contradicts an established rule (the multiplicative property).

I would consider an "illusion", in this context, to be a belief proven false by contradictions of evidence. For example the entasis and curved lines in some classical Greek architecture provide the illusion of straight lines: visually the columns and walls appear straight, but a more detailed measurement will show the curvature.

Now if mathematics is consistent, as it is by my definition, there will be no contradictions between rules and valid statements. Neither will there be contradictions between any two rules. Therefore no evidence of a contradiction can be found between a valid statement and a rule, nor between any two rules. Therefore neither mathematics nor a valid mathematical statement can be an illusion, as I define it.

The third question is ambiguous. I can't identify the subject or direct object for the last clause, which reads "... or is [what?] negatively affected by [which?] illusion?" I'm guessing it was supposed to be "is the mind negatively affected by the illusion of mathematics?" Regardless, I have asserted that mathematics is not an illusion so this third question would not apply.

Responses to comments

@jobermark has expressed concern that my definition of mathematics would apply to physics and other basic sciences, whereas he asserts these fields are not mathematics but only make use of mathematics.

I do believe the formulas of physics and other sciences, if expressed in mathematics, are in fact a form of mathematics. I stand by my opinion that no valid mathematical statement - even in the context of physics or another science - can be an illusion in its own right.

I agree with jobermark's implication that physics and other sciences are more than the pure mathematics they employ. The object of science is usually not to describe imagined systems but to describe real, physical systems and objects. As such it is common for a scientist to say, not only is this a valid mathematical statement but it also is a useful statement that describes the real world! This would be a new rule which introduces falsifiability, since presumably the scientist did not add a complete and accurate set of rules to precisely describe the real world.

Sometimes the mathematics used, while valid on its own, just doesn't apply to the situation as advertised. A physicist of old may have said, velocity is the product of speed and time. It is implied that this relation applies to reality, not just some imagined universe. A physicist today may retort, this equation does not hold at speeds approaching c. We have conducted experiments and gathered evidence in the real world which contradicts that formula's applicability to reality.

I would compare this misapplication of mathematics to using a false analogy in conversation with any other language. The language itself is perfectly valid, and one could indeed imagine a universe where a sentence such as "I am nine feet tall" is true. It may get the point across that I am a tall person, but it would be misleading (an illusion) to present that sentence as true when I am not actually nine feet tall.

@jobermark also expresses concern that my definition of illusion excludes any successful illusions, for example alchemical homeopathy.

For similar reasons to the above, I stand by my definition. Aelius Galenus may have proposed the relation, an excess of blood is positively correlated with optimism. Indeed such an equation could be valid in mathematical terms (optimism = blood + 6 liters, for example), but were Galen to assert that this relation applies to real people's blood and attitude (as he did), then that claim becomes falsifiable. The mathematics itself is not the illusion, only the application.

If jobermark meant to say that a valid mathematical statement might turn out to be an illusion because the rules change, I would say this is precluded by my definition of mathematics as following consistent rules. I don't think this was the intended argument, though.

  • Neither of your definitions really work: 1) Physics and other basic sciences are not mathematics. They are uses of it. Your definition wholly encloses all successful theories stated properly in mathematical terms. If a language includes its vocabulary, mathematics is only some core part of these languages. 2) Your definition of illusion excludes all successful illusions. Were principles like alchemical homeopathy not illusions for centuries only to become illusions lately? I don't think that fits the usage of the word you are trying to define. – user9166 Feb 27 at 17:04
  • @jobermark Thank you for the questions. I have edited my answer in response. – Max Feb 28 at 1:32
  • You can't change your definitions without changing them. You didn't say anything about the possibility of falsification, only about actual proved contradictions, that still implied what I said. You didn't do anything to separate the theoretical model from the application, again, you defined something, but it is not what you intended, waving your hands at it doesn't change anything. Your definitions are still wrong. – user9166 Feb 28 at 5:44
  • @jobermark I don't fully understand. With regard to 1) I still define mathematics as any of a set of languages with strict syntax and consistent rules. When a science such as physics includes, as a rule, that some statement should apply to the real world, they expose that statement to falsification via observation of the real world. If such observation disagrees with the statement you have a proved contradiction between the statement and that rule, therefore the statement is invalid, the rules are amended, or the science ceases to be mathematics. – Max Feb 28 at 6:33
  • With regard to 2) I would say that discredited sciences such as alchemy or humorism never were valid mathematics, but only generalizations. This is not to say they were without perceived utility, but in my opinion it is indeed an illusion that the mathematics of these subjects ever truthfully described something in reality. People may not have noticed or observed evidence showing the contradictions for a long time, which means to them it seemed like mathematics when in reality it was the illusion of mathematics. – Max Feb 28 at 6:48

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