We are often told (Feynman i.a.) that mathematics is different from science in that the results are not measurable.

We might take the speculation a bit further and wonder if indeed mathematics is completely separated from reality.

Perhaps some unknown –to-all mathematics is, but for the mathematics we know, i.e. human-formulated mathematics, is usually related to axioms which are formulated in a human language (or in symbols explained in a human language) developed in a heavily reality-burdened environment. I am not looking for a specific type of bias, that would tell us about specific alternative interpretation of the axioms of the axiom-related proven results, only argument s to deny a complete departure from reality.

When I google this I get all kind of entries about other thing than the above, but perhaps I need to formulate the question better.

I did find something: Timothy Williamson: “Absolute provability and the safe knowledge of axioms”: http://media.philosophy.ox.ac.uk/assets/pdf_file/0004/35338/provabilityfinal.pdf The author seems to point to variability of interpretation of an axiom, but I am not sure it directly addresses my concern.

Surely there must be some research about this?


1 Answer 1


Conifold is right; mathematics was originally invented to allow the number of goats in a herd to be tracked, etc. By cleverly investigating the properties of numbers in themselves (independent of goats), it became clear that there was nonobvious structure contained within the realm of numbers, and the more complex the number realm was allowed to be within this paradigm, the less obvious and more profound those structures became.

So although you cannot take the square root of a goat or the inverse of a goat or visualize what a negative goat would look like, each of those operations have essential and powerful applications in parts of our world that only indirectly have anything at all to do with something called a "goat".

It is one of the most amazing things in the history of math that parts of mathematics that what were once considered completely abstract structures (group theory, noneuclidean geometry, etc.) with no apparent connection with the real world turned out to be precisely what was needed to describe some newly-discovered part of the real world (particle physics, astrophysics, etc.).

  • I have two comments; 1) I agree with all comments on relating to “the number of goats”. Let me give a reference: in the article by Florian Luca and Filip Najman Titled “On the largest prime factor of x2 − 1 “ they first seem to have a result depending on the RH but later they seem to decouple that dependence: …thereby eliminating the apparent dependence of our results on the Generalized Riemann Hypothesis”. I think one should add that there has not been any decoupling over the development of math. regarding “the number of goats”. Jan 6, 2022 at 15:13
  • 2) I am disappointed that no one commented on my main reference to semantics that occurs in Williamson “Absolute provability and the safe knowledge of axioms”: .”However, there could in principle be mathematicians, perhaps non-human ones, who believe A and could not easily have come to believe ~A or any other falsehood in a relevantly similar way. As a by-product of the evolutionary history of their kind, their brains have come to be wired so as strongly to predispose them to accept as obvious any formula which they interpret as we actually interpret A, when the formula is presented to them.” Jan 6, 2022 at 15:19

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