**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.

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.

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 o... (read more)

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 m... (read more)

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 ... (read more)

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:

... (read more)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 tautological... (read more)

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?

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 likethought pattern S can reliably transform thought A into thought B, wherethought pattern Sis 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... (read more)

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 interpersonalone? 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 a... (read more)

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.

What is "twoness", physically?