This blog post is originally my comment on https://coelsblog.wordpress.com/2014/09/23/the-roots-of-empiricism-humes-fork-and-the-divide-between-knowledge-by-observation-and-by-reason/

Hi Coel,

In response to the following: <quote>This process tests the physics, but it also tests the maths and the logic. If the maths and the logic did not hold in the real world then the prediction would turn out false. By that process we arrive at correct maths and logic (where by “correct” I mean best modelling the real world). … … It follows from the above that axioms of maths and logic and laws of physics all have the same epistemological status. They were all adopted as concepts that model real-world behaviour. There is no other source for any of them (or indeed any other type of knowledge) other than from our contact with the real world, and thus from our experience of how the world behaves. Thus there is no basis for asserting any big epistemological distinction between them. </quote>

Essentially, your point is that you see no meaningful epistemological distinction between Logic/Maths and Physics. But I do think a distinction can be made. I’d describe the relationship between Logic/Maths and Physics this way: Logic/Maths is like the set of all Lego bricks ever made (this set of Lego bricks being the product of Logicians/Mathematicians). Physicists use Lego bricks to build models that correspond to the empirically observable world. When a model is found to incorrectly correspond to the real world, at least one of the following is true:

I: The Lego bricks used were not appropriate for the desired model.

II: The arrangement of bricks were not appropriate.

III: There does not exist Lego bricks that allow for the desired model to be built.

The Physicist works on I and II, while the Logician/Mathematician works on III.

Hence, one can say that the wrong bricks were used (I), or that the bricks were arranged wrongly (II). When III is true, the set of all Lego bricks ever made is said to be insufficient, which may motivate Mathematicians to develop new areas in Mathematics (i.e. create new kinds of Lego bricks) that can fulfill the needs of I and II.

A key point is that Mathematics need not work in the real world. There are areas of Mathematics that have yet to find any real-world application (https://www.quora.com/Is-there-a-modern-branch-or-area-of-pure-mathematics-with-no-presently-known-practical-application). Just because a large body of Mathematics has real-world applications doesn’t prove that _all_ Mathematics will have real-world applications. It could simply mean that the bulk of Mathematics were developed with real-world applications in mind.

Also, I’d add that Logic on a fundamental level could be thought of as what has been hard-wired into our brains (through evolution, etc.; the point is that Logic is thought of as a fixed feature of the human mind, and in this sense “analytic”) that enables us to perform logical operations on logical values. Mathematics then consists of all possible permutations of logical operations and logical values that human minds are capable of conceiving. Our logical minds (the Lego company) are continually inventing new ways of thinking, in the form of Logic/Maths (Lego bricks). Physicists then make use of the available Mathematics to try to model the real world.

Furthermore, regarding the following:

<quote>Quine’s famous paper on the synthetic/analytic divide starts by considering the statement that “No unmarried man is married”, which he declares to be “logically true”.

But it this genuinely a priori knowledge, true by logic alone, and entirely distinct from empiricism? In essence it is the basic logical law of non-contradiction, that something cannot be both itself and not-itself. But consider the following:

(1) No unmarried man is married.

(2) No dead cat is alive.

(3) No spin-up electron is spin-down.

</quote>

The truth of the statement “no unmarried man is married” relies on the assumption that “unmarried” is the logical negation of “married”. If in Physics a cat could be simultaneously “dead” and “alive”, then “the cat is dead” would not be the negation of “the cat is alive”, so that you have incorrectly applied the law of non-contradiction to a case that violates a crucial underlying assumption. I.e., by assigning “the cat is dead” a logical value of FALSE, and “the cat is alive” the logical value of TRUE, you are making the assumption that the two statements are logical negations (hence mutually exclusive). When Physics shows that both statements could simultaneously be true, then you should stop assuming that the two statements are logical negations (stop assuming that they are mutually exclusive), so that you are no longer allowed to use the law of non-contradiction to produce the statement “no dead cat is alive”.

Also, it’s worth saying that as long as there are meaningful differences between the fields of Logic/Maths and Physics, then a distinction can reasonably be made, so that some people may rightly choose to first affirm an analytic-synthetic distinction, then seek to define what they mean by that. This could even mean that people define “Mathematics” differently! (For example, my examples/explanations above provide one possible definition of Mathematics that allows for an analytic-synthetic distinction.) As a matter of fact, there is no universally accepted definition of “Mathematics” …

Finally, consider the “continuum hypothesis” in set theory. <quote>The continuum hypothesis was not the first statement shown to be independent of ZFC. An immediate consequence of Gödel’s incompleteness theorem, which was published in 1931, is that there is a formal statement (one for each appropriate Gödel numbering scheme) expressing the consistency of ZFC that is independent of ZFC, assuming that ZFC is consistent. The continuum hypothesis and the axiom of choice were among the first mathematical statements shown to be independent of ZF set theory. These proofs of independence were not completed until Paul Cohen developed forcing in the 1960s. They all rely on the assumption that ZF is consistent. These proofs are called proofs of relative consistency (see Forcing (mathematics)). … … In a related vein, Saharon Shelah wrote that he does “not agree with the pure Platonic view that the interesting problems in set theory can be decided, that we just have to discover the additional axiom. My mental picture is that we have many possible set theories, all conforming to ZFC.” (Shelah 2003).

</quote from https://en.wikipedia.org/wiki/Continuum_hypothesis#Independence_from_ZFC>

Simply put, to many Mathematicians, it is perfectly fine to have one version of “Mathematics” in which a particular axiom is true, and another version in which that same axiom is assumed to be false. This logically means that at most only one version of “Mathematics” will perfectly correspond to the real world. By your definition the other versions would be “wrong”, but then the fact remains that the other versions of “Mathematics” are possible in the sense that human minds are able to conceive such alternative versions. To declare that those versions of “Mathematics” that do not correspond perfectly with the real world should not be called “Mathematics”, would simply be a matter of choice.

Of course, there is the open question of whether the human mind will somehow continue to evolve and somehow adopt a more advanced logical system that does not allow such existence of multiple versions of “Mathematics”. But for now, there continues to exist very real and meaningful differences between Logic/Mathematics and Physics.

Having read your follow-up post (https://coelsblog.wordpress.com/2014/11/18/the-unity-of-maths-and-physics-revisited/), I have something to add:

You justify your “radical empiricism” by asserting that the mind is ultimately a product of experience (evolution, etc.), as seen in the following quote written in one of your comments: [quote]My counter would be to claim that nothing is known entirely independently of experience, and that all knowledge ultimately derives from experience. [/quote]

But the statement that “the mind is a product of real-world experience” doesn’t automatically imply that “all logically-consistent thought conceived by the mind must correspond to the real world”. The mind could easily conceive logically consistent but unrealistic worlds, i.e. fairy tales. We can imagine all kinds of crazy things that completely contradict real-world physics, and describe them by mathematical formulae, even though in practice nobody has any incentive to do so.