Yesterday we started a conversation about whether mathematics is invented or discovered. Today I’ll give you my three best arguments for “discovered”. And to focus the discussion, I’ll talk not about mathematics generally but about the natural numbers (0,1,2, and so forth) in particular.
I believe the natural numbers exist, quite independently of whether anyone’s around to think of them. Here’s why: First, we perceive them directly. Second, we know non-trivial facts about them. Third, they can explain the Universe. In more detail:
1. We perceive them directly. I believe that flowers, rainbows and headaches exist because I perceive them directly. I believe that the natural numbers exist for exactly the same reason. Almost anyone who has ever thought hard about higher arithmetic will tell you the same thing. Yesterday I quoted the Fields Medalist Alain Connes saying that when mathematicians contemplate arithmetic, “we run up against a reality every bit as uncontestable as physical reality”. Today I’ll quote Kurt Godel, the greatest logician of all time:
Despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves on us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e. in mathematical intuition, than in sense perception.
2. We know non-trivial facts about them. In 1637, Pierre de Fermat wondered whether you can find four positive numbers x, y, z and n, with n at least 3, that satisfy the equation
After 350 years, the question was settled by Gerhard Frey, Ken Ribet and Andrew Wiles in one of the most spectacular mathematical achievements of the twentieth century. The answer, as Fermat had believed, is no.
That’s certainly a meaningful statment: It means that no matter what four numbers you write down, we can predict with certainty that as long as they’re positive, and as long as n is at least 3, the equation I’ve just written down will never be true. But unlike the axioms that Godel was referring to, it’s hardly self-evident and it does not force itself on us as being true; that’s part of why it took 350 years to prove.
So how do we know that Fermat’s Last Theorem (i.e. the statement that the equation has no solutions) is true? The answer is not that it follows step by step from some list of self-evident axioms about the natural numbers. As far as I am aware, nobody has the foggiest idea whether Fermat’s Last Theorem follows from any set of reasonably self-evident axioms about arithmetic, such as the Peano axioms that I wrote about here. Instead, we know that Fermat’s Last Theorem is true via informal (but, to almost all mathematicians, completely convincing) arguments that are not about manipulating axioms but instead are about the properties of numbers themselves.
(It’s a virtual certainty that these informal arguments could be formalized in some language, but—again as far as I know—it’s quite unknown whether they could be formalized in the usual language of arithmetic.)
So I agree with Godel that the self-evident nature of the axioms is evidence that the natural numbers are real, but I also believe, quite separately, that the non-self-evident nature of statements like Fermat’s Last Theorem is additional evidence. Here we have a statement that is true, but it’s truth is not derived from axioms about arithmetic. Instead it’s true because it’s a correct statement about something. That “something” is the system of natural numbers.
3. They explain the Universe. This argument is surely more speculative than the others, but I cannot imagine any way to explain the existence of the Universe without the prior existence of the natural numbers. (This is more or less the same reason some people give for believing in God.) It seems to me that the most compelling question in philosophy is why anything exists at all. Any satisfactory answer has to start with something that must exist. The natural numbers fill that role admirably. In The Big Questions, I’ve sketched a story about how, once you’ve got some mathematical objects, the Universe can sort of bootstrap itself in existence from there; this is similar in spirit to the story advanced by the noted cosmologist Max Tegmark in his essay on The Mathematical Universe (cited in yesterday’s comments by Al V.)
Now admittedly, my inability to find any alternative explanation for the Universe does not prove that this explanation is correct. For that matter, my hunger for an explanation doesn’t mean there has to be one. Maybe the Universe just is. But when you’re facing a huge riddle and you can only think of one possible solution, you’ve got to at least contemplate the possibility that you’re on to something.
I’m missing something here:
You said that Fermat’s Last Theorem was evidence for the “reality” of numbers because it doesn’t follow from manipulating sets of axioms according to self-evident rules. But you also said, “It’s a virtual certainty that these informal arguments could be formalized in *some* language.”
If FLT can (almost certainly) be derived from mindless manipulation of symbols according to pre-defined rules in some hypothetical language, then why is it any more impressive than other mathematical statements that can be derived from the Peano axioms according to manipulation of the symbols?
Bennet, if you don’t like that example, what about something like Goodstein’s Theorem (or other examples of ‘natural’ Goedel sentences for PA)? That is undoubtedly true, in some important sense, and equally undoubtedly impossible to prove from the axioms of the Peano Arithmetic.
I believe the natural numbers exist, quite independently of whether anyone’s around to think of them.
What’s your take on moral realism? Are statements about ethics true or false independently of whether anyone’s around to think about them? Are moral rules discovered or invented?
Point 1:
No you don’t. When I cut vegetables, or drive a car, it feels like the tool is an extension of my body. But it isn’t. It is only that my mental model of my body has these appendages that my actual body does not. It’s a useful mental construct.
In a similar way I have mental models of various other things, which develop after practice, so that I can manipulate other tools, including invisible ones such as mathematical functions.
Point 2:
The several disciplines of theology are, I think, proof enough that knowing non-trivial facts about an object does not mean it exists other than as a mental construct.
I can recite Kubla Khan, which is not a trivial feat (maybe you think otherwise, but any five-year-old will correct you!), but does not prove that the maiden’s demon lover was not invented.
On your point 3:
The obvious sense in which mathematical objects exist, as consequences of axioms which are chosen by thinking beings, can’t apply there. Tregmark’s ERH proposes that our universe exists as a mathematical construct/model within another external reality, so that doesn’t supply an ultimate cause either.
We don’t know what preceded the first nanoseconds of the universe, and probably never will. We are also still very far from being able to make educated guesses. Believing we can discover things about the physical universe by reasoning from first principles is a habit we should have been cured of by now.
Here’s my view. It seems to me that you asked two questions:
Do mathematical objects exist?
Are mathematical objects discovered or invented?
Mathematical objects are the necessary consequences of axioms in a system of reasoning. Once the axioms are chosen, facts about the consequences are discovered.
To me it seems clear that the mathematical sense of the production “exists” is that an object is a neccessary consequence of chosen axioms or assumptions. I.e. as in the phrase “for each A there exists a B such that…”.
But the axioms are invented, and do not exist outside of the minds of thinking beings.
Algorithms and proofs on the other hand are wholely invented, and all the more creditable for that.
This debate has lasted at least 2 millenia and will probably last a few more.. I think it depends on what we mean by “exist”.. If we mean that the theorems of mathematics “exist” independently of human beings, I think not. If we mean “exist” independently of any particular individual, I think true.
As for Godel, etc, the fact that we can create a system so complex that we cannot formalize it completely truly is a surprise.
However, if we believe that this demonstrates that math somehow existed before people came along, and will exist when we are long gone, I am not sure that follows. How exactly have we shown this to be the case? What would be interesting would be a more substantive statement: What kind of existence is this? Did Math exist before the Big Bang? Would it survive a Big Crunch? How do we know?
Ben:
To me it seems clear that the mathematical sense of the production “exists” is that an object is a neccessary consequence of chosen axioms or assumptions. I.e. as in the phrase “for each A there exists a B such that…”.
So then you don’t think that the natural numbers exist?
autogen – The answer to your question “Did Math exist before the Big Bang?” depends on whether you believe math exists inside or outside of our universe. I believe, as do Steve and Max Tegmark, that math is “supra-universal”, and exists independently of our physical universe. It seems to me that the theorems of math are supra-universal, as they are provable, but Euclid’s axioms may exist in out universe only, because they can only be assumed.
Adding to my prior post, we’re trying to decide whether math is invented or discovered. And I questioned whether math is defined by our universe, or outside of our universe. It seems to me that there is a kind of decision tree.
– If math is invented, then it is necessarily in our universe, since we are in our universe.
– If math is discovered, then it can either be universal or supra-universal.
If math is invented, then math must be universal, or “of our universe”. Therefore, our mathematical system is a subsystem of our universe’s physics. Godel said that any mathematical system cannot be both consistent and complete. Hopefully, math is consistent, and it’s okay for it not to be complete, meaning that there are things in math that are unprovable. However, if math is universal, and a subsystem of our physical universe, then by Godel the implication is that there are laws of physics that are unprovable. Possible, but that makes be uncomfortable.
If math is supra-universal, then any incompleteness (unprovability) of our physical universe can be completed within our supra-universal mathematics, so it’s okay for our physical universe to be incomplete. That makes more sense to me.
I’m not sure what you mean here by “direct perception.” In the sense that I think of that phrase it relates to irreducible sensory impressions. I may perceive, as you said in the other entry, the separateness or individuality of an object – say Earth’s moon. But that perception is of an object around which I construct the notion of numbering and counting. I don’t (think I) perceive “one” in the same way I perceive “hot”.
Second, I’m not sure what you mean by “non-trivial facts.” Surely the notion of what is or is not trivial is itself homed within a theoretical framework. In order to assert that the proof of Fermat’s Last Theorem is “non-trivial” you need to undertake the assumption of a situation of importance of abstract mathematical proofs. It’s a wholly other sense of “trivial” than is meant in sentences such as “the rise in the national unemployment rate is non-trivial.” So if we know non-trivial things about the natural numbers within the world of mathematical theory that’s all well and good for that theory-world but it doesn’t seem to bear on the question of whether or not these numbers are invented versus discovered.
I am not in any sense a mathematical theorist so I rely only on my intuitive belief that the natural numbers were discovered and that they exist outside the specific frameworks used to talk about them, which I think are like most theories; that is, we have observations and/or intuitions about the way the universe is and we create theories to explain and predict around these things.
Point 1: Your quote does not support your conclusion. Godel concedes that numbers are remote from sense experience. Hence, he does not agree with your statement that we perceive numbers in the same way that we perceive flowers, rainbow, and headaches. Numbers are different. Instead, Godel points out that we can be confident in sets since they seem to exist independent of our wills just as much as physical facts. But this has little to do with discovery versus invention. Physical things, after all, can be invented too.
Point 2: We know non-trivial facts about them. But any good scientific theory about physical reality leads to the discovery of nontrivial facts about the world, even ones eventually proven “untrue,” e.g., Newton’s mechanics. The theory itself, however, is invented to describe the physical world. Often these theories are supplanted. Furthermore, the English language does a pretty good job of describing the universe, or, at least the things in the universe. Does that mean the English language was discovered rather than invented? That it must exist?
Point 3: Many theories explain the universe, but we know these theories are invented because they are eventually proven untrue and replaced by better theories. Why should natural numbers be any different? Furthermore, you claim that natural numbers must exist. Why? That seems to beg the question. I could very well argue that I must exist in order for the universe to exist. But that’s not very convincing is it.
If natural numbers merely exist, what’s the point of defining them?
My apologies if this has been covered elsewhere, but I’m not sure what you mean by “exist.” Do numbers exist in the same sense that beanie babies exist? If so, then I’m afraid you’ve lost me. If not, then explaining the different types of existence would probably end the argument.
I categorize math as a very useful concept. I have a hard time saying concepts exist, even if they are useful.
@Steve landsburg:
Natural numbers exist in the mathematical sense of “exist”, i.e. they can be derived from Lambda calculus or axiomatic set theory.
They don’t exist as concrete objects in the real world, which is the usual sense of “exist”. There may be two of something but it doesn’t have two-ness outside of the mind of an observer who knows the concept of two.
Ben: I’m not sure what you mean by “derived”. But if you’re suggesting that there would be no numbers in the absence of a formal language for describing them, I quite disagree.
Daniel Morgan: Numbers exist in the sense that they have properties. Fermat’s Last Theorem, for example, is a property of the natural numbers. It’s true, it was true before there was anyone around to think about it, and it would be true in any Universe, whatever the laws of physics in that Universe might be. It’s therefore not about the way our minds work and it’s not about anything physical. The natural numbers exist in the sense that they constitute the object that Fermat’s Last Theorem is talking about.
@ Steve Landsburg:
“But if you’re suggesting that there would be no numbers in the absence of a formal language for describing them, I quite disagree.”
Thanks for responding. I’m taking issue with your use of the word “be” in that sentence. It doesn’t make any sense to ask whether numbers exist without an agreed understanding of what existance entails. I think we have two different (though analogous) concepts with the same name.
The usual definition is that it is a concrete object in the physical world. Platypuses exist, unicorns don’t, even though we can have the idea of them. On the other hand, the idea of a unicorn exists, but only in minds. In this sense of “to exist”, the natural numbers either don’t exist or exist only as ideas in minds.
Since (to me) the mathematical sense of “to exist” is that it is entailed as a consequence of chosen axioms in a system of reasoning, it is clear that the natural numbers do exist, in the mathematical sense, for some systems of reasoning with certain axioms. That’s true whether or not anyone has ever thought of the axioms, since it is a statement about logic not about the universe.
But to conclude that this means they exist in the universe is to assume what you want to prove, namely that the universe obeys laws describable mathematically.
(You could argue that some numbers have concrete existance, but I think you are going to get stuck somewhere before 10^85, finding numbers with no concrete examples of exactly that many of anything)
Ben: First, with regard to your last sentence, it’s not individual numbers I’m talking about; it’s the the entire *system* of natural numbers; Fermat’s Last Theorem, for example, is a statement about the entire system.
Second, the notion that the natural numbers exist only in a sense entailed by axiomatic reasoning runs up against the fact that they have recognizable properties that are *not* entailed by reasoning from any set of axioms. This is really the crux of the matter.
I myself am sure of the Platonic existence of the Natural Numbers and even the Integers, Rationals, Reals and Complex (no matter what Kronecker said). Quaternions too – but Octonions – not so much.
@Steve Landsburg:
“the notion that the natural numbers exist only in a sense entailed by axiomatic reasoning runs up against the fact that they have recognizable properties that are *not* entailed by reasoning from any set of axioms. This is really the crux of the matter”
I’m not really sure what you have in mind. Surely it is either assumed axiomatically, entailed by reasoning from axioms, or is an analogical correspondence with an object in the real world, and so not actually a property of the number.
Is there anything else? Perhaps an example would help?
Ben: Fermat’s Last Theorem is an example.
What is an example of something that has properties that *are* entailed by reasoning from a set of axioms? Tautologies? Do mean to say that we know numbers were discovered rather than invented because we have reason to believe certain things about numbers that we cannot prove deductively?
SJA:
Do mean to say that we know numbers were discovered rather than invented because we have reason to believe certain things about numbers that we cannot prove deductively?
SJA: Yes, this is precisely what I mean to say.
You are quite right.. This is a religious question. :)
@Steve Landsburg, re. your statement “Fermat’s Last Theorem … would be true in any Universe, whatever the laws of physics in that Universe might be.” Tegmark hypothesizes different levels of universes. Level I are universes that obey our laws of physics. Level II are those that have the same laws of physics, but diffent physical constants. Level III are those that have different laws of physics, but the same mathematical foundation. And Level IV are those have have different mathematics. Thus, Fermat’s Last Theorem would be valid in Levels I, II, and III, but not at Level IV.
Of course, this is completely hypothetical.
@Steve Landsburg
I definitely agree with what you just said, but I still don’t want to use the word “exists.”
From where I am in the book, it seems like doing so has empirical consequences, but I’m not sure how this jives with the requirement of nonfalsifiability usually given to empirical claims.
Also, if math exists because it has properties, does it follow that nonsense math exists because it also has properties? “2 + 2 = 5” has the property of being incorrect. Does this mean that, if we accept your origin of the universe arguments from the book, that illogical mathematical systems are manifested as inconsistent parallel universes?
And, if so, how do we know that we’re not living in one now?
The problem with questions like “Were the natural numbers discovered or were they invented?” is that there is no obvious criteria for resolving the question. What do you mean by each? The distinction is hardly simple.
Consider an easier case. Was Newton’s law of gravity discovered or invented? The only sensible answer is “both.” Newton invented a mathematical model which captured much (but not all) of the behavior of gravity in the Universe. The inexactitude of the law proves that he didn’t just discover a pre-existing reality, he invented something a lot – but not exactly – like that reality. Is math – the natural numbers, say – like that? Come back when you can think of a criterion that might answer the question.
CIP: I understand your discomfort with the lack of a clear and fully articulated criterion. But I also think there is a clear difference between Newton’s law of gravity, which is subject to revision, and the various truths about the natural numbers, which are decidedly non-revisable.