Cross-posted on Quora under the question “Did Kant make a mistake when he said that mathematics is synthetic and a priori?”

The influential German philosopher Immanuel Kant famously claimed in his Critique of Pure Reason that mathematics contains a body of knowledge that is synthetic a priori. I believe this claim is wrong, and here is why.

First, it seems obvious to me that there is some mathematics involved in our everyday interaction with the physical world. Yet, to say that such mathematics is a priori and true becomes questionable as soon as we look more closely into how human brains “think” mathematics.

From an evolutionary biology and cognitive science perspective, our brains can “think” everyday mathematics mainly because:

1. Natural selection has preserved such hunting-and-gathering-related capacities as perceiving, abstracting, classifying, and grouping objects, which is coded in our genes and expressed as part of the biological development of our brains;
2. Through subjective experience in the real world, especially during early childhood, our brains have acquired stereotyped “principles” and “rules” in relation to objects and their mathematical properties, which are operationalized by physical/chemical reactions among neurons under specific arrangements/relations/patterns in our brains.

Now, to evaluate the claim that mathematics is synthetic a priori, we may need to review the particular definition of the term that Kant used. According to Wikipedia (Analytic–synthetic distinction):

In the Introduction to the Critique of Pure Reason, Kant contrasts his distinction between analytic and synthetic propositions with another distinction, the distinction between a priori and a posteriori propositions. He defines these terms as follows:

a priori proposition: a proposition whose justification does not rely upon experience. Moreover, the proposition can be validated by experience, but is not grounded in experience. Therefore, it is logically necessary.

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So basically an a priori mathematical proposition is justifiable independently of any experience whatsoever. For instance, and I follow Kant’s famous example, one can see that 5 + 7 = 12 is “true” by merely thinking about it, without referencing any experience such as 5 apples, 7 apples, 12 apples, and so on.

However, when we take the human brain’s inner workings into account, the fact that you know 5 + 7 = 12 is “true” by merely thinking about it actually involves the firing of neurons whose arrangements/relations/patterns have all been determined by the previous physical and experiential processes as I have described above. There is absolutely no guarantee that these processes will lead to truth. Indeed, it is a well-known fact that to the type of mathematics involved in the quantum world, the commutative law of multiplication (i.e., a × b = b × a) that we take for granted does not apply. There really is no a priori truth other than what you believe to be true mostly because of your heuristic experience with the world, which has been embedded in your thinking apparatus when you make the calculation.

To an outsider who can observe your neural activities, this whole “I know 5 + 7 = 12 by merely thinking about it” magic is no more different, in principle, than a mechanical Turing machine manipulating symbols on a strip of tape according to a predetermined table of rules, which will unmistakably give the result “5 + 7 = 12” (in its binary form) if requested. But there is nothing that prevents the rules from changing so that something like “5 + 7 = 21” becomes the result. And that is exactly why mathematics can be done analytically.