Paper Review: Mathematical Truth

by Alex Hollow3 min read30th May 202114 comments

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EpistemologyPhilosophyWorld Modeling
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Mathematical Truth (Benacerraf 1973), introduces a problem for mathematical knowledge that I find incredibly compelling. Benacerraf thinks that there are two requirements of a theory of mathematical truth: semantic uniformity, such that what we mean when we talk about math is similar to what we mean when we talk about everything else; and epistemic uniformity, such that the epistemology of math is similar to the epistemology of everything else. Benacerraf thinks that a good theory of mathematical truth has to meet both goals, and that the two goals are incompatible, which leaves mathematical truth in a rough spot.

Semantic Uniformity

On the topic of semantic uniformity, Benacerraf says that "A theory of truth for the language we speak, argue in, theorize in, mathematize in, etc., should ... provide similar truth conditions for similar sentences. The truth conditions as- signed to two sentences containing quantifiers should reflect in relevantly similar ways the contribution made by the quantifiers". For examples, he looks at these two sentences: 
   (1) There are at least three large cities older than New York. and 
   (2) There are at least three perfect numbers greater than 17. 
Then, he asks if these sentences have the same logical form: 
   (3) There are at least three F's that bear R to a.

Sentence (1) has the same logical form as sentence (3). This means that sentence (1) is true if there are at least three elements in the set of things that exist which satisfy the predicates “large city” and “older than New York”. For sentence (2), can we do the same thing? Benacerraf replies “That sounds like a silly question to which the obvious answer is “Of course.””. However, he continues, that requires numbers to be in the set of things that exist! A Platonic realist has no problem with this, because they posit the real existence of numbers. Benacerraf calls this the “standard view” – sentence (2) has the logical form of sentence (3), and numbers really exist so this is fine.

However, in intuitionist mathematics, sentence (2) would have a very different logical form than sentence (3). Benacerraf then turns to views where mathematical truth comes from being derivable from a set of axioms. Those theories give a truth-predicate for sentences that is strictly syntactic, allowing us to determine the truth or falsity of a sentence in a system of axioms without referencing anything in the real world. So, under syntactic (what Benacerraf calls “combinatorial”) views of mathematical truth, sentence (2)’s logical form is again unlike sentence (3). This is all just to say that it is controversial that sentence (2)’s logical form looks like sentence (3).

Benacerraf thinks “we shouldn’t be satisfied with an account that fails to treat (1) and (2) [the same]”.

Epistemic Uniformity

The second condition on a theory of mathematical truth, epistemic uniformity, requires that “we have mathematical knowledge, and that such knowledge is no less knowledge for being mathematical”. In other words, we need to be able to know whether at least some mathematical sentences are true or false. Mathematical knowledge “must fit into an over-all account of knowledge in a way that makes it intelligible how we have the mathematical knowledge that we have”.

Epistemic uniformity is hard! If we model sentence (2) in the form of sentence (3), it is a claim about properties of numbers. The “standard view”‘ says that numbers exist, so the sentence at least has a truth-value. Numbers are acausal. The existence or non-existence of the number 3 is not observable by humans. This is a problem for causal accounts of epistemology, where knowledge requires some causal link between the state of the world and your knowledge. Numbers are acausal, so we can’t have knowledge of them, and we can’t have mathematical knowledge more generally.

(Some people say that noticing that putting two items next to two items and seeing that you now have four items gives you some evidence that 2 + 2 = 4. However, “adding two items to two items gives you four items” being true or false has a causal effect on your belief after running the text, so you can have knowledge of it even if you can’t have knowledge of “2 + 2 = 4”.)

Dilemma

Here is the crux of the paper: those two properties can’t both be satisfied. If mathematical sentences are like normal sentences, they refer to objects. If mathematical sentences refer to objects, we can’t have causal knowledge of mathematical truth. Eek! This is a big problem for humans ever gaining knowledge of mathematical truth.

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There is no contradiction if you treat mathematical knowledge as a map of the world, not anything separate.

 2+2=4 is a useful model for counting sheep, not as useful for counting raindrops. 

Maps are neither true nor false, they have various degrees of accuracy (i.e. explaining existing observations and predicting new observations well) and applicability (a set of observations where they show good accuracy). 

Platonism makes the mistake of promoting an accurate and widely applicable model (a certain type of math) into its own special territory, and that's how it all goes wrong. Epistemic uniformity simply states that math is a useful model. Mathematical statements can be true or false internally, i.e. consistent or inconsistent with the axioms of the model, but they have no truth value as applied to the territory. None. Only usefulness in a certain domain of applicability. 

In this framework, semantic uniformity is a meaningless construct. You can only talk about truth value as being internal to its own model, and 1 and 2 are from different models of different parts of the territory. 3 is... nothing, it has no meaning without a context. There is no reason at all that 1 and 2 should have the form 3, unless they happen to be both submaps of the same map where 3 is a useful statement. For example, in the intuitionist view whether "There are at least three perfect numbers greater than 17" is discoverable experimentally (by proving a theorem or by finding the 3 numbers after some work), just like "There are at least three large cities older than New York" is discoverable experimentally (e.g. by checking in person or online). Again, I'm discounting Platonism, because it confuses map and territory.

Benacerraf convinced me that either mathematical sentences have different logical forms than non-mathematical sentences or that mathematical knowledge has a different form than non-mathematical knowledge. It sounds like your view is that mathematical sentences have different forms (they all have an implicit "within some mathematical system that is relevant, it is provable that..." before them), and also that mathematical knowledge is different (not real knowledge, just exists in a system). In other words, it sounds like you just think that epistemic uniformity and semantic uniformity are not important features of a theory of mathematical truth. That comes down to personal aesthetics and meta-beliefs about how theories should look, so I will just talk about what I think you're saying in this comment.

I think what you're trying to say is that mathematical statements are not true or false in an absolute sense, only true and false within a proof system. and their truth or falsehood is based entirely on whether they can be derived from within that system. 

If that's true, math is just a map, and maps are neither true nor false. If math is just a map, then there is no such thing as objective mathematical truth. So it sounds like you agree that knowledge about any mathematical object is impossible. But when you say that "Epistemic uniformity simply states that math is a useful model", I think that's a little different than what I intended it to mean. Epistemic uniformity says that evaluating the truth-value of a mathematical statement should be a similar process evaluating the truth-value of any other statement.

The issue here is that our non-mathematical statements aren't only internally true or false, they are actually true or false. If you asked someone to justify sentence (1), and they handed you a proof about New York and London, consistent on a set of city-axioms, you would probably be pretty confused. Epistemic uniformity says that a theory of mathematical truth should look relatively like the rest of our theories of truth - why should math be special?

I'm going to take a slight objection to using the phrase "discoverable experimentally" to describe proving a theorem and thinking up numbers, but let's talk about those examples. To me, it sounds like that is doing work within a system of math to determine whether a claim is consistent with axioms. There is some tension here between saying that math is just a tool and thinking that you can do experiments on it to discover facts about the world. No! It will tell us about the tool we are experimenting with. Doing math (under the intutionist paradigm) tells us whether something is provable within a mathematical system, but it has no bearing on whether it is true outside of our minds.

(Side note about intuitionism: 

I think it's important to prevent talking past each other by checking definitions, so I'd like to clarify what you mean by intuitionism. In the definition I'm aware of, intuitionism says roughly that math exists entirely in minds, and the corresponding account of mathematical truth is that a statement is true if someone has a mental construction proving it to be true. Please let me know if this is not what you meant!

My main objection with intuitionism is that it makes a lot of math time-dependent (e.x. 2+2 didn't equal 4 until someone proved it for the first time). Under an intuitionist account of mathematical truth, you can make sentence (2) true by finding three examples that fit. But then that statement's truthhood or falsehood is independent of whether the mathematical fact is really true or false (intuitionists usually don't think there exist universal mathematical truth). It seems to me that math is a real thing in the universe, it was real because humans comprehended it, and it will remain real after humans are gone. That view is incompatible with intuitionism.)

(Another note - can you be a bit more specific about the contradition you think is avoided by giving up Platonism? I think that you still don't have epistemic and semantic uniformity with an intuitionist/combinatorial theory of math)

First, I appreciate your thoughtful reply!

It sounds like your view is that mathematical sentences have different forms (they all have an implicit "within some mathematical system that is relevant, it is provable that..." before them)

Yes. And your paraphrasing matches what I tried to express pretty well, except when you use the term "knowledge".

math is just a map, and maps are neither true nor false. If math is just a map, then there is no such thing as objective mathematical truth.

Depends on your definition of "objective". It's a loaded term, and people vehemently disagree on its meaning.

So it sounds like you agree that knowledge about any mathematical object is impossible.

Not really, I just don't think you and I use the term "knowledge" the same way. I reject the old definition "justified true belief", because it has a weasel word "true" in it. Knowledge is an accurate map, nothing else.

Epistemic uniformity says that evaluating the truth-value of a mathematical statement should be a similar process evaluating the truth-value of any other statement.

I'd restrict the notion of "truth" to proved theorems. Not just provable, but actually proved. Which also means that different people have different mathematical truths. If I don't know what the eighth decimal digit of pi is, the statement that it is equal six is neither true nor false for me, not without additional evidence. In that sense, a set of mathematical axioms carves out a piece of territory that is in the model space. There is nothing particularly contradictory about it, we are all embedded agents, and any map is also a territory, in the minds of the agents. I agree that math is not very special, except insofar as it has a specific structure, a set of axioms that can be combined to prove theorems, and those theorems can sometimes serve as useful maps of the territory outside the math itself.

I am not sure what your objection is to the statement that mathematical truths can be discovered experimentally. Seems like we are saying the same thing?

Doing math (under the intutionist paradigm) tells us whether something is provable within a mathematical system, but it has no bearing on whether it is true outside of our minds.

It's worse than that. "Truth" is not a coherent concept outside of the parts of our minds that do math.

My main objection with intuitionism is that it makes a lot of math time-dependent (e.x. 2+2 didn't equal 4 until someone proved it for the first time).

A better way to state this is that the theorem 2+2=4 was not a part of whatever passed for math back then. We are in the process of continuous model building, some models work out and persist for a time, some don't and fade away quickly. Some models propagate through multiple human minds and take over as "truths", and others remain niche, even if they are accurate and useful. That depends on the memetic power of the model, not just on how accurate it is. Religions, for example, have a lot of memetic power, even if their predictions are wildly inaccurate. 

It seems to me that math is a real thing in the universe, it was real because humans comprehended it, and it will remain real after humans are gone. That view is incompatible with intuitionism.

Again, "real" does all the work here. Math is useful to humans. The model that "[math] will remain after humans are gone" is content-free unless you specify how it can be tested. And that requires a lot of assumptions, such as "what if another civilization arose, would it construct mathematics the way humans do?" -- and we have no way to test that, given that we know of no other civilizations.

can you be a bit more specific about the contradition you think is avoided by giving up Platonism? I think that you still don't have epistemic and semantic uniformity with an intuitionist/combinatorial theory of math

If you give up Platonism as some independent idea-realm, you don't have to worry about meaningless questions like "are numbers real?" but only about "are numbers useful?" Semantic uniformity disappears except as a model that is sometimes useful and sometimes not. In the examples given it is not useful. Epistemic uniformity is trivially true, since all mathematical "knowledge" is internal to the mathematical system in question.

We might be talking past each other though.

Hi! I really appreciate this reply, and I stewed on it for a bit. I think the crux of our disagreement comes down to definitions of things, and that we mostly agree except for definitions of some words.

Knowledge - I think knowledge has to be correct to be knowledge, otherwise you just think you have knowledge. It seems like we disagree here, and you think that knowledge just means a belief that is likely to be true (and for the right reason?). It's unclear to me how you would cash out "accurate map" for things that you can't physically observe like math, but I think I get the gist of your definition. Also, side note, justified true belief is not a widely held view in modern philosophy, most theories of truth go for justified true belief + something else.

Real - We both agree it doesn't matter for our day-to-day lives whether math is real or not. (It may matter for patent law, if it decides whether math is treated as an invention or a discovery!) I think that it would be nice to know whether math is real or not, and I try to understand the logical form of sentences I utter to know what fact about the world would make them true or false. So you say I "don't have to worry about" whether numbers are real, and I agree – their reality or non-reality is not causing me any problem, I'm just curious.

I also view epistemic uniformity as pretty important, because we should have the same standards of knowledge across all fields. You seem to think that mathematical knowledge doesn't exist, because mathematical "knowledge" is just what we have derived within a system. I can agree with that! The Benacerraf paper presents a big problem for realism, which you seem to buy - and you're willing to put up with losing semantic uniformity for it. 

I think our differences comes down to how much we want semantic uniformity in a theory of truth of math.

Interesting... My feeling is that we are not even using the same language. Probably because of something deep. It might be the definition of some words, but I doubt it. 

Knowledge - I think knowledge has to be correct to be knowledge, otherwise you just think you have knowledge.

what does it mean for knowledge to be correct? To me it means that it can be used to make good predictions. 

you think that knowledge just means a belief that is likely to be true (and for the right reason?)

well, that's the same thing, a model that makes good predictions. "The right reason" is just another way to say "the model's domain of applicability can be expanded without a significant loss of accuracy". 

It's unclear to me how you would cash out "accurate map" for things that you can't physically observe like math

You can "observe math", as much as you can observe anything. How do you observe something else that is not "plainly visible", like, say, UV radiation? 

We both agree it doesn't matter for our day-to-day lives whether math is real or not.

That is not quite what I said, I think. I meant that math is as real as, well, baseball.

You seem to think that mathematical knowledge doesn't exist, because mathematical "knowledge" is just what we have derived within a system.

I... was saying the opposite. That mathematical knowledge exists just as much as any other knowledge, it just comes equipped with its own unique rigging, like proven theorems being "true", or, in GEB's language, a collection of valid strings or something. I don't want to go deeper, since math is not my area.

In general, the concept of existence and reality, while useful, has a limited applicability and even lifetime. One can say that some models exist more than others, or are more real than others. 

I also view epistemic uniformity as pretty important, because we should have the same standards of knowledge across all fields.

I agree with that, but those standards are not linguistic, the way (your review of) Benacerraf's paper describes it, that they should have the same form (semantic uniformity). The standards are whether the models are accurate (in terms of their observational value) in the domain of their applicability, and how well they can be extended to other domains. Semantic uniformity is sometimes useful and sometimes not, and there is no reason that I can see that it should be universally valid.

Not sure if this made sense... Most people don't naturally think in the way I described.

Thoroughgoing anti realism gives you a kind of semantic uniformity , at the expense of having the same level of anti realism about non mathematical entities. Do you want to give up believing in electrons?

I see no reason why the semantic uniformity property should hold for mathematics. Having sentences of similar form certainly does not require that they have the same semantics even for sentences that have nothing to do with mathematics. English has polysemous words, and the meaning of "exists" in reference to mathematical concepts is different from the meaning of "exists" in reference to physical entities.

If this was for some reason a big problem, then an obvious solution would be to use different words or sentence structure for mathematics. To some degree this is already the usual convention: Mathematics has its own notations, which are translated loosely into English and other natural languages. There are many examples of students taking these vaguer translations overly literally to their own detriment.

This is definitely not a "big problem" in that we can use math regardless of what the outcome is. 

It sounds like you're arguing that semantic uniformity doesn't matter, because we can change what "exists" means. But once you change what "exists" means, you will likely run into epistemological issues. If your mathematical objects aren't physical entities capable of interacting with the world, how can you have knowledge that is causally related to those entities? That's the dilemma of the argument above - it seems possible to get semantic uniformity at the expense of epistemological uniformity, or vice versa, but having both together is difficult.

Nobody has to believe that ordinary non mathematical langage contains a single well defined meaning of "exists". And if "exists" is polysemous , then one of its meanings could be the meaning of mathematically-exists ... it doesn't have to have a unique meaning. Fictivism is an example: the theory that mathematically-exists means fictionally-exists, since ordinary language allows truth and existence to be used in reference to fictional worlds.

To provide a more concrete example, I would say that the claim "There are at least three Jedi older than Anakin Skywalker" is, in the interpretation of the sentence almost anyone who understands it would use, a true statement, even though Jedi certainly do not exist in the same sense that New York City exists. I would not say that the fact that Jedi and New York exist in very different ways really violates Semantic Uniformity in any substantial way. 

I'm not super up-to-date on fictionalism, but I think I have a reasonable response to this.

When we are talking about fictional worlds, we understand that we have entered a new form of reasoning. In cases of fictional worlds, all parties usually understand that we are not talking about the standard predicate, "exists", we are talking about some other predicate, "fictionally-exists". You can detect this because if you ask people "do those three Jedi really exist?", they will probably say no. 

However, with math, it's less clear that we are talking fictionally or talking only about propositions within an axiomatic system. We could swap out the "exists" predicate with something like "mathematically-exists" (within some specific axiom system), but it's less clear what the motivation is compared to fictional cases. People talk as if 2+2 does really equal 4, not just that its useful to pretend that it's true. 

The main difference between mathematics and most other works of fiction is that mathematics is based on what you can derive when you follow certain sets of rules. The sets of rules are in principle just as arbitrary as any artistic creation, but some are very much more interesting in their own right or useful in the real world than others.

As I see it, the sense in which 2+2 "really" equals 4 is that we agree on a foundational set of definitions and rules taught at a very young age in today's cultures, following those rules leads to that result, and that such rules have been incredibly useful for thousands of years in nearly every known culture.

There are "mathematical truths" that don't share this history and aren't talked about in the same way.

The motivation to me seems exactly the same as with fiction: we're talking about things other than physical objects or whatever.

Talking about mathematics, qua fictions, cant possibly have less motivation than talking about fictions qua fictions.

People talk as if 2+2 does really equal 4, not just that its useful to pretend that it’s true.

People also tend to regard their own tribal myths as really true, as well.