Meanings of Mathematical Truths
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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 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.
This has not been my experience in mathematics. There's lots and lots of theorems that I know where I don't have any physical counterpart for the claim. Cantor's diagonalization argument, for instance. The irrationality of the square root of two. The fundamental theorem of arithmetic, aka the unique prime factorization theorem.
For natural-number arithmetic, geometry, and calculus, it's often easy to find close physical analogs. For the rest of math, I don't see it.
Are you using "physical tools" or "linked to" in a much broader sense than I understand?
This has not been my experience in mathematics. There's lots and lots of theorems that I know where I don't have any physical counterpart for the claim.
You have your physical brain (I did mention this option in the comment). The "counterpart" can follow a syntactic description of the semantic fact, for example, which is how the axiomatic method normally works (but there are other options, the important thing is that you (correctly) believe the tool to be controlled by the abstract fact, as you believe your mathematical reasoning to be controlled by abstract facts).
There was a thought related to this that I've been trying to untangle. Assuming the brain is computable, which seems to me to be quite a safe assumption, doesn't that imply that all of mathematics should be able to be framed computationally? It seems obvious to me that there should be some algorithm that could create mathematics by organizing a stream of input data in some fashion. After all, there is a (presumably) computable subsystem of Earth that does that, namely the mathematics community.
That said, in what light are we to consider "incomputable" mathematics? It seems like this is drawing a meaningless distinction given the computable universe hypothesis. Perhaps it can be thought of as a sort of counterfactual reasoning? This certainly seems to be the case with, say, accelerated Turing machines, the counterfactual being "what if there were no universal speed cap?"
Apologies if I'm not making sense, I've been awake for quite a while.
The position that makes the most sense to me is that uncomputable mathematics exist in a platonic sense, we just happen to inhabit a part of the ultimate ensemble that appears computable, but (1) there's a chance that our universe is actually uncomputable (or alternatively, we're "simultaneously" in computable and uncomputable parts of the multiverse), and (2) we can use computable machinery to learn some facts about uncomputable math.
The alternative of saying that only computable math exists seems to imply that all of the work done on understanding uncomputable math (such as Turing degrees and the like) is merely a game of manipulating meaningless symbols. In other words, those mathematicians who think they are reasoning about uncomputable math are not actually doing anything meaningful. That is hard to swallow.
What do you mean by computable or uncomputable math? In one sense, all math is computable (you can enumerate all valid proofs in a formal system, even if it talks about something mind-bogglingly huge like Grothendieck universes). In another sense, all math is uncomputable (you can't enumerate all true facts about the integers, never mind more complex stuff). Is there some well-defined third way of cashing out the concept?
I don't have an answer for that, since I'm not proposing to make a distinction based on computable vs uncomputable math. But I think Juergen Schmidhuber has defended some version of "only computable math exists" on the everything-list, so you can try searching the archives there.
It seems like your conception is isomorphic to mine to the extent that the ultimate ensemble can be considered as a set of counterfactual (with respect to local physical law) universes. You'll see below that I brought up Zeno machines, and noted that they seem to follow this paradigm. Does this seem accurate?
The alternative of saying that only computable math exists seems to imply that all of the work done on understanding uncomputable math (such as Turing degrees and the like) is merely a game of manipulating meaningless symbols. In other words, those mathematicians who think they are reasoning about uncomputable math are not actually doing anything meaningful. That is hard to swallow.
Even the most abstract symbol manipulation can turn out to be functionally equivalent to a more concrete mode of reasoning and hence be quite useful. It is still an exploration of some sort of algorithmic reasoning, and coming from the view that mathematics is something like the art of predicting the outcome of various manipulations of structured data, such abstract games are not totally invalid.
My intuition is currently along the lines of thinking that uncomputable mathematics is disguised computable math, since the manipulation and creation of the system itself is computable (again, given the assumption that the human mind is computable), and so it is perfectly valid but misinterpreted in some sense.
I feel that I should say that my primary motivation for this line of thought is a desire to think about how a machine might create mathematics from scratch. I've started reading Simon Colton's work, it seems quite interesting. Perhaps you could offer some insight into this sort of issue?
Talking or reasoning about the uncomputable isn't the same as "computing" the uncomputable. The first may very well be computable while the second obviously isn't.
Talking or reasoning about the uncomputable isn't the same as "computing" the uncomputable.
Obviously, but I didn't mean to imply that it was. My question is what we are actually reasoning about when we are reasoning about the uncomputable. Apologies if this wasn't clear.
It seems to me that oracle computations and accelerated Turing machines, seem to be related to counterfactual reasoning in the sense that they suppose things that are not the case such as Galilean velocity addition or that we can obtain the result of a non-halting computation in finite time and use that to compute further results.
You may have heard of Chaitin's constant, an uncomputable number. It's easy to define the number, but it's impossible for machines or people to compute it: For every person, there is a number N such that that person will never be able to figure out what the Nth digit of Chaitin's constant is. I hope this helps.
Well, as I understand it Chaitin numbers are computable with a Zeno machine or an oracle machine with a halting oracle as they are in the arithmetic hierarchy. This still seems to follow my previous line of thinking; that reasoning about incomputable mathematics is itself computable (of course), but (more interestingly) can be framed as reasoning with counterfactuals about physical law (what if I could break the speed of light, in this case).
I would also note that the reals would be countable if it weren't for incomputable numbers. Any number that can be approximated to an arbitrary degree by an algorithm can of course be represented by that algorithm, and thus the set of all such numbers is equivalent to a subset of Turing machines, which are countable.
All this considered, I'm not really sure how Chaitin's constant (at least, in particular, beyond the fact that it is incomputable) bears on the issue.
"Controlled by abstract fact" as in Controlling Constant Programs idea?
Since this notion of our brains being calculators of - and thereby providing evidence about - certain abstract mathematical facts seems transplanted from Eliezer's metaethics, I wonder if there are any important differences between these two ideas, i.e. between trying to answer "5324 + 2326 = ?" and "is X really the right thing to do?".
In other words, are moral questions just a subset of math living in a particularly complex formal system (our "moral frame of reference") or are they a beast of a different-but-similar-enough-to-become-confused kind? Is the apparent similarity just a consequence of both "using" the same (metaphorical) math/truth-discovering brain module?
"Controlled by abstract fact" as in Controlling Constant Programs idea?
That post gives a toy model, but the objects of acausal control can be more general: they are not necessarily natural numbers (computed by some program), they can be other kinds of mathematical structures; their definitions don't have to come in the form of programs that compute them, or syntactic axioms; we might have no formal definitions that are themselves understood, so that we can only use the definitions, without knowing how they work; and we might even have no ability to refer to the exact object of control, and instead work with an approximation.
Also, consider that even when you talk of "explicit" tools for accessing abstract facts, such as apples or calculators or theorems written of a sheet of paper, these tools are still immensely complicated physical objects, and what you mean by saying that they are simple is that you understand how they are related to simple abstract descriptions, which you understand. So the only difference between understanding what a "real number" is "directly", and pointing to an axiomatic definition, is that you are comfortable with intuitive understanding of "axiomatic definition", but it too is an abstract idea that you could further describe using another level of axiomatic definition. And in fact considering syntactic tools as mathematical structures led to generalizing from finite logical statements to infinite ones, see infinitary logic.
This kind of thinking (that I actually quite like) presupposes the existence of abstract mathematical facts, but I'm at loss figuring out in what part of the territory are they stored, and if this is a wrong question to ask, what precisely does it make it so.
Does postulating their existence buy us anything else than TDT's nice decision procedure?
This kind of thinking (that I actually quite like) presupposes the existence of abstract mathematical facts
I can infer stuff about mathematical facts, and some of the facts are morally relevant. Whether they "exist" is of unclear relevance.
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.)
A note on language: ...
Corrected, thanks.
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.
It doesn't seem to me that the levels can be easily separated. Even if you verify that "2+2=4" using your brain, pencil or computer, I can always object that you have not verified the proposition itself, but that the system of brain, pencil or the specific computer program behaves according to the laws of integer addition as defined in ZFC set theory. Now the computer program was probably written to simulate the said laws of integer addition, while the apples were added before mathematics was formalised, but that's a remark of merely historical interest.
Even if you verify that "2+2=4" using your brain, pencil or computer, I can always object that you have not verified the proposition itself, but that the system of brain, pencil or the specific computer program behaves according to the laws of integer addition as defined in ZFC set theory.
Actually, no: if you wanted to make an "objection" like this, you'd have to pass to the next level up. You'd have to say something like "we've just verified that the brain/pencil/computer behaves according to the laws of proof-verification as defined in [some formalization of metamathematics]". This is not the same -- and, crucially, is not incompatible with also having verified "2+2=4". The two notions of verification are different: verifying "2+2 = 4" has to do specifically with producing a proof, while verifying that "X behaves according to laws Y" involves empirically observing X in general, in a way that directly depends on what Y is.
(To help illustrate the distinction, note for example that to verify that "2+2 =4", you only need to produce a single proof, in a single medium; whereas verifying that "X behaves according to Y" would usually require observing a multitude of instances.)
So this "objection" wouldn't support the first view, which holds that "2+2=4" is verified in the latter way, by observing apples (etc.), and denies the necessity of a formal proof intervening for verification to take place. (To the holder of the first view, a formal proof would just be a kind of instrument for predicting the behavior of apples and the like, and is not necessary to the meaning of the proposition.)
Of course I couldn't say "we've just verified that the brain/pencil/computer behaves according to the laws of proof-verification as defined in [some formalization of metamathematics]" after producing a proof of "2+2=4". To be able to say that, I would need to produce many different proofs and ascertain that they are correct from an independent source.
The idea of my "objection" was that I have to trust the medium of verification that it behaves according to the laws of proof-verification as defined in some formalization of metamathematics in order to use it for proving the theorem. But the medium itself is always part of the physical world, and there is no fundamental difference between proving the theorem using apples and proving it by drawing squiggles on a paper.
But the medium itself is always part of the physical world, and there is no fundamental difference between proving the theorem using apples and proving it by drawing squiggles on a paper.
The second statement does not follow from the first.
There is a difference between empirically observing that combining two apples with two apples yields four apples, and observing a sequence of squiggles on paper that constitutes a formal proof that 2+2=4. Yes, I'll grant you that the difference isn't written on the atoms making up the apples or the paper; rather it's a matter of semantics, i.e. how these observations are interpreted by human minds. (You could presumably write out a formal proof using apples too -- and then it would be just as different from the observation about combining pairs of apples as the squiggles on paper are.)
There is only one "level" of reality, but our model of reality can be organized into distinct levels. In particular, we use some parts of the physical world to model others; and when we do so, we have to be careful to distinguish discourse about the model from discourse within the model (i.e not to "confuse the map and the territory").
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.
The distinction between true and valid in this sense seems useful. In practice, my first interpretation says that "5324+2326=7650" is true, while the second says it is valid.
Logic is useful because it produces true beliefs.
I'd rather say it conserves true beliefs that were put into the system at the start, but these were, in turn, produced inductively.
[math statements] would be just as valid in any other universe
I've often heard this bit of conventional wisdom but I'm not totally convinced it's actually true. How would we even know?
Well, what if in some other universe every process isomorphic to a statement "2 + 2" concludes that it equals "3" instead of "4" - would this mean that the abstract fact "2 + 2 = 4" is false/invalid in that universe?
As far as I can see, this boils down to a question about where are these abstract mathematical facts stored, or perhaps, what controls these facts if not the deep physical laws of the universe that contains the calculators that try to discern these facts...
Perhaps to say "valid in any other universe" is a language mismatch, since validity doesn't refer to universes, only to the rules of the system.
Well, what if in some other universe every process isomorphic to a statement "2 + 2" concludes that it equals "3" instead of "4" - would this mean that the abstract fact "2 + 2 = 4" is false/invalid in that universe?
If it concluded 3 instead of 4, it would not be isomorphic to our "2+2". Both systems of mathematics (in our and the other universes) have to be isomorphic as a whole to enable translation between them.
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?
The joke response would be to let 'n' represent that number.
n+1.
Somewhat related (not really at all, but your post reminded me of it) is Scott Aaronson's digression on "who can name the biggest number": http://www.scottaaronson.com/writings/bignumbers.html
Reality is far too interesting to allow such a limit on itself.
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."
A thought (and I might just be being crazy here): if we think of mathematics as a specific case of analogical reasoning a la Hofstadter or Gentner it seems that we could think of mathematics as layered analogies.
More concretely; geometry, arithmetic and algebra have obvious physical analogues and seem to have been derived by generalizing some sorts of action protocols. Basic algebra allows one to generalize about which transactions are beneficial, geometry allows one to generalize about relative sizes of things and, well, a lot of more complicated sorts of things like architecture.
Mathematics can be thought of as a sort of protocol logic. We use protocols to reason about protocols, and so we can devise a protocol logic for types of protocol logics. This seems to be what many more abstract areas of mathematics really are. They reason analogically from other domains of mathematics, borrowing similar tricks, and apply them to thinking about other parts of mathematics. In this way mathematics acts as its own subject matter and builds on itself recursively.
Take mathematical logic (from an historical perspective) for example. Mathematical logicians look at what mathematicians actually do, they take the black box “doing math” and devise a rule set that captures it; they search for a representative protocol. N logicians could devise N hypotheses and see where the hypotheses diverge from the black box (‘inconsistent!’ one may shout, ‘underpowered, cannot prove this known result!’ yet another might say). Like any other endeavor, we cannot expect that we have hit the correct hypothesis, and indeed new set theories and logics are still being toyed with today.
Just take Ross Brady's work on universal logic. He devised an alternative logic in which to build a set theory that allowed for an unrestricted axiom of comprehension, nearly one hundred years after Russell's paradox.
It seems to me that ultimately a mathematical logician should desire to obtain a mechanical understanding of mathematics; the task of building a machine that can create new mathematics (as opposed to simple searching the space of known mathematics, or simpler still the space of known analytic functions) requires this understanding.
I expect a machine to take its input data, and arrange expected changes into some sort of logical protocols so that it can compute counterfactuals. I expect that recurrent protocols of this sort should be cached and consolidated by some process, which seems very hard to actually define algorithmically.
This actually makes quite a bit of sense (to me, of course) in terms of outcomes, it would explain why mathematics is so applicable; it is all about analogical reasoning and reasoning about certain types of protocols.
So, am I crazy? Did that spiel make any damned sense?
Just take Ross Brady's work on universal logic. He devised an alternative logic in which to build a set theory that allowed for an unrestricted axiom of comprehension, nearly one hundred years after Russell's paradox.
I don't know the book, but here's a review. Unrestricted comprehension, at the expense of restricted logic, which is an inevitable tradeoff ever since Russell torpedoed Frege's system. It's like one of those sliding-block puzzles. However you slide the blocks around, there's always a hole, and I don't see much philosophical significance in where the hole gets shifted to.
Yes, I've read that review and you're correct. Probably a bad example. Anyway, my general point was that mathematics is built from concrete subject matter, and mathematics itself, being a neurological phenomenon, is as concrete a subject matter as any other. We take examples from our daily comings and goings and look at the logic (in the colloquial sense) of them to devise mathematics. The activity of doing mathematics itself is one part of those comings and goings, and this seems to me to be the source of many of the seemingly intractable abstractions that make ideas like Platonism so appealing.
Does that seem correct to you?
You would find Lakoff and Nuñez's Where Mathematics Comes From interesting. Their thesis is along these lines. I read the first chapter and I got a lot out of it.
It seems that you prefer the second interpretation of mathematical statements. But the first one, that which refers to physical world, isn't completely unattractive either.
For example I can use multiplication only for calculating areas of rectangles; if so, I would probably hold that "5x3" means "area of a rectangle whose sides measure 5 and 3", and "there is a real number which multiplied by itself equals two" means "there is a square of area 2". I would say that 5 times 3 equals 15 if and only if it was true for all real rectangles which I have met, and after seeing a counter-example I would abandon the belief that "5x3=15".
Or I can mean "if I add together five groups of three apples each, I would find fifteen objects". If this was my understanding of what the proposition means, a counter-example consisting of a 5x3 rectangle with area of 27 wouldn't persuade me to abandon the abstract belief, because multiplication is not inherently about rectangles.
In the real world people are familiar with both uses of multiplication and many others, so any counter-example in one area is likely to be perceived as evidence that multiplication isn't a good model of that process, rather than that we have to update our understanding of multiplication.
Math is exact in the sense that once the rules of inference are given there is no freedom but to follow them, and unobjectionable in the sense that it is futile to dispute the axioms. Any axiomatic system is like that. But most mathematical models we actually used have one advantage over that: they are not arbitrary, but rather designed to be useful to describe plethora of different real-world situations. Removing a single application of multiplication doesn't shatter the abstract truth, but if you consecutively realised that multiplication does capture neither calculating areas nor putting groups of objects together nor any other physical process, what content would remain in propositions like "5x3=15"? They would be meaningless strings produced by an arbitrary prescription.
For example I can use multiplication only for calculating areas of rectangles; if so, I would probably hold that "5x3" means "area of a rectangle whose sides measure 5 and 3", and "there is a real number which multiplied by itself equals two" means "there is a square of area 2".
Or I can mean "if I add together five groups of three apples each, I would find fifteen objects".
As a quick aside, I think these two interpretations are actually the same thing in disguise. Areas as measurements have units attached to the numbers. Specifically, the units are squares whose sides measure one "unit length". So when you're looking at a rectangle that measures 5x3, you're noting that there are five groups of three squares (or three groups of five squares, depending on how you want to interpret the roles of the factors). Otherwise it's hard to see why the area would be a result of multiplying the lengths of the sides.
I think perhaps a better example would be the difference between partitive and quotative division. Partitive ("equal-sharing") says "I have X things to divide equally between N groups. How many things does each group get?" Quotative ("measurement" or "repeated subtraction") says "I have X things, and I want to make sure that each group gets N of those things. How many groups will there be?" This is the source of not a small amount of confusion for children who are taught only the partitive interpretation and are given a jumble of partitive and quotative division word problems. It's not immediately obvious why these two different ideas would result in the same numerical computation; it's actually a result of the commutativity of multiplication and the fact that division is inverse multiplication. So there's a deep structure here that's invisible even to participants that still guides their activities and understanding.
Math is exact in the sense that once the rules of inference are given there is no freedom but to follow them, and unobjectionable in the sense that it is futile to dispute the axioms. Any axiomatic system is like that.
I agree that axiomatic systems are like that, but I don't think the essence of math is axiomatic. That's one method by which people explore mathematics. But there are others, and they dominate at least as much as the axiomatic method.
For instance, Walter Rudin's book Real and Complex Analysis goes through a marvelously clean and well-organized axiomatic-style exposé of measure theory and Lebesgue integration. But I remember struggling with several of my classmates while going through that class trying to make sense of what is "really going on". If math were just axiomatic, there wouldn't be anything left to ask once we had recognized that the proofs really do prove the theorems in question. But there's still a sense of there being something left to understand, and it certainly seems to go beyond matters of classification.
What finally made it all "click" for me was Henri Lebesgue's own description of his integral. I can't seem to find the original quote, but in short he provided an analogy of being a shopkeeper counting your revenue at the end of the day. One way, akin to the Riemann integral, is to count the money in the order in which it was received and add it up as you go. The second, akin to Lebesgue integration, is to sort the money by value - $1 bills, $5 bills, etc. - and then count how many are in each pile (i.e. the measure of the piles). This suddenly made everything we were doing make tremendously more sense to me; for instance, I could see how the proofs were conceived, even though my insight didn't actually change anything about how I perceived the axiomatic logic of the proofs.
The fact that some people saw this without Lebesgue's analogy is beside the point. The point is that there's an extra something that seems to need to be added in order to feel like the material is understood.
I'm going to some lengths to point this out because the idea of math as perfect and axiomatic just isn't the mathematics that humans practice or know. It can look that way, but the truth seems to be more complicated than that.
I think perhaps a better example would be the difference between partitive and quotative division.
Maybe even easier example is the commutativity of multiplication itself. It is not a priori clear that 5 group of 3 objects each are the same as 3 groups of 5 objects each. When I was a child I was feeling confused why addition and multiplication are commutative while exponentiation isn't.
I'm going to some lengths to point this out because the idea of math as perfect and axiomatic just isn't the mathematics that humans practice or know. It can look that way, but the truth seems to be more complicated than that.
Yes, we have powerful (sometimes astoundingly powerful, as in case of Ramanujan) intuitions built in our brains that allow us to do high-level operations. Mathematics is practically never done on the lowest level of formal manipulation. There is certainly large difference between mathematics as an axiomatic system and the art of mathematics as a human endeavour - if there weren't, mathematicians were replaced by machines long ago. But that doesn't seem much relevant to the question of truth of mathematical theorems. Whatever intuitive thought had lead to its discovery, people will agree that it is valid iff there is a formal proof.
Maybe even easier example is the commutativity of multiplication itself.
That's a good point! I avoided that example because there's a pretty easy and convincing "proof" of the commutativity of multiplication, namely that turning a rectangle on its side doesn't change how many things constitute it So, it doesn't matter whether you count how many are in each row and then count how many rows there are, or if you do that with columns instead.
I think it's terribly sad that they don't encourage children to notice that or something like it. But there are a lot of things about education I find terribly sad and that I'm doing my damnest to fix.
But that doesn't seem much relevant to the question of truth of mathematical theorems. Whatever intuitive thought had lead to its discovery, people will agree that it is valid iff there is a formal proof.
Agreed, though there's no objective definition of what constitutes a "formal proof". Despite what it might seem like from the outside, there's no one axiomatic system and deductive set of rules to which all subfields of mathematics pay homage.
In the actual physical world we live in, statements like "there is a square of area two" might not be exactly true. There certainly is no evidence that they are exactly true. (Whereas, in contrast, two plus two really gives you exactly four bananas.)
It's certainly true that much human-developed math was developed to serve practical purposes, and therefore does accurately model aspects of the real world. But the math isn't made less true because the physical world deviates slightly from it; likewise the math that's less tied to the physical world isn't less true. There are lots and lots of theorems that are interesting, and even useful, but that don't seem to have much to do with anything physical. (E.g., number theory, or abstract algebra.)
We should maybe taboo the word "true", since for a mathematical theorem to be true is not exactly the same as for an interpreted sentence about the physical world. How would you then formulate the sentence "the math that's less tied to the physical world isn't less true"?
In this case, I mean something like "if you start off with consistent and true beliefs, adding more true beliefs won't lead to self contradiction." I can define self-contradiction formally, as asserting both a statement and its formal negation.
This may seem slightly circular, but I think it's still a useful definition that captures what I want. I also think some circularity is useful to capture what we mean by an axiomatic system.
“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?
All I have postulated about that strange world was the assumption about addition of apples. I have no consistent model how that world works besides that and thus I can't answer your question.
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?
Then is there a fact of the matter about this claim?
There is no guarantee that any statement of arithmetics would have a sensible direct physical interpretation. However many propositions of form "for all n, P(n)" have such an interpretation. For instance, starting from the original example we can replace it by "for all n, putting one apple to a group of n apples produces a group of n+1 apples" which may be physically checked for different values of n.
In some sense the two interpretations get closer to each other concerning quantified propositions. Having "for all n, P(n)" not only one can check the outcome of the corresponding physical process (having in mind the first interpretation) but it also seems reasonable to check "P(n)" formally for each n, using a subsystem of arithmetics which doesn't include quantifiers. If we found a single n allowing to prove "not P(n)", arithmetics would be inconsistent. Since we don't know for sure that arithmetics is consistent, to test it in such a way is not completely useless.