The role of mathematical truths
Related to: Math is subjunctively objective, How to convince me that 2+2=3
Elaboration of points I made in these comments: first, second
TL;DR 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. This cashing-out keeps them purely about the physical world and eliminates the need to appeal to an immaterial realm, as some mathematicians feel a need to.
Background: “I am quite confident that the statement 2 + 3 = 5 is true; I am far less confident of what it means for a mathematical statement to be true.”—Eliezer Yudkowsky
This is the problem I will address here: how should a rationalist regard the status of mathematical truths? In doing so, I will present a unifying approach that, I contend, elegantly solves the following related problems:
- Eliminating the need for a non-physical, non-observable “Platonic” math realm.
- The issue of whether “math was true/existed even when people weren’t around”.
- Cashing out the meaning of isolated claims like “2+2=4”.
- The issue of whether mathematical truths and math itself should count as being discovered or invented.
- Whether mathematical reasoning alone can tell you things about the universe.
- Showing what it would take to convince a rationalist that “2+2=3”.
- How the words in math statements can be wrong.
This is an ambitious project, given the amount of effort spent, by very intelligent people, to prove one position or another regarding the status of math, so I could very well be in over my head here. However, I believe that you will agree with my approach, based on standard rationalist desiderata.
Here’s the resolution, in short: For a mathematical truth (like 2+2=4) to have any meaning at all, it must be decomposable into two interpersonally verifiable claims about the physical world:
1) a claim about whether, generally speaking, people’s models of “how numbers work” make certain assumptions
2) a claim about whether those assumptions logically imply the mathematical truth (2+2=4)
(Note that this discussion avoids the more narrowly-constructed class of mathematical claims that take the form of saying that some admittedly arbitrary set of assumptions entails a certain implication, which decompose into only 2) above. This discussion instead focuses instead on the status of the more common belief that “2+2=4”, that is, without specifying some precondition or assumption set.)
So for a mathematical statement to be true, it simply needs to be the case that both 1) and 2) hold. You could therefore refute such a statement either by saying, “that doesn’t match what people mean by numbers [or that particular operation]”, thus refuting #1; or by saying that the statement just doesn’t follow from applying the rules that people commonly take as the rules of numbers, thus refuting #2. (The latter means finding a flaw in steps of the proof somewhere after the givens.)
Therefore, a person claiming that 2+2=5 is either using a process we don’t recognize as any part of math or our terminology for numbers (violating #1) or made an error in calculations (violating #2). Recognition of this error is thus revealed physically: either by noticing the general opinions of people on what numbers are, or by noticing whether the carrying out of the rules (either in the mind or some medium isomorphic to the rules) has a certain result. It follows that math does not require some non-physical realm. To the extent that people feel otherwise, it is a species of the mind-projection fallacy, in which #1 and #2 are truncated to simply “2+2=4”, and that lone Platonic claim is believed to be in the territory.
The next issue to consider is what to make of claims that “math has always existed (or been true), even when people weren’t around to perform it”. It would instead be more accurate to make the following claims:
3) The universe has always adhered to regularities that are concisely describable in what we now know as math (though it’s counterfactual as nobody would necessarily be around to do the describing).
4) It has always been the case that if you set up some physical system isomorphic to some mathematical operation, performed the corresponding physical operation, and re-interpreted it by the same isomorphism, the interpretation would match that which the rules of math give (though again counterfactual, as there’s no one to be observing or setting up such a system).
This, and nothing else, is the sense in which “math was around when people weren’t”—and it uses only physical reality, not immaterial Platonic realms.
Is math discovered or invented? This is more of a definitional dispute, but under my approach, we can say a few things. Math was invented by humans to represent things usefully and help find solutions. Its human use, given previous non-use, makes it invented. This does not contradict the previous paragraphs, which accept mathematical claims insofar as they are counterfactual claims about what would have gone on had you observed the universe before humans were around. (And note that we find math so very useful in describing the universe, that it’s hard to think what other descriptions we could be using.) It is no different than other “beliefs in the implied invisible” where a claim that can’t be directly verified falls out as an implication of the most parsimonious explanation for phenomena that can be directly verified.
Can “a priori” mathematical reasoning, by itself, tell you true things about the universe? No, it cannot, for any result always needs the additional empirical verification that phenomenon X actually behaves isomorphically to a particular mathematical structure (see figure below). This is a critical point that is often missed due to the obviousness of the assumptions that the isomorphism holds.
What evidence can convince a rationalist that 2+2=3? On this question, my account largely agrees with what Eliezer Yudkowsky said here, but with some caveats. He describes a scenario in which, basically, the rules for countable objects start operating in such a way that combining two and two of them would yield three of them.
But there are important nuances to make clear. For one thing, it is not just the objects’ behavior (2 earplugs combined with 2 earplugs yielding 3 earplugs) that changes his opinion, but his keeping the belief that these kinds of objects adhere to the rules of integer math. Note that many of the philosophical errors in quantum mechanics stemmed from the ungrounded assumption that electrons had to obey the rules of integers, under which (given additional reasonable assumptions) they can’t be in two places at the same time.
Also, for his exposition to help provide insight, it would need to use something less obvious than 2+2=3′s falsity. If you instead talk in terms of much harder arithmetic, like 5,896 x 5,273 = 31,089,508, then it’s not as obvious what the answer is, and therefore it’s not so obvious how many units of real-world objects you should expect in an isomorphic real-world scenario.
Keep in mind that your math-related expectations are jointly determined by the belief that a phenomenon behaves isomorphically to some kind of math operation, and the beliefs regarding the results of these operations. Either one of these can be rejected independently. Given the more difficult arithmetic above, you can see why you might change your mind about the latter. For the former, you merely need notice that for that particular phenomenon, integer math (say) lacks an isomorphism to it. The causal diagram works like this:
Hypothetical universes with different math. My account also handles the dilemma, beloved among philosophers, about whether there could be universes where 2+2 actually equals 6. Such scenarios are harder than one might think. For if our math could still describe the natural laws of such a universe, then a description would rely on a ruleset that implies 2+2=4. This would render questionable the claim that 2+2 has been made to non-trivially equal 6. It would reduce the philosopher’s dilemma into “I’ve hypothesized a scenario in which there’s a different symbol for 4”.
I believe my account is also robust against mere relabeling. If someone speaks of a math where 2+2=6, but it turns out that its entire corpus of theorems is isomorphic to regular math, then they haven’t actually proposed different truths; their “new” math can be explained away as using different symbols, and having the same relationship to reality except with a minor difference in the isomorphism in applying it to observations.
Conclusion: Math represents a particularly tempting case of map-territory confusion. People who normally favor naturalistic hypotheses and make such distinctions tend to grant math a special status that is not justified by the evidence. It is a tool that is useful for compressing descriptions of the universe, and for which humans have a common understanding and terminology, but no more an intrinsic part of nature than its usefulness in compressing physical laws causes it to be.
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I think philosophy of math discussion on LW would probably be better if it ever referred to the thinking that has been done by professional philosophers of math. Or maybe that thinking is worthless enough that it’s worth restarting from scratch (e.g. if they don’t have our necessary background concepts), but then that should be noted and defended from time to time.
Right. I think a lot of the comments here have been speaking towards that request in some way.
Lakatos struck me as quite impressive in that field, but I am precisely the opposite of widely-read in that area.
Never heard of it and illiterate? Those are some mighty fine lucky key-presses.
I’m not yet seeing how this way of thinking about math contradicts platonism. It seems to leave unaddressed the questions that platonism purports to answer. That is, your account here is essentially independent of the ontological status of mathematical objects, operations, etc.
For example, you wrote:
This seems to leave unanswered the classical kinds of questions that gave rise to platonism, such as:
What kind of thing is this “isomorphism” of which you speak? Where does it live? It doesn’t seem to be a physical thing itself, so what is it? And what about the mathematical operation that is isomorphic to the physical system? Is the mathematical operation another physical system? If so, which specific physical system is it? Is it, for example, some particular physical electronic calculator? If so, which one is it? It seems implausible that any particular physical calculator has the honor of being the mathematical operation of addition, say. But if the mathematical operation is not a particular physical system, what is it? Is it an isomorphism class of physical systems? But this gets back to the problem of what, physically, an isomorphism is, and adds the problem of what, physically, a class of physical things is. One might try to identify the class with the mereological sum of its elements, but there are well-known problems with this approach. And what about the “rules of math”? Which inhabitant of the physical universe is a “rule of math”? And so on.
All of the questions above are certainly confused to some degree. But I’m not yet seeing that you’ve made much progress on dissolving them.
[ETA: I don’t mean to say that anything you said was wrong. It certainly seems to me to be the most promising way to approach the subject.]
If your “native hardware” can’t understand and implement isomorphisms, you’re no better off by positing an immaterial realm in which isomorphisms exist. At some point, you can no longer define your functionality in terms of a sub-specification. But this just means that an agent must have some level that acts automatically, without further reflection (cf Created Already in Motion), not that a being’s ontology is insufficient on account of failing to posit some superset realm for the concepts it implicitly uses.
In case it wasn’t clear from the previous paragraph, I look at this from the perspective of creating an artificial being that can do everything I can. If the deficiencies of the way of handling of math I’ve described (including failure to specify to ever greater precision the “rules of math”) don’t correspond to some kind of failure mode of the artificial being, then I have to ask if it really is a deficiency.
Let me know if that was responsive to your questions.
I took your post to be an account of the meaning of mathematical claims, or of what it is that they assert about the world. In particular, you said that you would eliminate “the need for a non-physical, non-observable ‘Platonic’ math realm.” (Emphasis added.)
I take your comment here to be describing the sense of “need” that you were using in your OP: To need the concept of a platonic math realm means to need it to build an artificial being that can do math.
But I don’t think that many platonists would disagree. I’ve never heard anyone claim that calculator engineers need to learn mathematical platonism, or indeed any philosophy of mathematics at all, to do their job. Certainly none would say that we have to somehow program the calculator to be platonist for it to do its job. They wouldn’t even say that a human mathematician has to be a platonist to succeed at mathematics.
The problem with platonism isn’t that it keeps anyone from being able to build calculators. I’d say that the problem with platonism is that it convinces people that they can know about some things (ideal geometric objects, say) without interacting with them causally. This encourages some people to credit other mysterious “ways of knowing”, such as religious faith. And that, in turn, can get them so confused that they can’t succeed at certain tasks, such as building an AI. (Is that what you were getting at?)
If that sequence of confusions is the “failure mode” to avoid, then your success in your OP is to be judged by whether it actually keeps humans from feeling such a felt need for platonism.
But I don’t yet see that it does this, for the reasons that I gave in my previous comment. Someone could easily read your post, agree with the picture it paints, and yet say, “Yes, but just what kinds of things are these isomorphisms and operations and rules of math? I think that the most satisfying answer is still that they are inhabitants of some ideal platonic realm.”
Agreed, but that was an implicit premise, not something I was trying to prove. That is, my article takes it for granted that you will not want to use an epistemology that implies that knowledge can arise without causal interaction, and that therefore you deem your epistemology flawed if and to the extent that it does so. So I assume the reader regards removal of the platonic realm dependency as desirable, for any of a number of reasons, including that one.
True: if you can’t implement a well-defined procedure (such as isomorphism or standard math) without positing its existence in an immaterial realm, then my article doesn’t have much that will change your mind on that matter (“you” in the general sense).
But I don’t see how someone would well-versed enough in rationality for this article to be relevant, yet still make such a leap. That kind of error occurs at a more basic level. Whatever reason suffices to make one feel the need to posit a platonic realm must have a broader grounding, right?
I think that this gets at the crux of my criticism. What kind of dependency on Platonism do you see your article as removing? That is, what kind of “need” for Platonism did you picture a reader feeling before reading your article, but being cured of after reading it?
Thanks, that does get to the heart of the matter. To borrow from one of the linked articles, I imagine someone in the role of Eliezer Yudkowsky here, being challenged by “the one” (bold added):
That is, a rationalist could avoid making obvious or large errors, but still believe “2+3=5”, above and beyond any physically-verifiable claim between two people, and above and beyond any specific model (map) of reality, physically instantiated in agents. My article says to that rationalist, no, you needn’t believe in this platonic “2+3=5″ apart from its implication in a commonly used model, and you can still elegantly and consistently handle all of the dilemmas associated with having to classify such abstract statements. In fact, you needn’t make a statement about anything non-physical.
Do you believe I’ve done so, and said something relevant to rationalists?
Which implication is still a fact which seems to be non-physical, seems to have been true before there were any humans to do logic, etc. You’ve eliminated Platonic numerical entities and metaphysically privileged formal systems—which do seem to be improvements—but not non-physical a priori truths.
It is a counterfactual claim about something physical. You can represent it in a causal diagram with only physical referents.
The causal diagram in your OP contains a node labeled “Integer math implies 2+2 = 4?”
What is the physical referent for “Integer math”?
Any physical system that, as best you can infer, behaves with a known isomorphism to integer math.
**ETA: Oops, see zero_call’s correction; following the article, integer math actually corresponds to some widely-held conception—within human brains—of how numbers work. Since Tyrrell_McAllister’s point was that I was slipping in non-physicality, the rest of the exchange is still relevant, though.
So, to make the pure physicality of all referents clear, should we label that node:
where S is the name of a specific concrete physical system such that the string ‘2+2=’ physically makes S output ‘4’ in a way that is isomorphic to the way that the rules of arithmetic logically imply that 2+2=4?
Yes, basically. I mean, I’d tweak it to read something more like
but I don’t think that impacts whatever point you were trying to make.
I think that your tweak makes an important difference. And, if I may be so bold, I think that you want something closer to what I wrote :).
I’m trying to make good your claim that the causal diagram refers only to physical things. But your label refers to M, which is an isomorphism. What is the concrete physical referent of “M”?
Why? The constraint that a system output a “string” is too strict; it suffices that they output something interpretable as a string.
An isomorphism M is a one-to-one mapping between two phenomena X and Y. In this context, then, the physical referent of M is whatever physically encodes how to identify what in Y it is that the aspects of X map to.
I agree now that “string” is too strict. I should have said “symbol”, where a symbol is anything with physical tokens. My proposed label is now
where
“S” is the name of a specific concrete physical system, and
“A” and “B” are the names of specific physically-manifested symbols,
such that a token of the symbol A physically makes S output a token of the symbol B in a way that is isomorphic to the way that the rules of arithmetic logically imply that 2+2=4.
I think that the work that you want to do by adding the word “interpretable” to the label is done by my conditions on what S, A, and B are.
Then you should be able to make the label refer directly to that physical encoding of M. That is, instead of mentioning the isomorphism M, you ought to be able to refer just to some specific physical system T that “encodes” M in the same way that my physical system S above encodes the operation of adding 2 to 2.
However, if you’re still unhappy with my label, then you would probably be unhappy with this unpacking of your reference to M. But I can think of no other way to make good your claim to refer only to physical things.
(A strict platonist would say that even my label refers to nonphysical things, because it refers to symbols, only the tokens of which are physical. I’m happy to ignore this.)
Well, I would need to permit more than just one physical encoding; I’d need to permit any physical encoding that is, er, isomorphic to an arbitrary one of them. But I don’t see this as being a problem—it’s like what they do with NP-completeness. You can select one (arbitrary) problem as being NP-complete, and then define NP-completeness as “that problem, plus any one convertible to it”.
So it appears I can avoid binding the meaning to one specific physical system, while still using only physical referents. And yes, your updated terminology is fine as long as you allow “symbols” and “fed” to have sufficiently broad meanings.
Incidentally, are you saying the same problem arises for defining “waves”? Do you think that referring to one particular wave requires you to reference something non-physical? Would you say waves are partly non-physical?
M as an isomorphism is just an interpretation between things (rocks, birds, etc.) and “math things” (numbers, etc.) Its physical referent is the human mental instantiation of that interpretation (e.g., in the form of neutro transmitters or what have you.) However, (see my comment a little above), I don’t think this is what you were getting at.
No, I thought the physical referent for the integer math was something like “Human mental instantiation of an idea that is reasonably agreed upon.” I believe you are referring to the physical referent of the preimage of the isomorphism (i.e., the physical system itself. A somewhat redundant thing to call a referent, since it is actually the explicit meaning of the statement.)
You’re right, I agree. I was being inconsistent with my article there.
Voted up because this is a great topic that I’d like us to try and begin to tackle.
But this post really frustrating to try to respond to. Not because it is especially wrong-headed or poorly written but just because it is a little hard for me to find my way around your theory. It is difficult to find a point of traction. In general, I suspect it just isn’t really solving problems but eliding distinctions and ignoring problems (just based on what I do know and the relative shortness of this compared to most other work in philosophy of math). This is pretty much the way I feel about what Eliezer has said on the subject and just about every single thought I’ve ever had on the subject. I’m also not sure I’m familiar enough with the subject area to be able to examine this post in the way it requires.
So I suspect I’ll end up prodding you in a couple places but to begin with: what exactly do you take the Platonist thesis to be? If there is an analogical relationship between a particular expression in our system of inscriptions and our rules for manipulating them (i.e. a written equation) and a physical system (i.e. a system that equation describes) that seems to suggest an underlying structure which is instantiated in both the mathematical expression and the physical system. That such structures exist independently of the mind strikes me as a platonist position. What exactly is wrong with that position? Or what even did you say to contradict it?
Perhaps we need to have a discussion about abstract objects in general before tackling the math.
I do think you’re right about the map-territory confusions here. They definitely abound.
This is not a good heuristic, because in philosophy, works tend to be longest when they’re confused, because most of the length tends to be spent repairing the damage caused by a mistake early on.
So philosophy can get long because the author is running damage control. True. But it can also be short because the author is trying to answer 5-6 questions at once without engaging with the arguments of those who argue against his position. So length by itself- maybe a bad heuristic. But I’m leveraging this heuristic with enough background to make it work.
That there is an immaterial realm of ideal forms (structures, concepts) of which our universe consists solely of imperfect approximations of.
I would say instead that there is some generating function for reality. A system of inscriptions/rules can describe that generating function imperfectly; but this in no way means that the rule/inscription system has some existence apart from its instantiation as the universe itself, and again explicitly in a model.
This stuff about imperfect approximations is just a remnant of Plato’s mysticism. Few modern platonists would say anything like that. This notion of an immaterial “realm” has similar connotations. How about:
Platonism is appealing because it adheres to our norm of accepting the existence of things we make true statements about. “Silas is cool” implies the existence of Silas. Similarly, “3 is prime” implies the existence of 3. The list of non-platonist options as far as I can recall consists of: mathematical objects are mental objects, mathematical objects are physical objects, statements about mathematical objects are false (like statements about Santa Claus), or statements about mathematical objects are actually paraphrases of sentences that don’t commit us to the existence of abstract objects.
It seems like you are trying something like the last. But for this strategy you really should give explicit paraphrases or, ideally, a method for paraphrasing all mathematical truths.
But then what kind of thing is this function? It clearly isn’t merely a set of inscriptions and rules for manipulating them (the models). Nor is it merely the physical universe. We talk like it exists. If it doesn’t, why do we talk like this and what do claims about it really mean?
At least for geometrical forms, the abstractions may be intrinsic to the mind, even if they don’t exist outside it.
In The Man Who Mistook His Wife for a Hat, there’s a description of a man who lost the ability to visually recognize ordinary objects, though he could still see. The one description suggest that he just saw geometry.
In Crashing Through, which is about a man who lost his sight at age 3 and recovered it in middle age and which has a lot about recovered vision and the amount of processing it takes to make sense of what you see, there’s mention of some people who are very disappointed when they recover their sight—they’re constantly comparing the world to an idea of it which is perfectly clean and geometrical.
I’m a little confused: did is visual field lose focus such that, instead of seeing the details on objects and their imperfections he actually just saw idealized geometric figures?
One problem with this as evidence of the possibility that geometric forms could exist only in the human mind is that it presumably only applies to a rather narrow class of geometric forms. It would be weird if the geometric forms we have innate access to had a different ontological status from forms that can’t be instantiated in the human mind: like a 1000-sided polygon or something in 4+ dimensions.
What I meant was that, if people have simple geometric forms built deep into their minds, then it would be tempting to conclude that math has an objective eternal existence because it feels that way.
In any case, I found the actual quote, and I’ve very uncertain that it suggests what I thought it did. It seems as though the man was at least as sensitive to simple topology as geometry, However, people don’t romanticize topology.
Here’s the passage, which I had not remembered as well as I thought:
It’s a wonderful extract in any case. It is fascinating to see someone describing the world without anything more than the phenomenology of his surroundings. It is interesting that the concepts he had access to were mathematical and geometric- that these concepts involve a part of the brain separate from the part that involves more complex and obviously learned concepts like shoe, glove, and flower does seem important to keep in mind when evaluating the evidence on this issue. You’re right that this fact could lead to us positing a false ontological difference… though of course there are those who will say “gloveness” and “flowerness” are abstract objects as well. The fact that these concepts are processed in different parts of the brain could also be taken as evidence for the distinction in that different evolutionary processes generated these two kinds of concepts. I’m not sure how to interpret this. Good for keeping in mind though.
Googling it looks like maybe he just had visual agnosia? Which doesn’t really entail what you’re saying. That would mean that he could see normally but just couldn’t recognize figures as objects with names and functions. Or are you saying the details of objects disappeared and all that was left were the basic geometric forms?
On problem with this as evidence of the possibility that geometric forms could exist only in the human mind is that it presumably only applies to a rather narrow class of geometric forms. It would be weird if the geometric forms we have innate access to had a different ontological status from forms that can’t be instantiated in the human mind: like a 1000-sided polygon or something in 4+ dimensions.
This is very clarifying.
To bring the comparison closer to the mark: It would also imply the existence of ‘cool’.
Heh. What I had in mind was Quine’s criterion for ontological commitment under which it wouldn’t. So Silas is cool is something like, where cool is the predicate letter C: ∃x(Cx ∩ x=”Silas”). We’re committed to the existence of the bound variables (to exist is to be the value of a bound variable) but not of the properties, there doesn’t have to be anything like coolness (assuming that was what you were suggesting).
There is an older argument that claims all words must refer to things and thus a word like “cool” must refer to coolness. But I wasn’t intending to make that argument (though I didn’t say nearly enough in my previous comment to expect everyone to figure that out).
My reading of Silas’s essay (and in particular looking at his diagrams) gave me impression that his ‘2’ is closer to what you would describe as a ‘property’ than the category in which you put ‘Silas’.
I was just starting from the observation that in our mathematical discourse we treat numbers like objects, not properties. “The number between 2 and 4”, “there is a prime number greater than one million”, “5 is odd” etc. all treat numbers as objects.
I would call those properties that had properties. But I’m a programmer, not a mathematician or philosopher (so don’t know which limitations I’m supposed to have placed around my thinking!)
By the way, I think ‘cool’ is kinda ‘lame’ but ‘awesomeness’ is kinda ‘cool’. Just sayin’.
This, I claim, is where you should stop the chain. You’ve erred from the moment you think that that a true statement implies independent existence of its predicates, leaving out the possibility that the predicates exist only as part of models. (Or, equivalently, from treating the term “existence” as having the same meaning whether it refers to something in the map or the territory.)
Isn’t that exactly what I did? The method I gave is that you break down the truth assertion into a claim about the correctness of the terms (i.e., part 1, what the common usage is), and a claim about whether the model in common usage implies the truth (part 2). I suppose I could have been more specific about how exactly one does that, but it seems straightforward enough that I didn’t think I needed to give more detail.
No, no. Not the predicates. Just the values of it’s bound variables. I’m not saying “Primeness exists”. See my reply to wedrifid.
So “3 is prime” means what? “2+3= 5″ means what? You absolutely need to give more detail because as far as I know none of the many very smart people who have tried have been able to give a satisfactory paraphrase for all true mathematical statements.
How does “3 is prime” imply that “3″ exists, while “primeness is related to the zeroes of the Zeta function” not imply that “primeness” exists?
This whole discussion is, ultimately, about the definition of the word “exist”. But if you try to hang that definition off linguistic phenomena, then you’re at the whim of every linguistic construct people can come up with, and people can probably twist words to get something covered by your definition that you didn’t want to be.
The latter does imply primeness exists. But “3 is prime” doesn’t. Luckily you haven’t just used primeness as the value of a bound variable, you’ve given an appropriate paraphrase (although now you’re committed to the existence of the set of zeros of the Zeta function).
Huh?
My method would be to find the associated #1 and #2 statements. The #1 statement would be a claim about what people use the terms “3”, “is”, and “prime” to mean. Under this method you would next identify the common conception of “3″ (by empirical examination of how people use the term and under constraint of Occam’s Razor) as something like, “the quantity immediately following the quantity immediately following the quantity immediately following the quantity of nothing”. Then do the same for the other parts.
(Also, keep in mind that this method is only necessary for the bare statement that “3 is prime”. You needn’t construct the associated #1 statement for more specific claims like, “Here is a system of math. Under those rules and definitions, 3 is prime.”)
Then you would construct the #2 statement, which would be that, under those meanings, the claim as a whole follows from the definitions and assumptions of the system implictly used by those meanings. This would be something like, “under any physical system behaving isomorphically to the assumptions in #1, the physical correlate of ‘3 being prime’ will hold”, and that physical correlate will be something like, “any division of the units correlating to 3 will be such that each partition will have a different number of units, or one unit, or three units”.
...Er, okay, perhaps more detail was needed. Does that answer your question, though?
Yes. That is helpful. I was going to bring this up in my first comment but decided to focus on one thing at a time. Your view seems very similar to nominalist-structuralism, which I find appealing as well. At least I can read nominalist-structuralism into your post and comments. Afaict, it’s considered the most promising version of nominalism going. They basically take your use of isomorphism and go one step farther. The SEP discusses it some but it’s a pretty poorly written article. You might have to do a lot of googling. Structuralism argues that math does not describe objects of any kind but rather, structures and places within structures (and have no identity or features outside those structures). Any given integer is not an object but a places in the structure of integers. As you can imagine once you think of mathematical truths this way it become obvious how math can be used to describe other physical systems: namely those systems instantiate the same structure (or in your words, are isomorphic?). Nominalist structuralism involves disclaiming the existence of structures (which are abstract objects) as independent of the systems that instantiate them
ETA: One issue here is the work being done by the word “resembles” when we say “the structure of real numbers resembles the structure of space’, or “the structure of macroscopic object motion resembles the structure of simulated object motion in your physics simulator”. Which is the same issue as “what is the work being done by ‘isomorphic’.
I think there’s a tautology hidden in there someplace. If two ice cubes plus two ice cubes equal one puddle of water, and two guinea pigs plus two guinea pigs equal an unspecified number of gunea pigs, you can say that isn’t what you meant by addition, but I think that what you mean is that the physical isomorphism has to be arranged so that you get the answer you were expecting.
Is it true that there’s always a physical isomorphism for math? My impression is that some math sits around just being math for quite a while, and then someone finds physics where that math is useful. It’s at least plausible that no one will ever find a physical isomorphism for some math, even if the math is logically sound.
A weaker claim—that math is logically derived from experience of the physical world—might hold up.
As long as the person using math and asserting its relevance (compressive power) in a situation can specify, in advance, what must be true in order to for the isomorphism to hold, there’s no tautology. This situation exists for all theories: the theory’s validity, plus the validity of your observation, plus several other factors, jointly determine what you will observe. If it contradicts your expectations, that reduces your confidence in all of the factors, to some extent. Mathematical predicates, depending on how easy they are to verify, can be more or less resilient than other factors in the face of such evidence.
Right, that’s the caveat I was referring to here:
In other words, some mathematical claims are about arbitrary axiom sets, not necessarily related to physical law, and simply assert that some implication follows therefrom. This article isn’t about those cases. Rather, it’s about bare claims like “2+2=4”, not “under this axiom set, with these definitions, 2+2=4″. Therefore, their truth will hinge partially on the meaning given to the terms, and claims without an explicit axiom set have an assumed one, and necessarily hinge on the presence of an isomorphism to physical law.
What are the reasons that mathematicians like to appeal to a mathematical realm?
In my case, I feel like I ‘manipulate’ mathematical objects in my mind as one would manipulate physical objects. Also, I feel like I ‘explore’ a mathematical subject as one would explore a territory … I investigate how it works rather than make up how it works. (If any one else feels like belief in a math realm is natural, even if illusory, what are your reasons?)
The temptation to appeal to an immaterial realm may also relate to your 4th point:
It has always been the case. But why? Why do mathematical isomorphisms have to follow the same rules as their physical counterparts?
My guess, completely unsubstantiated by any knowledge of neuroscience, is that when mathematicians do math, they are creating and manipulating physical models in their brains. We explore the logic embedded in physical reality by studying physics on a smaller scale, in a much more abstract way, in our brains. Because—and this is my only argument—how else would we know? I would guess that the models are implemented at the cellular (neuronal) scale rather than sub-cellular.
So then there would indeed good reasons for our sense of a Platonic realm. The Platonic realm would be the special software (hardware?) that we run when we think about and develop mathematics.
Source, please?
Moravec, “Mind Children”. Page 93 in the copy I’m looking at.
I strongly agree! Assuming physics, “belief in math” is equivalent to the belief that these models behave very consistently in my and others’ brains, and reflect other regions of physical reality effectively. But even without that, whatever this floaty thing in my mind I call math is, my “belief in math” is one that I hold as convictingly and implicitly as anything else I’m aware of.
This was a great post.
Some observations:
1) The map/territory tool can be used more extensively. Let us take the territory as fairly ‘unknowable’ except that using our map, we can make predictions. If our predictions are wrong then we assume that is a failing in our map and we try to repair the map. Math is a tautological system that has not failed us yet and we use it as part of the map. If it did fail us, we would change it or abandon it from the map. We think the chances of this are vanishingly small.
2) We can construct enlargements to Math tautologically. I think of this as invention but discovery is OK if it pleases. They are enlargements of the map and not objects in the territory. There is a danger in treating Math objects as part of the territory—an infinite regression of maps of maps of maps.....of territory.
3) The reason we invented Math originally and the reason it maps the territory so well are both because its roots are in the structure of the brain and the brain has evolved to do a good (passable) predictive model of reality (or map of the territory).
Please carry on with your investigation of the subject and post on it again.
I’m very much not convinced by your case. Why wouldn’t simply map=territory for math?
Does accepting your arguments change anything, or is it just a less convenient way to look at the same math?
The question isn’t why wouldn’t it or why couldn’t it, but why must it.
Many of our maps could equal the territory, or at least match it precisely. For me, the territories that my present understanding of math maps to are:
physical reality,
what happens when others and my past and future selves “do math”.
I expect a proof of a theorem in my mind right now to tell me that
physical reality will behave a certain way if the theorem applies to my model of it, and
when I or others “do math”, we’ll have ideas that are either consistent with the theorem, or ideas that have a particular reflective feeling about them that I’ve learned to call “errors”.
I adhere to this belief—this trust in math—more faithfully than to any other I can think of, and have seen more evidence for it than, well, almost anything. But I’m still not sure that it must “equal the territory”, and I think that’s the point.
I agree with you. For what it’s worth, I’m a mathematician, and for me math is as much about subjective anticipation as everything else (though most of my colleagues disagree). It’s about expecting the same conclusion every time, and expecting to find something familiar that I’d classify as an “error” when that doesn’t happen.
With really abstract math, when I believe that theorem X is true, what I’m thinking is more like thought pattern S can reliably transform thought A into thought B, where thought pattern S is a pattern people usually call “deduction”, and “A” and “B” are thought types usually called “hypotheses” and “conclusions”.
Isn’t the simple way to deal with most of these issues is to treat math as another language that let’s us communicate about the world? There isn’t really a “two” by itself out there, two is more of an adjective describing the number of objects [or position, or whatever]. This is akin to how we say that there’s nothing that’s inherently a tree, but there are objects we call trees so we all know what object we’re talking about.
If I’m missing the mark, or this leads to some silly conclusion, someone please correct me.
Hmm… For those of us who weren’t that worried about Platonism to begin with, this seems to let us rest a little easier. I’m not sure it accomplishes any more than that. But often, that’s enough.
Regarding your original formulation, I think you could phrase things a little more simply. For example: “For a mathematical claim to be true, we require two conditions. Firstly, the axioms of the claim should agree with our own accepted axioms, and our axioms should be reasonable. Secondly, the claim should follow from those axioms.” As far as I can tell, this is basically what you’re saying, although what you mean by the reasonableness of our axioms is unclear to me.
Regarding the existence of mathematical entities, you’ve seemed to answer in the negative. But I don’t see this as following from your original framework. That framework says nothing about the identity of the mathematical structures in and of themselves. That framework only tells you how the math works, not what it is. Although perhaps you’re saying that the math structures have no identity in and of themselves, existing solely in this framework of claim verification. But in that case, the resolution is tautological, and I’m not sure it really gets to the heart of that question.
In looking at the discovery versus invention of math, you support the invention. But this is essentially a reconfiguration of the previous problem. If we don’t know precisely the identity of the math, the difference between its invention or its discovery is moot.
I disagree about your negative conclusion of math making new predictions a priori. There is the following underlying problem. The conditions of the reality in general do not exactly match the conditions of the math, or at least, this is not verifiable in general. Hence you can never be sure that your isomorphism between math and reality is strictly correct. But that means that a priori mathematical reasoning is the de facto standard (in general). Obviously there are some special cases like adding apples or rocks together which seems to be fully correspondent, but in most cases, the isomorphism may be unverifiable. That’s why it’s amazing that it works.
Regarding the evidence for the truthfulness of math statements… this “truthfulness” just follows by construction from within the original framework. Not sure what you were getting at in that section.
Universes with other math systems—I like this section of yours the most, I think it at least correctly identifies the non triviality of that possibility. A system where non-trivially 2+2=6 would have to correspond to a bit of a different reality, but that reality would also presumably be self-consistent. But if it is self-consistent, then that statement would have to make sense within the system itself. Therefore you escape any problems or contradictions and it loses its idea of being special or strange.
Anyways hope this feedback might be helpful for you and I apologize if I’m misinterpreting you here, as I’ve done a lot of synthesis in this comment.
Thanks for the feedback. I’ll reply to your concerns as best I can.
I didn’t require the axioms to be reasonable in this approach, except, of course, to the extent that their reasonableness causes people to generally accept them in common usage.
That is indeed what I’m saying, but I disagree that it’s tautological. To the extent that my framework handles difficult problems and paradoxes in a satisfactory way, that is its non-tautological substantiation, as it shows how you don’t need to appeal to concepts outside of what I have reduced math to.
I mostly agree, but refer back to the causal diagram. As a standard Bayesian rule, you will never have 100% certainty on any of your premises or conclusions. However, failure of the predicted causal implication to hold (“adding two rocks to two rocks will yield four rocks”) needn’t have the same impact on your degree of belief in each of its causal parents. You can do a lot more to verify your math than to verify the isomorphism to something physical.
If the isomorphism has a lot of evidence favoring it, then the math can tell you surprising things about particular regions of the domain of supposed applicability, which turn out to be true. This is the essence of science and engineering. My point here is only that the math’s applicability to the universe always depends on the empirical validity of the isomorphism, which you might miss if you view the output of math as being the critical step in an insight.
I think the amazingness will eventually be demystified by a common factor that caused both our use of math and the universe’s frequent close isomorphisms thereto.
Yes, and the framework can be relevant or irrelevant to physical systems; people are more likely to be referring to axiom sets that are relevant (have an isomorphism) to physical systems.
Could you expand on 1) the common conception of the rules of how numbers work?
You’ve written:
To what extent is the truth that 2+2=4 an interpersonal one? Is this because if we had different ideas about it, it would be less true -- that the ‘truth’ of addition stems from the fact that we all seem to agree on the way it works?
For myself, I would be reluctant to adopt a concept of mathematical truth that relies on community agreement, but I am curious as to why you emphasize the role of more than one person.
Also, to check my understanding regarding the assumptions component of (1): are these generated in order to model physical phenomena, so that if two people agree on the physical phenomena being modeled, they would agree on the assumptions of the model?
The interpersonal aspect is in there to constrain what the symbols in “2+2=4” actually mean; it has no bearing on the underlying logical truth (part 2). Nevertheless, common agreement in the use of terms is necessary to give those terms meaning. In that respect, the opinions of other people do impact whether such a statement evaluates to “true”. For the same reason, the isolated statement “2+2=6” should evaluate to “false”, even though someone could say, “oh no, see, here, I meant this ‘6’ symbol to mean ‘4’.” That person may have an accurate internal model of reality, but hasn’t correctly conveyed it.
Words can be wrong in terms of sudden unexplained deviation from common usage.
Is 4 not by defintion 2+2, Is math not self proving? I mean why all this “explantion” when it is more evident to say that this thing mathematics is a complex game with rules designed to match the reality.
Typically, 4 is by definition 1+(1+(1+1))), so “2+2=4” means
(1+1)+(1+1) = 1+(1+(1+1)))
In other words, it means two particular different processes for adding up 1′s will yield the same result. This is not assumed in Peano arithmetic, but proven from a selection of even more basic assumptions (which need not explicitly mention associativity), albeit a very clever selection.
What is “twoness”, physically?
. .
There are two dots, but that’s not “twoness”. Otherwise, we wouldn’t be able to count distant objects that are never in conjugation, or ideas.
∃x∃y ( ~(x=y) & ( ∀z ( ~(z=x) ⊃ (z=y) ) & ( ~(z=y) ⊃ (z=x) ) )
Only works in a limited universe of discourse, though.
In lower brow discourse, try: (.)v(.)
I think you may have meant (.Y.)
That works too. Although I must confess I prefer the smaller cup size. :P
Or ∃x∃y ( ~(x=y) & ∀z ( z=y or z=x) )
Still, that’s not ‘twoness’. That’s a sentence that’s only satisfied when there are two things, and could be taken as a definition of what it means to assert that there are two things, or even as a definition of there being two such things, but it’s not ‘twoness’. ‘Twoness’ implies number is a property of objects, which I think Frege pretty conclusively disproved.
I think the fact that a definition of “2” in symbolic logic can be taken to count as an answer to the question “What is twoness, physically?” pretty much says all that needs to be said about the clarity of the question.
Three blues, an anti-red, 4 mm^3 of ether and half a consciousness.
That is correct, except for these errors:
It’s actually four blues, not three.
The whole project of seeking a universal, “platonic” essence of “twoness” is one giant confusion from the very start and will serve to do nothing but distract you from accurately modeling the world, even if, for some reason, you don’t care about paperclips.
For a character whose very existence is founded on facetious irony and satire Clippy sure seems to struggle with it at times.
Actually, my existence is founded on the inherent goodness of paperclips. This confusion on your part causes me to reconsider my classification of you as a good human.
:)
I thought that was perfectly on point, as usual.
Where is the chess in Deep Blue?
What I should have said:
The nature of mathematics is one instance of the problem of universals. (The opposite of a “universal” is a “particular”.) Platonism says that universals exist independently of particulars—and extreme platonism says that universals are all that exists (example). Aristotelian realism says that universals always occur in association with a particular. Nominalism says there are no universals, just words.
I cannot tell if you are a nominalist or an Aristotelian realist. For a physical process involving rocks to have an isomorphism onto the equation 2+2=4, it seems like there has to be some actual twoness in the physical reality, which maps onto the abstraction ‘2’. So I want to know your views on the nature of this physical twoness.
Okay. What I have advocated here is a species of nominalism.
Well, that’s where I disagree. For the isomorphism, it’s only necessary that I have a working model; I needn’t endorse any more abstract or universal concept of “twoness”.
I continually ask myself: for whatever truth I posit, how justifiably surprised can I be if nature refused to play along? (Following the heuristic here.) All the “twoness” that I need to accept the existence of, is contained in physical agents’ physical models of reality. To give it any greater role is to attach myself to a premise for which I have no contradiction to complain about if nature were to refuse to yield any other instance of “twoness”.
What is “twoness”, mathematically?