Supernatural Math

All quarks are tiny, but only some quarks are strange.

My (perhaps naive) take on this proceeds from the assumptions that

1. we can define mathematical operations any way we like (provided we build up consistently from axioms etc.)
2. the mathematical operations so defined may provide us with more or less useful models of the world.

Parts of this discussion seem to me as though they're conflating these two issues.

We can axiomatize things such that 2+2=4 is, in a particular sense, 'true'. We could do this independently of whether or not it were generally the case that if we took 2 bananas and 2 more bananas and stuck them together we consistently ended up with 4 bananas. If, whenever we took 2 bananas and stuck them together with 2 more bananas, we ended up with 3 bananas, 2+2=4 would still be 'true' in the abstract sense that it proceeds naturally from the axioms, but it would no longer be a useful model to apply to the real world situation of taking-2-bananas-and-then-taking-2-more-bananas-and-sticking-them-together. It would, in that particular sense, be 'false'.

(dfn) \a. 0 + a = a
(dfn) \ab. Sb + a = b + Sa
\a. SS0 + a = S0 + Sa = 0 + SSa = SSa
SS0 + SS0 = SSS0

 N = the set of strings of the form S*0
 0 = the string "0"
 S = the function that prepends "S" to its argument

Now, notwithstanding the above, does 2+2 really equal 4, independent of any human thoughts about it? The answer is we can't know.

We can and do know that, though we can't be absolutely certain. There are different shades of certainty, not at all blurred together in the uniform "we can't know".

I would like to agree with this, but still feel that the difference between necessary and contingent truth needs to be accounted for somehow. I can, like Eliezer, vaguely imagine a world in which 2+2=3 in Peano arithmetic, but I don't think I could precisely model it in the way I could precisely model a world with different laws of physics. I might be able to, by a huge amount of Dark-Lords-of-the-Matrix-type meddling, precisely model a world in which some things whose closest equivalents in our world would be implementations of Peano arithmetic output 2+2=3, but (a) that just seems too much like cheating to mean much (b) the world's fundamental laws would still have to be consistent with 2+2=4 – if it had physics like ours, 2 joules and 2 more joules would still make 4 joules; if it were a cellular automaton, 2 living cells and 2 more living cells bordering a cell would be 4 living cells.

(See also: Math is Subjunctively Objective.)

The disappointing truth is that bananas are quarks, and by amazing good fortune, the properties of everyday macroscopic objects are sufficiently related to those of other physical phenomena that a few lucky humans can just barely manage to crudely adapt their banana-counting brain hardware to work in those other domains.

Amazing good fortune implies amazing coincidence, which implies strong disconfirmation relative to any alternatives that don't need such a coincidence.

Philosophy of math is hard, and I'm not sure we can get very far trying to figure it out on LW.

I would say

• 4 = 3 + 1 by definition
• 3 = 2 + 1 by definition
• 2 = 1 + 1 by definition
• 4 = (2+1) + 1 by substitution
• 4 = 2 + (1 + 1) by association
• 4 = 2 + 2 by substitution

2, 2, 4, +, and = are things that we define. Given those definitions, 2 + 2 = 4. Problem solved. Don't want to define them that way? Then you're talking about something else.

Mathematical statements are kinds of representations.

What exactly is being represented? What exactly is the status of the representation itself, and how does it relate to what is represented?

We don't have a solid theoretical understanding of these matters, and I until we do, I reckon we're going to continue to argue about it, and probably not very productively. At the moment we have trouble finding ways to describe what mean when we talk about the various aspects of representations.

Yeah, this is getting repetitive now. At least you haven't cleared up my "confusion". When I check to see if 2+2=4 I don't count any objects at all. I just look at what the terms mean and the sentence is self-evidently true. If 2+2=3 then you have redefined at least one of the terms.

I think either math is merely a language we have to explain and describe events in the external world (in which case mathematical operations have definitions and usage rules just like language). Either this language is just really helpful or its principles are actually embedded in reality. I'm not sure that question can ever be answered. None of this requires us to believe in supernatural math or Platonic heaven.

When I say that "2 + 2 = 4" is a fact independent of the physical universe, I mean that it is the sort of information that does not tell us (directly) which universe we are in, what the laws of physics are. (Though it can be used in a line of reasoning evaluating propositions that are about our particular universe.) I do not mean that it is a supernatural force, or a mystical energy field controlling our destinies.

And the whole purpose of making that point was to frame the question of how well the physical systems in our brains represent these logical facts, and how we can evaluate our uncertainty about the answer to that question.

Personally, I find talking about words and language to be more interesting than numbers. The word "banana" is easier to attack than the number "2". When you say:

I can put 2 bananas on a table, then put down 2 more bananas, and count out 4 bananas.

This implies that the perception of "2 + 2 = 4" is built on your perceptions of "bananas". In my opinion, this has nothing to do with math and everything to do with language. "2 + 2 = 4" is relatively useless unless everyone agrees 2 + 2 = 4. The same is true of "banana". The word is useless unless enough people agree about what it means. To ask if 2 + 2 = 4 is "true" is linguistically irrelevant.

This is not to say that the question is irrelevant in all contexts, but when talking about it, asking about its truth may be a misnomer.

Abstract philosophy is problematic. Please indulge me for a few paragraphs as I work my way up from the factory floor.

How do you gain power over the world? You could do stuff and remember how it works out, but experience is a dear teacher. The holy grail is to be able to predict. Instead of doing A, B, and C and remembering for next time that C worked best, predict the consequences of doing A, B, and C, and then do only C.

We can hand crank simple models of reality, yielding predictions that turn out to be wrong because the models were too simple. We respond with more elaborate models. We cannot rigorously work out what they predict, because they are not simple enough, but waving our hands and talking fast we reconcile them after the fact to the way things turned out.

That serves for both craftsmanship and politics. The craftsman knows the right answer from the traditions of the craft, so he can always guide the derivation of predictions with extra, informal ingredients from craft-lore so that they give the right answers. The politician has only to persuade. Having persuaded that his theory of society explains what is wrong he will win election. Having explained away why his theory didn't work first time he may yet win a second election, but seldom a third.

The engineer faces a stiffer test. He must scale up the design for his bridge, his engine, or his chemical plant by a factor of 10 or 100. Some projects are only plausible on an expensive scale and engineering is then a theory-based leap into the unknown. His backers will not fund the trial and error development of a craft tradition; his equations must predict accurately.

This is where mathematics comes in. We have simple, unrealistic models we can solve, and complicated, realistic models that we cannot solve. Mathematics pushes the boundary, always trying to find ways to solve models a little more complicated than current understanding permits. The essence of this enterprise is rigour. Once you say "obviously", "who can doubt", "as every-one knows", "as every-one agrees", the point is lost.

The value of mathematics lies in its refractory nature. Once the physicist or the engineer has proposed that a piece of mathematics models nature, the model, if humanly soluble, gives him a single prediction that is either right or wrong. If right, he can test it some more and may find himself in possession of the holy grail, a tool for prediction.

Notice the sharp contrast between this and the ordinary notions of theory and prediction. Ordinarily there is a degree of hand waving involved in getting from theory to prediction. We believe our theory because it "predicts" things that have already happened, because knowing the outcome in advance we can guide our hands to paper over the gaps in the theory. But when it comes to predicting things that haven't happened yet, we are careful not to be held accountable. We know, even as we dodge fully conscious acknowledgment, that those who were careless about this often get their come-uppance.

I have painted a picture of where mathematics fits into practical reality. Mathematics offers practical men a selection of soluble models that are not so simple as they would be restricted to if confined within their personal limits. Physicists select a model and an interpretation of the model which jointly agree with reality. Engineers use this to design artifacts which then work.

What is the cause of this miracle? Why do the predictions come true? I don't really know, but it seems clear that the refractory nature of mathematics plays an essential role. A mathematician sets up a formal system, such as Peano arithmetic and, like Baron Frankenstein, he finds that the creature escapes from his control. Perhaps the formal system turns out to be useful for modeling and prediction in real life, perhaps it doesn't, but the crucial point seems to be that the mathematician cannot fudge it after the fact. If it doesn't apply to reality the mathematician can invent/discover something else, or the physicist can try applying it to a different phenomenon, but there is no avoiding the fact that the interpretation of the formal system didn't agree with reality. That is somehow linked to the fact that formal systems that do have interpretations that agree with reality are rare and precious and give predictions that come true.

I'm unconvinced by the top level posts ontology because it seems to have no place for the applied mathematician's failed formal systems. These formal models horrify and fascinate us for the same reason that Frankenstein's monster does: they have a mind of their own. The applied mathematician was attempting to model the real world and created his formal system to do so, but it defied him, and followed its own rules, to its creator's sorrow. When we respond to this stubborn independence of mind by expelling the perpetrator from really, its not ontology, its pique.

I argued, elsewhere in this thread, that you could not evolve a mathematical understanding that 2+2=3, because “2+2=4” is mandatory, independent of any physical reality or observation, any context whatsoever. This was not whole-heartedly accepted so let me again try to defend the inviolability and universality of mathematics.

I assert my case with three definitions: a definition of mathematics, a definition of logically follows, and a definition of consistent.

Mathematics is: you define things and then determine what logically follows from those things.

As fallible humans, we may not know if something really logically follows. But we just define what we mean by "logically follow": logically following means follows necessarily, mandatorily, independently of everything else.

If an axiomatic system is consistent it means that if x logically follows, then “not x” does not logically follow.

Maybe you don't like these definitions. But that is what they are. To the extent to which I can speak for mathematics, if you change these definitions, you’re not talking about mathematics any more.

Unlike science, the definitions precede the observation. Here, imagine: humans in the savannas, dressed in their animal skins, shake their clubs at the empirical world and the unreliability of the senses. They wield the power by defining exactly what they mean with no heed whatsoever to what actually is. This is the difference between math and science, this is why mathematics is trustworthy even when all else -- all empirical sense -- might fail.

Consider Peano arithmetic. We understand that 2+2=4 logically follows from the axioms in Peano arithmetic in our current universe, context C1. If "2+2 is not equal to 4" is deduced by the Peano axioms in some other context C2, then at least one of the following must be true:

(a) 2+2=4 does not logically follow in context C2, so 2+2=4 did not really logically follow from the Peano axioms -- by definition of logically follow, as it is context independent,

(b) 2+2=4 does logically follow in context C2, so the Peano arithmetic is not consistent -- by definition of consistent

Thus, mathematical truth is independent of context, including the physical world. This is why mathematicians love saying, "it is true by definition". It is the precise source of the omnipotence of mathematics. That's it. Everything mathematically true is true by definition. It’s our terms, our game. Unlike the empirical world where we don’t get to define anything, where the existence of things precede our observation.

Regarding the suggested arbitrariness of “2+2=4”: I disagree that a mathematical understanding could evolve that “2+2=3”: mathematics is not a random guess that turns out to be correct. Mathematical understanding is more than knowing/guessing that “2+2=4”: it is the understanding of why this is true and why it has to be that way. In some sense, it has to be that way independently of any physical reality/observation. This is why mathematics is really quite different from science.

Edit: The rest of this comment was removed because the subsequent points weren't worth arguing if the above isn't taken for granted.

Asking "Now, notwithstanding the above, does 2+2 really equal 4, independent of any human thoughts about it?" requires human thoughts. There would still be four bananas without anyone to count them because I'm here imagining that world in my human thoughts. It's like trying to imagine what death feels like. You fail by trying. Just like the Game (muhahaha!).

Hey, we no longer need to grant mathematical statements special transcendent status now that we have computers and the axiomatic method. Math theorems predict outputs of specified computer programs given specified inputs, period, end of story. Thus it all folds back neatly into experimental science and directly observable facts. And if some proof can't be thus interpreted - can't be single-stepped through an axiomatic checker even in principle - then it's not yet math.

Of course this leaves unanswered the question of why math not only works internally, but also describes our world so well. Maybe we'll learn the answer in time.