In the Beginning God Created Mathematics

Edit: As Ritwik Priya points out to me, there’s a whole school of philosophy making my argument below. Like I said, I’m no philosophy-literate, but hopefully that makes my sketch a little more… accessible.

John Aziz doesn’t agree. The debate started over this little tweet:

All theory is founded on empirical observation, whether we admit it is or not.

I agree, for the most part. Even the most basic economics, the idea that consumers purchase more goods and services at lower price levels, is behaviorally founded. Our contention is whether “pure mathematics” itself is prior to empirics. I haven’t studied Wittgenstein, Godel, or any philosophy of mathematics. But I do think about these things a lot, and here’s a sketch of my intuitions.

Let’s start with numbers. Historically, it’s important to distinguish between the rationals and reals. The Greeks and other ancients very intuitively understood the distinction between continuous processes (like time) and quanta, like the discrete number of apples Herodotus owns.

Until the advent of real analysis at the turn of the 19th century, continuous variables were very empirically-founded. For example, the amount of water in a swimming pool isn’t discrete – outside of a molecular level which, obviously, the ancients did not understand. So it could be measured as the rational number of cups that sum total of water in the pool approximated.

But counting required no such precepts. Aziz notes that even that models the idea that 2 apples and 2 apples make 4 apples: but this just restates the internal consistence of mathematics and is hence a slight tautology. Further, 2 apples and 2 bottles make 4 (apples or bottles). This empirical foundation is infinitely abstracted until we hit the notion of “objects” or, colloquially, “things”. What is the  connection so abstract a concept has with empirics?

The discrete units of existence is something that is prior to everything else. It is not possible for the human mind to concoct a world in which discrete objects do not exist: at least not one with humans in it. That is sufficiently fundamental to think that philosophy of something discrete is prior.

If there are discrete quantities, a number system must exist, if only in binary. Certainly the idea of binary states must be similarly prior, that is the knowledge whether something is living or dead. Is this empirically-founded? Even in a world of utter sameness – think a man in a black void – there is a sense of the thinking man, and everything else. Life and not-life. Furthermore, the relevance of mathematics is not associated at all by the physical reality of the world around us. Sure, the applications in physics, chemistry, and economics are worked by empirical microfoundations. But what such foundations does math itself have? What empirical reality do axioms of logic follow?

Here’s a test. Can you think of any possible world or set of circumstances in which logic itself does not model what it does on Earth? The application thereof in physics and economics are empirically and/or behaviorally founded. But those are only applied extensions of something inherent.

This brings about a meta-level deconstruction of empirical foundations. If empirical foundations are prior, then knowledge derived thereof can only be accepted through a deductive framework. This assumes the correctness of deduction which, then, cannot be empirically founded at all. Hence the position becomes that either pure logic is prior to everything, or nothing is prior at all.

I assert the latter cannot be true for two reasons. One, it would void any possibility of absolute knowledge. Aziz seems to go with this, claiming “I do think absolute knowledge is impossible… logic and empirics frameworks toward degrees of certainty, though”. But if absolute knowledge is impossible, then we can’t know at all the latter part of that sentence. And if we can’t know that empirics and logic together aid us in accumulation of knowledge, then we can know nothing.

This is a deep philosophical argument, I sense, one which many scholastic and Continental theorists have written treatises on. Regardless, unless one is willing to accept that we all live in a delusion and know nothing, they must take logic as prior to empirics. Remember, there are no caveats to absolution, and the comfort such a position grants is undone by the infinite constraints thereof. Because I think most people would agree we do know some things. That is justified, true, belief on a given event is not impossible.

But there’s another problem with this:

My method: take empirical observation for granted, with caveat that I could be delusional/hallucinating.

We cannot take empirical observation for granted. Or, more precisely, we cannot convince ourselves of the significant associations present in empirical observation without statistical tests. But the theory of statistics, in no irony, is not empirically founded at all, but based on pure mathematics. We convince ourselves that there’s a 95% chance that the association between two events isn’t random.

And if we can’t convince ourselves of that, we cannot even interpret empirical observations at a rigorous level. Forsaking this requirement of “being convinced” would be the death of scientific enquiry as a whole. And yet to be convinced we must take pure math as prior.

As I told him last night (and the reader has undoubtedly figured out by now), I’m way too sober to have this argument in any cogent fashion.

Addendum: Most of this post considered cardinal numbers, but ordinality is equally prior. Is there a circumstance in which n + 1n? If so, what are the empirical foundations altered that brought about this “delusion” of an understanding? More is always more than less. And therefore logic exists with or without the physical world.

1. I’m curious, what do you think about the math and logic in this argument;

‘David’s claim was that with QWERTY, “markets drove the industry prematurely into standardization on the wrong system” …. The onus is not on Paul David to show that QWERTY was inferior to one possible keyboard, Dvorak. The onus is on his critics to show that QWERTY is superior to all other possible keyboards. I calculate that there are 2,658,271,574,788, 448,768,043,625,811,014,615,890,319,638,527,999, 999,999 of these, an admittedly large number (some 1054). So even if someone finally and convincingly proves Dvorak inferior to QWERTY, this only goes part way toward proving QWERTY’s superiority, the critics still have all the other cases before them.’

• Path dependency is obvious, here.

• So, you believe that making assertion is enough to justify its acceptance? That the asserter is under no obligation to provide evidence for its truth?

2. Aziz said:

I am not drunk enough to have this discussion either!

We cannot take empirical observation for granted. Or, more precisely, we cannot convince ourselves of the significant associations present in empirical observation without statistical tests.

My foundation for empiricism is basically just what I see, hear, touch and taste. I’ve been alive and interacting with the physical world for thousands of days, so as a rule of thumb I am willing to assume up to a very high confidence threshold that the real world is real, with the caveat that I could be wrong. But if I am wrong and the real world is either vastly different from what I — and potentially the entire human species — is perceiving, or if the real world does not exist at all, then my observations about the real world are irrelevant. Any conclusions I may have drawn are contingent to whatever I am perceiving, and if what I am perceiving is not the real world, I have no capacity to draw any conclusions about reality.

The historical record suggests our senses have been finely honed over billions of years of evolution, so I am pretty confident that we are either experiencing reality, or at least that we are experiencing something that allows us to have some useful interaction with reality!

Is there a circumstance in which n + 1 < n?

By definition are there not numbers in the infinities and continua that possess these properties? In a truly infinite set there include the numbers JohnAziz and AshokRao, an infinite number of permutations of which will satisfy the property n+1<n. Otherwise, if there are numbers that do not satisfy these properties, it is not a truly infinite set.

• “By definition are there not numbers in the infinities and continua that possess these properties? In a truly infinite set there include the numbers JohnAziz and AshokRao, an infinite number of permutations of which will satisfy the property n+1<n. Otherwise, if there are numbers that do not satisfy these properties, it is not a truly infinite set."

Interesting.. I'll have to think about this one. What I was getting at, of course, was that "n < n +1" is intuitively impossible and hence our notions of ordinality are based on this. That paradoxes are possible within our system of math perhaps increases the argument that abstract math itself is not empirically founded?

Anyway, I think you said it best on Twitter – mathematics is an experiment of the mind.