Mathematical investigations as parallel to other sciences

There is a common sentiment that mathematical truths are distinct in nature from truths of other kind (in the sciences or otherwise) in that mathematical truth is eternal and unchanging. I would like to pose an alternative view to this considers mathematical truth as essentially being on the same footing as other kinds of truth.

To take an example of physics, a common example is that the laws of physics are not eternal and in fact we have gone through several revisions of them from Newtonian physics to general relativity and quantum mechanics. This is in stark contrast to Pythagoras’ theorem that is as true today as when it was first proven over two thousand years ago.

Through this lens, it certainly seems as though mathematical truths are different in nature from other kinds of truths. I would argue that there are two confusions here.

First, in physics and other sciences, the true facts are not the laws of physics but rather the body of observations that the laws of physics aim to explain. In other words, the true statements of physics are the observations we make about the world: The sun rose in the east today, yesterday and so on as far back as we can remember or that balls have fallen to the surface of the earth with a certain measurable speed for as long as we have measured them.

In this context, the laws of physics are a way of compressing these observations and moreover, a method for generating predictions about such observations. In this sense, our body of facts can never be wrong. If the sun fails to rise tomorrow, it will still be true that sun rose in the east today! While it is possible to imagine that a prediction of our model (laws of physics) will turn out to be wrong, it is not possible to imagine that an observation already made turns out to be wrong.

Second, coming to mathematics, the true facts are facts about the natural numbers (or other similar structures). Taking Fermat’s Last Theorem as an example, we observe that no matter how many examples we look at, we cannot get two cubes of natural numbers to add up to a third distinct cube of a natural number. This is an experimental fact about the world in that given our procedures for adding and multiplying numbers, it states the impossibility of achieving something.

Now Fermat’s Last Theorem is a statement about a model (the natural numbers and a set of axioms that describe it) that encodes our procedures for adding and multiplying numbers. Even given Fermat’s Last Theorem, it is still possible for us to imagine that I routinely set about carrying out the procedures for addition and multiplication and arrive at two non zero numbers whose cubes are also a cube.

Assuming that the proof of Fermat’s Last theorem is valid, this simply means that the axioms we choose to encode our procedures for addition and multiplication in (the Peano axioms) were the wrong ones to choose! Our model for the natural numbers turned out to be incorrect in describing the physical processes of addition and multiplication the same way Newton’s Laws turned out to be incorrect in describing our physical world.

In general, a set of axioms is designed to capture certain aspects of interesting “mathematical objects” arising from patterns we observe in the world but it is always possible that our axioms are an incomplete or incorrect description. This has in fact happened in the past. Sufficiently unclear foundations of analysis, set theory and algebraic geometry led to mistakes and paradoxes in mathematics during the time of Weierstrass, Cantor and the Italian school of Algebraic Geometry.

I would argue that these instances are similar to the revolutions in physics in relativity and quantum mechanics.  The underlying objects/patterns we study did not really undergo a change but the axioms we use to describe them did. This is exactly similar to how the underlying observations we try to explain in physics haven’t changed but our axioms/model for the explanations have.

Continued: Ray 2.1, Ray 2.2

Asvin G.

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