Meanings of Mathematical Truths

The only way in which we can access an abstract mathematical fact, learn about its truth, is through some physical tools, such that we believe their behavior to be linked to the abstract statement.

In this case, we believe that if 5324 + 2326 = 7650, then adding 5324 apples and 2326 apples gives 7650 apples; and if 5324 + 2326 = 9650, then adding 5324 apples and 2326 apples gives 9650 apples; and if adding 5324 apples and 2326 apples gives 7650 apples, then 5324 + 2326 = 7650.

If instead of normal apples, we have magical apples that don't add, then we probably wouldn't believe that this particular abstract statement is related to the apples, and that observing the apples tells us something about the statement. On the other hand, we'd still need some tool to access the abstract fact in order to reason about it, perhaps a brain or weird-physics-based calculator if no simple objects comply. If such is not available, then we can't answer (or even ask) the question.

The second view is the correct one, but your characterization of it is not quite right:

The second intuition is based on the self-consistence of mathematics: mathematics have its own ways how to decide between truth and falsity, and those ways never defer to the external world; therefore the first interpretation must be false.

It is not that the methods of determining mathematical truth "never defer to the external world"; it is that they do not depend on the external world on the object level. You do not verify that 2+2=4 by observing apples, and then earplugs, and then generalizing; instead, you verify it by writing down -- on paper, or in your mind, or wherever and however, but yes, somewhere in the actual world -- a formal proof within a formal system. You may then observe the behavior of apples to conclude that that formal system is a useful model of (some aspect of) the world -- but that proposition is a distinct, meta-level proposition, and it is not the same as the object-level, within-the-formal-theory proposition that "2+2=4".

Now, back in the days of the Babylonians, Egyptians, etc., before an independent discipline of "mathematics" existed, the first view was correct. Then, statements of arithmetic such as "2+2 = 4" referred directly to abstracted physics: they were generalized propositions about apples and cats and dogs and so on. Subsequently, however, the Greeks came along and invented this new thing called "mathematics". Ever since then, civilization has had a specific discipline where what you do is study the behavior of formal systems, and this discipline is now the appropriate context for the interpretation of statements such as "2+2=4".

"2+2=4" is not a physical claim; a physical claim would be something like "apples behave according to the laws of integer addition as defined in ZFC set theory". If the behavior of apples were to suddenly change, that would not change the truth-value of "2+2=4"; instead what would (or might) happen would be that funding agencies would demand that mathematicians stop studying ZFC set theory so much and start studying some other system that better modeled the new behavior of apples.

That does not mean, however, that "2+2=4" is "independent of the physical world". The dependence simply occurs at a different logical level: the level where the instruments of formal verification (brains, computers, pencils, papers) are considered to be physical objects.

A note on language: "arithmetic" in English is singular. When I started reading your post I thought you were using the plural "arithmetics" to refer to distinct formalizations of arithmetic (there's Peano Arithmetic, and so presumably other people have their own "Arithmetics", or versions of arithmetic); but it soon became apparent that you simply thought the word was plural, analogously to "mathematics". This is not the case.

(Also note that even though "mathematics" is plural in form, it acts grammatically as singular: "mathematics has", not "mathematics have", in your last paragraph.)

Since being introduced to Less Wrong and clarifying that 'truth' is a property of beliefs corresponding to how accurately they let you predict the world, I've separated 'validity' from 'truth'.

The syllogism "All cups are green; Socrates is a cup; therefore Socrates is green" is valid within the standard system of logic, but it doesn't correspond to anything meaningful. But the reason that we view logic as more than a curiosity is that we can use logic and true premises to reach true conclusions. Logic is useful because it produces true beliefs.

Some mathematical statements follow the rules of math; we call them valid, and they would be just as valid in any other universe. Math as a system is useful because (in our universe) we can use mathematical models to arrive at predictively accurate conclusions.

Bringing 'truth' into it is just confusing.

It's because of discussions like these that I wrote this (low-rated) article.

Summary: Mathematical truths can be cashed out as combined claims about 1) the common conception of the rules of how numbers work, and 2) whether the rules imply a particular truth. Together, one's beliefs about mathematical claims AND beliefs about whether real-world systems are ismorphic to their axioms, imply your expectations. If such expectations turn out to be wrong, then either your computation was wrong, or you were wrong that the system's dynamics are isomorphic to that particular axiom system.

I think it's dangerously easy to get lost in contemplating the "true nature" of mathematics. Math gives some very strong subjective impressions about its nature, such as that its truths are eternal and universal. And like any strong subjective impression, this feeling lends itself to the mind projection fallacy. That isn't to say that these impressions are wrong, but that even if and when they're right we tend to trust them for the wrong reasons. And, thus, we don't notice when those impressions really are wrong.

I don't claim to have a complete answer to this conundrum. I do, however, see many key pieces that seem to go a long way to dispelling this confusion.

First, as just an empirical observation, it seems that mathematical objects are reifications. If you watch little kids learning how to add, they go through a predictable sequence of development. First they count out objects, put them together, and then count the whole collection:

Here's one, two, three, four, five. And here's one, two three. That's, um, one, two three, four, five, six, seven, eight!

After a while of doing this - and "a while" can be a surprisingly long time - they realize that they can compress the first quantity by jumping to the end:

Here's one, two, three, four, five. And here's one, two, three. So that one is five, and then six, seven, eight!

After doing this for a while, they start to think about the process of counting "one, two, three, four, five" in terms of the final state ("five"). This lets them manipulate the process as an object.

Ah, but once this happens, this triggers the parietal cortex to apply the idea of object permanence to "five". Suddenly there's this sense that "five" is there even when the child doesn't see it. And behold, the eternal entity 5 as a mathematical object is born in the child's mind.

We're so used to thinking this way that we don't really see it in ourselves anymore. But it's still there and shows up in oddities in how we think about even basic math. For instance, what does it mean to add 5 and 3? You put 5 and 3 together... somehow... and suddenly an 8 pops out of nowhere. What happened to 5 and 3? If we pause and think about it, we can make sense of it with visualizations or other mental tricks, but there's this slight-of-hand we do to ourselves before we pause to think about that process in which we treat 5 and 3 as objects but don't think to ask how they combine to create the object 8. They just "merge" somehow, and a whole entity - not just a composite, but something thought of as an object - appears.

What really seems to be going on is that we have a built-in capacity from birth to subitize quantities less than 4, and then we build on those in order to perform rituals of ordered synchronization of movement and speech. This is why children find it so important to actually touch the objects they're counting as they're speaking the magic words "one, two, three...". This is also the best current explanation I'm aware of for the seemingly unrelated symptoms of Gerstmann syndrome: when people can no longer distinguish between their fingers, they don't have the proprioceptive bind to the verbal counting ritual that's needed to understand numbers greater than three. After a while, the parietal cortex provides a shortcut to dealing with familiar processes by treating the end-state as an object that can stand in for having done the process.

So it might very well be that mathematical truths are not so much "encoded in reality" as that our descriptions of these truths are embodied characterizations of the world. It might be that they seem eternal as an accidental side-effect of our using our parietal cortices to simplify computations. They're seemingly universal because the universe we're capable of experiencing is the one in which our bodies work - and notice that in places where our bodies do not work normally (e.g. dreams), mathematics doesn't seem to work quite so well either.

I'm taking the time to point this out because it's way too easy to waste tremendous amounts of time wondering about where mathematics "is". Even if there's some objective essence of math that is somehow lurking within and guiding the physical world unseen, the question remains as to how we, with our physical brains and bodies, can come to understand those truths. We can't understand some semi-Platonic Idea in its raw form; we have to use the material tools from which we are constructed in order to model those ideas. Therefore, the only mathematics we can ever possibly know about is that which is governed by the structure of our minds. This makes the origin of mathematics really a question of psychology, not philosophy - which is thankful because psychology has the blessing of being empirical!

Wouldn't it then follow that there is a largest number bounded by the ability of the universe to store information? That this number is decadent: entropy is constantly eroding its value? And that values like pi have terminating decimals since that same limit would apply to storing those as well?

There are elementary statements about arithmetic that don't have obvious real-world equivalents, or where the obvious physical implication is false.

Suppose I tell you, for instance, that there's a real number that when multiplied by itself equals 2. There's a natural geometric interpretation of this in terms of ratios; it says that given two segments, A and B, there's a segment C such that A:C is the same ratio as C:B.

But it could very easily be the case that this is not true in the actual geometry of the universe around us; measurements are noisy, space is curved, space isn't quite continuous, etc. But the math is unobjectionable and exact.

So that suggests that at least some elementary math can be, and has to be, justified in some way other than "true for the universe around us."

“Pure logical thinking cannot yield us any knowledge of the empirical world; all knowledge of reality starts from experience and ends in it. Propositions arrived at by pure logical means are completely empty of reality.” –Albert Einstein

I don't agree with Al here, but it's a nice quote I wanted to share.

So let's say you're in a world where putting 5324 and 2326 apples together produces 157 apples (due to weird physics). In that world, what would a calculator (which is itself a physical object), or a brain, for that matter, that produces "5324 + 2326 = 7650", look like?

Could you have the same sort of expansive mathematical framework (that we have today), that would be represented (computed) by a fairly "simple" object such as a calculator/computer? And how much (if at all) more complex (in terms of the physical representation/structure of various sorts of mathematical operations) would such a computer necessarily have to be?

Certainly it is possible to argue about what the meaning of "5324 + 2326 = 7650" is, but this is not where the really interesting weaknesses of arithmetic are to be found (or at least, to be looked for). The statements that should excite the most controversy all begin with "for all numbers n..." or "there is a number n..." That is where your interpretations 1. and 2. really start to diverge.

For instance, "for all integers n, there is an integer m such that applying the Goodstein operation m times to n yields 0." This certainly can be given a formal syntactical meaning in Peano arithmetic, and a formal proof as well (but not in PA!), so we are good as far as interpretation 2 goes.

But the only physical meaning we can try to give it is that a certain process (fighting hydras) will stop eventually. Since there's no guarantee it will stop before the universe ends, this claim can't be tested. Then is there a fact of the matter about this claim?