Related to: How to Convince Me That 2 + 2 = 3
This started as a reply to this thread, but it would have been offtopic and I think the subject is important enough for a top-level post, as there's apparently still significant confusion about it.
How do we know that two and two make four? We have two possible sources of knowledge on the subject. Note that both happen to be entirely physical systems that run on the same merely ordinary entropy that makes car engines go.
First, evolution. Animals whose subitizing apparatus output 2+2=3 were selected out.
Second, personal observation; that is, operation of our sense organs. I can put 2 bananas on a table, then put down 2 more bananas, and count out 4 bananas; my schoolteachers told me 2+2 is 4; I can type 2+2 into a calculator and get 4; etc.
Now, notwithstanding the above, does 2+2 really equal 4, independent of any human thoughts about it? This way lies madness. If there is some kind of pure essence of math that never physically impinges upon the stuff inside our heads (or, worse, exists "outside the physical universe"), there's no sensible way we can know about it. It's a dragon in the garage.
The fact that our faculty for counting bananas can also be used to make predictions about, say, the behavior of quarks is extremely surprising to our savannah-adapted brains. After all, bananas are ordinary things we can hold in our hands and eat, and quarks are tiny and strange and definitely not ordinary at all. So, of course, the obvious thing that comes to mind to explain this is a supernatural force. How else could such dissimilar things be governed by the same laws?
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. No supernatural math required.
All quarks are tiny, but only some quarks are strange.
I think it was an intentional pun by OP
My (perhaps naive) take on this proceeds from the assumptions that
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'.
I'm not so sure of that. If putting 2 S's next to 2 S's got us 3 S's, we could prove 2+2=3 in PA with the usual definition of addition:
Depending on the universe's other rules for putting n things next to m things, we might also be able to derive "2+2=4". In this case, we would decide that PA is inconsistent! Whatever the other rules are, this already shows that the "abstract" conclusions we can draw from a set of axioms depend on the way symbol manipulation works in our world.
I don't think this is really a problem for your argument, but it's an interesting complication. Many (most?) physical facts seem to have no influence on the symbolic manipulations we can use to derive them. For instance, symbolically computing a series for pi doesn't seem to involve any actual circles the way shuffling symbols to add 2 and 2 in PA involves putting SS next to SS.
Nitpick but important: we couldn't actually prove it, just produce a convincing (in that world) false proof (that is actually a proof of a theorem in some other, inconsistent, system with slightly different inference rules).
We could figure out that our symbolic manipulation is inconsistent with the axioms based on the quirk you consider.
There are more axioms needed to define Peano Arithmetic. Taking axioms 7 and 8 from Wikipedia, translated into your notation:
\a. Sa != 0 \ab. Sa = Sb -> a = b
(I also use the symmetry and transitivity of equality.)
Note, from the axioms you stated:
\a. S0 + a = 0 + Sa = Sa
So, axiom 8 can be restated as:
\ab. S0 + a = S0 + b -> a = b
So, starting with your result:
SSS0 = SS0 + SS0 = S0 + SSS0
S0 + SS0 = SSS0 = S0 + SSS0
So, by the restatement of Axiom 8:
SS0 = SSS0
And then using the original form of Axiom 8 twice:
S0 = SS0 0 = S0
We have a contradiction of Axiom 7.
Thus, it is proven that our symbol manipulation does not follow the Peano Axioms. This does not invalidate the Peano Axioms. It simply means that a given physical system does not follow them. Of course, it would be difficult for people living in an alternate universe where symbols really behaved this way to notice the distinction. And they likely would not have to, if all objects behaved that way; they would figure out some other math to represent their situation. And at some point, mathematicians in their ivory towers would develop this weird math that is really hard to write down and has no known application in the real world (both properties bringing great joy to these academicians).
You're right. PA is still consistent (i.e. has a model) even if
fails to be one because of the way string concatenation works. There's nothing mathematically special about theories that can use physical objects as a model.
(Minor quibble: the definition of addition isn't an axiom. It's just a relation definable in the first-order theory of arithmetic.)
The naive take is correct. I'm arguing against assertions like "math exists outside the universe" or is somehow not a physical phenomenon. When we think about axioms that don't correspond to real objects, it's still happening inside a brain right here in our universe, and we know which axioms correspond to real objects the same way we know other facts, not through some supernatural math-sense.
Consider the function f on the real numbers such that for any real number x, f(x) is 1 if x is irrational and f(x) is 0 otherwise.
We have an axiomatic measure theory, which tells us that the integral over values of x from 0 to 1 of f(x) is 1.
I don't believe there is any physical system, in our brains or otherwise, that represents this function or its integral. There is a physical system in our brains that represents the rules of logical inference, and it is through this physical system that we can form beliefs about mathematical facts that do not model any known physical system. That is, we can make statements of the form: if there were a physical system that satisfied this proposition, it would also satisfy this other proposition, regardless of whether that physical system actually exists.
The facts that we call mathematical and logical truths can be represented in any universe capable of representing the rules of inference (and having sufficient memory and processing power). In this sense, they are independent of our particular physical universe.
Not that I disagree with your conclusion (or agree – mostly I'm just confused), but:
Including the representation in your computer, or your brain, of the phrase "the function f on the real numbers such that for any real number x, f(x) is 1 if x is irrational and f(x) is 0 otherwise"?
Ah, I should clarify that point, it is confusing as I wrote it.
I meant that there is no physical domain over which some point wise property varies discontinuously as a function of whether the point, in some measure, has a rational or irrational distance from some reference, and that the only physical systems that in any sense represent the function do so indirectly, by representing propositions about it (such as the examples you gave).
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.)
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
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.
substitution, association, etc work? Amazing!
Stuff is consistent, we never have a true contradiction, never have A AND NOT A? incredible!
That's the sense in which I'd say math is "real"... the reality is tied to the fact that stuff sure seems to actually be consistent. I can't really conceive of what it would mean for that not to be so, but still, it seems like there's something I'm confused about here.
Yeah, I'm beginning to think this discussion could use a domain expert, not to tell us the answer but to clarify the issues. Anyone know someone who works in Philosophy of Math?
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.
No, you run the sentence "2+2=4" through a processor and note that the resulting evaluation is "correct".
You then predict that you will produce the same evaluation again in the future, and that others will get the same result when they try it.
What would you do if your processor returned "correct" some of the time and "false" the rest?
Greg Egan's short story "Luminous" is something the people reading this thread should take a look at.
I thought that being rational -- whatever it means on Less Wrong -- departed from extreme epistemological nihilism. Suppose I were to meet you in nihilism land. How can we tell the difference between beliefs in the flying spaghetti monster and whether 2+2 can be 3 or 4? That's why the nihilistic position is "silly". While Jack, out there in making-sense-land, says that he knows 2+2=4.
I'm sorry, but I don't think that you understand.
The fact that you reach a conclusion does not force the universe to conform to your conclusion. It may be perfectly obvious to you that 2+2=4, just as it was perfectly obvious to the Greeks that Euclid's Fifth Axiom was true or that there was universal time was clear to Newton. Nevertheless, they were wrong, as you may be wrong.
jBelief is not conviction. Conviction is not knowledge. Knowledge isn't truth.
That seems at best trivially true, in that "knowledge is truth" seems to be committing a category mistake. A common-enough epistemological position is that knowledge is about true propositions.
More specifically, many in epistemology will define knowledge as "justified true belief". So by this way of thinking, if "S knows X" is true, then X is true.
Nobody defines knowledge simply as justified true belief anymore. Everybody needs a workaround for Gettier cases.
Well, while I agree with your sentiment, surely your statement is technically false. Indeed, one way to get around Gettier cases is to simply make "justified" a more difficult credential to obtain (not that I think that's a good solution).
Also, many philosophers no longer think definitions need to specify necessary or sufficient conditions, and so would happily claim 'justified true belief' is a 'good enough' definition of knowledge.
I'm probably overgeneralizing from the professors I've had. Your point is well taken.
Do you believe in truth, in general, at all? If so, how do humans experience truth?
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:
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.
While this might seem overly pedantic (hardly something to complain about in this discussion), I'd like to point out that this definition only matches the usual one if you also accept the law of non-contradiction. More precisely, a system is consistent if it does not contain a contradiction.
Also, your definitions of Mathematics and "logically follows" don't seem very... good. Did you make them up?
EDIT: changed "excluded middle" to "non-contradiction". duh.
ETA: Yeah, my point was stupid. Nevermind.
What we mean by a contradiction is the truth of a statement of the form "x and not x", or, in Bynerma's terms, that there is a statement x such that both x and "not x" logically follow (from the axiomatic system). (Things that we (correctly) intuitively recognize as contradictions allow us to derive a statement of that form.) So, although Bynerma used slightly unusual language, she correctly expressed the content of the usual definition.
The Law of the Excluded Middle is not needed here. It says that statements of the form "x or not x" are true (provable even, according to Wikipedia).
Absolutely correct - note edit
I see what I was doing here. The law of excluded middle is equivalent to the law of non-contradiction. Probably why I had the two confused. Example in sentential logic:
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.
You appear to be arguing for a meta-logic that governs the logic or math that can exist. I don't see any reason why this is necessary. All we know about math is that our math, derived from observations (evolutionary or personal) of the universe, describes the universe. If the universe worked a different way, such that 2+2=3 (and whatever followed from that, in that scheme), you'd end up arguing that 2+2=3 independently of physical reality there, too, right?
Wrong, these are different objects, (universe1 :: 2+2) and (universe2 :: 2+2).
But even saying that implies that there's some meta framework from which we can consider both. From inside our reality, there's no reason to presume that we can know what physical laws, logic, math, or anything holds in another reality (or even that such a thing as an "other reality" exists). I switched to using "reality", by the way, to make it clear that I'm not using this in the way that scientists sometimes talk about "other universes", which, if we could know anything about, would necessarily be part of our reality.
Do you hold that there is a possible world in which 2+2 does not = 4? Just trying to translate your position to my vocabulary.
No, I'm not holding that there actually is such a world, only that there would be no reason to apply our reality's rules to such a world. My real point is that the logical follows in actual historical fact from the physical, rather than being some sort of special knowledge that can be deduced without reference to anything physical.
Really, our beliefs about the logical follow from the physical. Don't confuse the map with the territory.
Heh. This is precisely the question, isn't it? Are logic and mathematics in the territory somewhere or are they the language of the map?
The way I see it is that logic is a territory, my beliefs about logic form a corresponding map, and that map is useful for constructing maps of other territories (and the accuracy of those maps is evidence of the accuracy of the logic map).
If nothing else this is a really well phrased statement of position. Maybe I'm just committing the philosopher's fallacy (deciding things are necessary because I'm not creative enough to think up alternatives) but I really just can't see what it would mean for there to be a world in which A didn't = A, in which the middle wasn't excluded, in which triangles are round, etc. What criteria are you using to decide on one view over the other?
Until the time of writing this comment, I did believe that there would be a meta-logic that governs the math or logic that "can exist". However, I concede that such a statement requires proof and I do not have enough (read: any) background in logic to know if such a proof is forthcoming.
This is the question: if there were two physical universes realizing two distinct sets of mathematical ideas such that at least one mathematical idea is realized in one set and not the other, then would mathematics still provide a meta-framework for both sets of mathematical ideas?
More succinctly: Given any two models, can you always find a meta-model that includes them both? Or maybe it can be proven that this is unprovable within a given model?
A logician's input on this would be helpful.
I think this is a confused question. The answer is yes, but when I tell you what the meta-model is, I don't think your underlying curiosity will be satisfied.
Given 2 models A and B, we can construct a model C. The set of objects in C will be the union of the set of elements of the form (A, a) for all objects a in the model A, and the set of elements of the form (B, b) for all objects b in the model B. Then every proposition about A can can correspond to a proposition about C, particularly about its objects of the form (A, a), by simple substitution of (A, a) for a; and similarly propositions about B correspond to propositions about the objects of the form (B, b). This is, of course, a trivial combination in which the parts that correspond to the original models do not interact with each other at all, hence my belief that you will not be satisfied.
Perhaps a better question to ask would be: is there a meta framework that distinguishes some mathematical systems as "good" and others as "bad"? I think the answer to this question is the criteria that a mathematical system be self-consistent, that is, it does not produce contradictions. Of course, this criteria is not always possible to verify.
Yes, the question was confused. I got distracted thinking about stuff I don't know about (asking "how" instead of "whether").
You don't need "meta-logic", whatever that might be, to know that 2+2=3 cannot be consistent with Peano arithmetic.
Here’s the “proof”. We know that 2+2=4 in “our” Peano Arithmetic. Suppose that 2+2=3 in “another” Peano Arithmetic in an alternate reality. Then the two Peano Arithemetics are actually different because they have a different set of "trues". When we say that 2+2=4 in “our” Peano Arithmetic, we mean “our” PA and not the other. Whatever distinguishes the two arithemetics can be incorporated in our PA as an axiom – indeed it was included implicitly in what we meant by PA all along even if we lacked the imagination to explicitly identify it.
But still, just to be clear, with a meta-logic or not, you can't have 2+2=3 be consistent with Peano arithmetic in a different physical reality.
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.
Aargh! Every time I come across this argument, I am frustrated that people don't see how this 'problem' is resolved. Even Einstein is said to have remarked on the 'mystery' of why the world is knowable.
We ditch the 'explanations' that don't describe our world well, and keep those that do. That's why our models end up looking like the world. When new data arrives that isn't compatible with those models, eventually we end up discarding them and creating new ones.
There is never any guarantee that some phenomenon we come across won't be beyond our ability to understand. There is never any guarantee that any model we possess is an accurate one, no matter how useful it's been in the past or how well it accounts for known data.
There is no mystery as to how we can know the world. We don't.
Keeping "models that work" won't help in a world of chaos, and it's a useless characterization of intelligence. You fail to properly argue a position on the problem of induction.
This is just silly.
There is no "problem of induction" because induction doesn't do the things that argument requires it to do.
There's doubly no "problem of induction" because there's no contrast between deduction and induction. Deduction is a subset of induction. The former can do nothing that the latter cannot.
Frankly, I don't believe you really grasp what a "world of chaos" would look like.
No, it's simply a truth you haven't developed enough to grasp yet. The beginning of wisdom is recognizing that you know nothing, in the sense that people often use 'know'.
Thanks! I stand 50% corrected. Yes, we keep those models that work. But math seems an unreasonably effective model even after accounting for the selection effect. Why did conic sections turn out useful for describing planetary orbits 2000 years later, and why did Hilbert spaces turn out useful for quantum mechanics 10 years later?
That misses the point. Conic sections are useless for how many things? Likewise for Hilbert spaces. Likewise for all of mathematics. A mathematical construct is useful for the things it is useful for, and useless for everything else.
Mathematics isn't a model. (Well, it is, but not in the sense that you mean it.) It's what we use to build models out of, what makes them possible.
If a branch of mathematics exists, and someone finds a way to use it to describe a set of relationships they find in the world, we call that branch 'useful'. If its behavior doesn't match the relationships we're interested in studying, we ignore it. And if it was needed, but doesn't exist yet, we never realize it.
Thanks, I stand 100% corrected.
Do you have a citation for this analysis, or did you make it up? (or non-excluded middle)