### Basic Arithmetic: On What There Is

This is an extremely elementary post about numbers. (“Numbers” means the natural numbers 0,1,2 and so forth.) It is a sort of sequel to my three recent posts on basic arithmetic, which are here, here and here. But it can be read separately from those posts.

Today’s question is: Do numbers exist? The answer is: Of course, and I don’t believe there’s much in the way of serious doubt about this. You were familiar with numbers when you were five years old, and you’ve been discovering their properties ever since. Extreme skepticism on this point is almost unheard of among mathematicians or philosophers, though it seems to be fairly common among denizens of the Internet who have gotten it into their head that extreme skepticism makes them look sophisticated.

Let me be clear that I am not (yet) asking in what sense the natural numbers exist — whether they have existed since the beginning of time, or whether they exist outside of time, or whether they exist only in our minds. Those are questions that reasonable people disagree about (and that other reasonable people find more or less meaningless.) We can — and will — come back to those questions in future posts. For now, the only question: Do the natural numbers exist? And the answer is yes. Or better yet — if you believe the answer is no, then there’s obviously no point in thinking about them, so why are you reading this post?

“Existence” here is used in the ordinary everyday sense of the word, according to which rocks and trees exist, you and I exist, your hopes and dreams exist, and the idea of a unicorn exists. Unicorns themselves do not exist and therefore it makes no sense to study their properties. (Though you can have fun inventing some properties for them.) By contrast, it makes perfect sense for geologists to study the properties of rocks, for botanists to study the properties of trees, for folklorists to study the properties of the idea of a unicorn, and for mathematicians to study the properties of the natural numbers.

An extreme skeptic might deny the existence of rocks. The only possible answers are: a) It’s hard to believe you’re serious, since you’ve been encountering rocks — just like you’ve been encountering numbers — your entire life. b) If you really are serious, I suppose your best strategy is to stop thinking about rocks, and leave them to those of us who find geology interesting. And c) Do not fool yourself into believing that your position is anywhere close to any mainstream school of thought.

Another extreme skeptic might deny the existence of numbers. I’ll leave it to my readers to replace rocks with numbers in the above retorts.

What else might one say to an extreme skeptic? Answer: One might attempt to acquaint him with Godel’s Completeness Theorem. (This is not the same as the far more famous Godel’s Incompleteness Theorem.) Here is (part of) what the Completeness Theorem says: First, without making any assumptions about existence, write down a list of axioms for the natural numbers. For example, write down the Peano Axioms. Then the Completeness Theorem tells you that as long as those axioms are consistent, there must be some mathematical structure that obeys those axioms. (Note that “be” is a synonym for “exist”.) The smallest of those structures (known as “models”) is our good old friend the natural numbers.

In other words, Godel’s Theorem tells you that if the Peano axioms are consistent, then the natural numbers must exist. (Don’t confuse the map with the territory! “Consistency” applies to the axioms; “existence” applies to the natural numbers themselves.)

On the other hand, we can also argue in the opposite direction: If the natural numbers exist, then the Peano axioms, being true statements about existing objects, must be consistent. An accurate map of an existing territory cannot contradict itself.

So — We know that the natural numbers exist because we know the Peano axioms are consistent. And we know that the Peano axioms are consistent because we know that the natural numbers exist. Does that sound circular? It’s not. Here’s the point: We have extremely good reasons for believing in the existence of the natural numbers (beginning with intuition, lifelong familiarity, and the fact that we seem to be able to discover their properties). We have (partly) separate extremely good reasons for believing in the consistency of the Peano axioms (beginning with intuition and the fact that they’ve never yet led us to a contradiction). The fact that our two beliefs reinforce each other — that if either is true, then so must be the other — should build up our confidence that the whole picture hangs together.

Now let’s get back to our extreme skeptic. He denies the existence of the natural numbers. We respond that Godel’s Completeness Theorem proves the existence of the natural numbers, as a consequence of the consistency of the Peano axioms. He now has only two recourses (other than to concede defeat). One is to deny the consistency of the Peano axioms, and the other is to deny the accuracy of Godel’s Completeness Theorem. Let’s see how those strategies are likely to work out for him.

Should he doubt the consistency of the axioms? The Peano Axioms lay out the rules of arithmetic that you’ve used your whole life; they say things like “Every number has exactly one immediate successor” and “x + (y+1) = (x+y) + 1”. People (and to some extent animals) have been applying these axioms, explicitly or implicitly, since long before the dawn of history and no contradiction has ever arisen; moreover, for what it’s worth, the consistency of these simple axioms is instantly clear to most people’s intuitions. If we were to jettison our belief that these axioms are consistent, then we’d pretty much have to give up all quantitative reasoning.

Well, then, should our skeptic doubt Godel’s Completeness Theorem? The theorem is proved using elementary notions about sets — the idea that it’s possible to talk about sets of things and about membership in a set, that it’s possible to form the union of two sets, and so on. This has nothing to do with the more esoteric subject of “axiomatic set theory”; instead, it uses only the most fundamental notions associated with forming collections of things. (These notions, in fact, are prerequisite for axiomatic set theory and therefore cannot depend on it.) Once again, if you were to abandon this sort of reasoning, you’d pretty much have to abandon reasoning altogether.

For anyone who accepts the simplest sorts of combinatorial reasoning, there is no longer an out. The natural numbers are real. Again, this says nothing about where they came from — be it Plato’s heaven, the minds of humans or the mind of God. We’ll get back to that in the next installment of this occasional series.

#### 58 Responses to “Basic Arithmetic: On What There Is”

1. 1 1 Ben

“Let me be clear that I am not (yet) asking in what sense the natural numbers exist”

Then

” We respond that Godel’s Completeness Theorem proves the existence of the natural numbers, as a consequence of the consistency of the Peano axioms.”

You can’t prove that they exist until you decide in what *sense* you are going to prove they exist, surely?

Or are you here explaining the sense in which they exist?

Saying you have proved something exists without saying what you mean by that is like refereeing a game of Calvinball.

2. 2 2 Steve Landsburg

Ben: The Completeness Theorem shows (roughly) that the natural numbers exist in (at least) any sense in which sets exist. (These are “sets” in the naive sense of the word, not in any technical sense).

3. 3 3 Chris

You say (I’m paraphrasing) that no mainstream school of thought would deny the existence of rocks. Are anti-realists really that uncommon? As far as I was aware, the philosophical debate between the various forms of realism and anti-realism was far from over.

4. 4 4 Harold

Someone once said something like “I think therfore I am”. I take this to mean that the only thing we can be absolutely sure about is that we have some sort of existence, otherwise we could not think. Everything else is filtered through our experience, and thus we cannot be quite so confident.

We are forced to act as though we “know” other things, otherwise, as Steve says, we could not really do much at all. We all act as though the world exists, rocks exist. We live our lives as though this was the case, but we do not actually know it to be so.

I am happy to accept the reality of the world, and the reality of Goedels completeness theorem, and thus the reality of natural numbers. I accept that they exist, because to do otherwise gets nowhere. But somewhere deep down is an acknowledgement that it is all based on an assumption somewhere. This makes absolutely no difference in all practical situations.

5. 5 5 Alan Wexelblat

I think I’m missing something in the circular nature of your self-reinforcing propositions. I know that a theorem can be tested. And I know that a set of axioms can be mathematically shown to be consistent or inconsistent. Therefore I have no problem accepting that “Godel’s Completeness Theorem proves the existence of the natural numbers”. All good and fine by me.

Where I fall behind is when you run the logic in the other direction. You write “we know that the Peano axioms are consistent because we know that the natural numbers exist.” But you’ve just used the consistency of the Peano axioms (call that CP) to prove the existence of the natural numbers (call that EN).

I’ve never been very good at logic but I think I remember that statements of the form “If CP then EN” don’t allow you to turn around and say “since EN then CP.” You need an independent source of truth for EN, don’t you? Or am I being too restrictive in trying to use first-order predicate logic here?

Finally, you say we (humans and animals) “have been applying these axioms, explicitly or implicitly”. I recall a lecture by Feynman in which he talked about quantum electrodynamics and part of his point was that QED was an explanatory framework for what happens when you toss a ball through an open window (as are Newton’s equations) and the truth or applicability of QED and Newton’s laws does not depend on us understanding them. Feynman argued that the theoretical frameworks applied regardless of our use of them.

Are you arguing the opposite? That is, we are applying the Peano axioms? Or are the Peano axioms like QED in that they’re an explanatory framework for what we do? (Unlike yesterday I’m not trying to tweak you here; I really don’t have an opinion on the answers to my questions.)

6. 6 6 Steve Landsburg

Alan Wexelblat: The “circular reasoning” is this:

a) Consistency of the Peano Axioms (plus Godel’s Theorem) implies the existence of the natural numbers
b) Existence of the natural numbers implies the consistency of the Peano axioms

These are separate assertions with separate reasoning behind them. Either implication stands on its own. Taken together, they say that if you believe in either of these two things, then you must believe in the other.

That is, we are applying the Peano axioms? Or are the Peano axioms like QED in that they’re an explanatory framework for what we do?

If I understand your question, the answer is clearly the latter. People have been working with numbers for thousands of years, but the Peano Axioms are only a hundred years old. The natural numbers are the (ancient) territory; the Peano Axioms are the (modern) map.

7. 7 7 Chris

Harold: That was Descartes in case you’re interested. You’re right, for the practical purpose of everyday life people abduce that rocks exist, even those who’re not entirely convinced they do. But we are hardly talking about normal circumstances if we are questioning the existence of numbers.
It just seems to me that Steve was slightly ridiculing anyone who isn’t a realist.

8. 8 8 KarlC

I read that using Löb’s theorem you can prove that 1+2=5. Would you write a post/comment explaining what this means and how it relates to the consistency of Peano arithmetic?

Thank you!

P.S.
I love your insightful explorations into the foundations of mathematics.

9. 9 9 Silas Barta

Let me be clear that I am not (yet) asking in what sense the natural numbers exist — whether they have existed since the beginning of time, or whether they exist outside of time, or whether they exist only in our minds. Those are questions that reasonable people disagree about (and that other reasonable people find more or less meaningless.)

But,

a) You can’t defend the proposition unless you know in what sense you mean “exist”.

b) The denizens I think you’re referring to were only disputing it in these “permissible” senses you’re referring to.

c) This is a completely different approach than you take in The Big Questions, in which your definition of exist is only that (from memory) “It exists if it’s consistent” (at least when the concept is applied to math), and then you port over whatever implications you want from other definitions of existence, such as when you criticize others, who may be using different definitions — see the link above.

I wish I could make money from this level of reasoning!

10. 10 10 Silas Barta

Ben: The Completeness Theorem shows (roughly) that the natural numbers exist in (at least) any sense in which sets exist. (These are “sets” in the naive sense of the word, not in any technical sense).

I’m pretty sure that technical proofs only prove things about technical meanings, not naive ones.

11. 11 11 jj

Steve, I don’t think the “extreme skeptics” believe anything different than you do. It seems like here you’ve just set up an easy straw-man to shoot down. All you’ve said is that ideas exist, natural numbers are an idea, therefore natural numbers exist. Nobody disagrees with that. The skeptics are interested in the SENSE of existence, that is, the distinction between the existence of “rocks and trees”, and “your hopes and dreams”. Which you just said it’s reasonable to disagree about.

————————

One thing I would like to know, which I’m unclear if you’re saying here, is: Do the Peano axioms exclusively and comprehensively define the natural numbers? Is there any other possible definition of natural numbers that would also agree with “the simplest sorts of combinatorial reasoning” but disagree with the Peano axioms?

12. 12 12 Ben

“The Completeness Theorem shows (roughly) that the natural numbers exist in (at least) any sense in which sets exist”

In what (naive, non technical, non mathematical) sense *do* sets exist? Haven’t you just replaced one fuzzy meaning of the word with another (And then made it more fuzzy with the word “roughly”)?

Possibly you mean in the sense of “the set of all Sumatran Tigers”.
But that would mean “as social/mental constructs that people constantly argue about”:
http://en.wikipedia.org/wiki/Species_problem

What I am saying is that in the naive sense, the word “set” is not well defined. It is only well defined in the technical, i.e. mathematical, sense.

Or to put it another way, I don’t deny the existence of rocks, but there is a fuzzy edge where “shale” gives way to “clay” (the process by which sedimentary rocks are formed makes that obvious I hope).

Doesn’t that apply to pretty much all naive, non-technical definitions of words?

13. 13 13 Super-Fly

Even if I assume that my mind is the only thing that exists, isn’t Godel’s Completeness Theorem still true? The existence of sensory experience is still true (well, my sensory experience at least). I suppose, if you really wanted to, you could deny the existence of sets, logic, or anything you want, but I’ve never seen a non-ridiculous denial of logic before.

14. 14 14 Steve Landsburg

Silas Barta: You attempt to paraphrase something from The Big Questions as follows:

“It exists if it’s consistent”

This is not the first time you’ve said this, and it’s as inaccurate as it ever was. Consistency applies to theories, whereas existence applies to structures. It makes no sense to say “It exists if it’s consistent”, because “It” is either a theory (in which case existence is not an issue) or a structure (in which case consistency is not an issue).

The actual argument you are so badly mangling went more like this:

a) The Peano Axioms appear to be consistent.
b) If they are in fact consistent, then their consistency would be explained by the existence of the natural numbers.
c) Therefore the consistency of the Peano Axioms is evidence for the existence of the natural numbers.

One could also argue:

a) The Peano Axioms appear to be consistent
b) The completeness theorem tells us that if the Peano Axioms are consistent then there exists a structure satisfying those axioms
c) Therefore the consistency of the Peano Axioms (together with the simple combinatorial reasoning behind the completeness theorem) implies the existence of the natural numbers.

I have on several occasions and in several forums tried to explain to you that these arguments are very different from “It exists if it is consistent” and that the latter is, in fact, not wrong but incoherent. Nevertheless you keep right on repeating the same damn misreading. You really do appear to be remarkably dense.

15. 15 15 Steve Landsburg

Ben:

What I am saying is that in the naive sense, the word “set” is not well defined. It is only well defined in the technical, i.e. mathematical, sense.

This is surely false. It is quite impossible to study, say, axiomatic set theory without having naive set theory as a basis. Naive set theory captures simple principles of reasoning, and it is the *starting* place for higher mathematics. There *is* no technical definition.

16. 16 16 Steve Landsburg

jj: I agree with you that essentially nothing in this post is controversial.

I am not sure I understand your final question. There are many different systems of axioms that accurately describe the natural numbers, though none of them describes the natural numbers completely. And for any axiomatic system that describes the natural numbers, there will be many other mathematical structures that it also describes.

Is that the answer to what you were asking?

17. 17 17 Silas Barta

@Steve_Landsburg:

I have on several occasions and in several forums tried to explain to you that these arguments are very different from “It exists if it is consistent” and that the latter is, in fact, not wrong but incoherent.

Then maybe you shouldn’t have grounded your proposition that natural numbers exist by equating this claim with their consistency. (Don’t have the book on me, but I can cite your sleight-of-mind when I get the chance — or you can save us the time and quote the passage I’m referring to? j/k, I don’t expect you to help anyone find passages in your book that could make you look bad.)

18. 18 18 Jonathan Kariv

I’m going along with pretty much everyone else who commented here and contending that most of your disagreement is over what exactly the word “exist” means. I mean the argument you make seems to convlude that the natural numbers exist in the sense of being ideas. Well OK but if I am using “exist” to mean something else then we’re really arguing semantics.

19. 19 19 Silas Barta

jj: I agree with you that essentially nothing in this post is controversial.

I’m pretty sure he said a bit more than that, such as identifying your criticisms as being directed at strawmen.

20. 20 20 Roger Schlafly

Steve, I think that your opinions in this post are controversial, and would not be shared by very many logicians.

You say that the consistency of Peano axioms must be taken without proof, and accepted as true based on experience. Then you use that consistency to prove the existence of the natural numbers. But there is the Gentzen’s consistency proof of Peano, and it is much more satisfactory that relying on the completeness theorem.

You also deny that there is a technical definition of a set. The whole point of ZFC is to give such a definition, and to give an axiomatic framework for using the definition. Naive set theory is only satisfactory if you are willing to ignore the sorts of issues that you raise in your posts on this subject.

21. 21 21 Seth

I’ve seen no evidence that suggests natural numbers don’t exist.

22. 22 22 Steve Landsburg

Silas Barta:

GOD, you’re dense:

your proposition that natural numbers exist by equating this claim with their consistency.

For the 9999999th, time: “The consistency of the natural numbers” is a meaningless phrase. You seem to think that a) it means something and b) I referred to it. NEITHER OF THESE THINGS IS TRUE. I have no idea what you think the phrase means, or what you think I think it means, or why you think I said it, other than the fact that you at some point got it into your head and you are too damn dumb to re-examine it.

23. 23 23 Steve Landsburg

Roger Schlafly: Gentzen’s proof relies on induction at least as high as epsilon-nought, which, for most people, is harder to swallow than the consistency of Peano arithmetic.

There is certainly no technical definition of a set. ZFC is a system of axioms for the universe of sets, but it doesn’t tell you what a set is. (Just as the axioms of group theory talk about the elements of groups, but do not tell you what those elements are.) To study anything interesting about ZFC, you have to study models of ZFC, and those models are constructed out of naive set theory.

24. 24 24 Silas Barta

@Steve_Landsburg: Sorry, the “consistency of natural numbers” should instead be something like “the consistency of arithmetic” — you know what passage I’m referring to (if you don’t, I’ll remind you when I get my hands on the book), and it’s _still_ wrong to equate existence with consistency by definitional fiat.

25. 25 25 nobody.really

I often find metaphysical discussions unilluminating. I can only get my hands around an issue when it has a context. Here’s the context of existence Landsburg has articulated so far:

Godel’s [Completeness] Theorem tells you that if the Peano axioms are consistent, then the natural numbers must exist.

Thus, I currently understand Landsburg to argue that natural numbers “exist” in the sense that “existence” is used in the proof of the Completeness Theorem. Whatever that may be.

At the risk of getting ahead of Landsburg’s discussion, I might better understand Landsburg’s concept of existence if he could answer the following question: So what?

For example, does Landsburg wish to expound upon the special, quantum nature of natural numbers as distinct from the continuous nature of rational numbers, real numbers, complex numbers, etc? And what propositions can Landsburg expound upon if we believe that natural numbers exist that Landsburg could not expound upon if were merely assume an understanding of natural numbers?

By comparing the consequences of a belief in existence to the consequences of some alternative (or lack of belief), I might better understand how Landsburg wants us to understand existence and what makes this concept important to this discussion.

26. 26 26 Steve Landsburg

Silas:

Sorry, the “consistency of natural numbers” should instead be something like “the consistency of arithmetic”

This is the very last time I am going to explain this to you.

The “consistency of arithmetic”, taken in context, means the consistency of Peano arithmetic, which is a theory.

I am offering the consistency of Peano arithmetic NOT as an argument for the existence of Peano arithmetic (we know Peano arithmetic exists, because it’s a series of marks on paper that we can write down and look at). Instead I am offering it as evidence for the existence of the natural numbers, which form a structure.

PEANO ARITHMETIC AND THE NATURAL NUMBERS ARE TWO VERY DIFFERENT THINGS. To say of any *one* thing that its consistency implies its existence would be incoherent. To say that the consistency of ONE is related to the existence of the OTHER is a meaningful statement.

That’s your first mistake.

Your second mistake is more important. You keep saying that I’ve *defined* existence via consistency. THAT’S NOT TRUE. I’ve said that the consistency of Peano arithmetic is EVIDENCE for the existence of the natural numbers. This is the case for two reasons:

1) The existence of the natural numbers implies the consistency of Peano arithmetic. If Peano arithmetic is consistent, then anything that implies its consistency is more likely to be true.

2) The consistency of Peano arithmetic, together with the completeness theorem, implies the existence of the natural numbers. If you believe in the completeness theorem and the consistency of Peano arithmetic, then you must believe in the existence of the natural numbers.

Those are not, I think, difficult arguments to grasp. (Argument 1) is the one I stress in the book, though I think I mention argument 2) there as well.) But you seem completely oblivious to the distinction between MAKING AN ARGUMENT and STATING A DEFINITION.

It is exactly as if I had said “Such and such a website provides evidence that crime is on the rise”, and you had decided that I’d defined “crime” to mean “website”. And then repeated it, ad nauseum, despite 11 gazillion attempts to correct you. I think maybe you are just too dumb to think about this subject.

27. 27 27 Ben

So when you said “”sets” in the naive sense of the word” you mean sets as in what I think of as set theory – sorry I didn’t get that. I thought by “non-technical” you meant in an informal, non-mathematical sense (hence my tiger and rock examples).

If I have understood you correctly. your statement:
“natural numbers exist in (at least) any sense in which sets exist”

translates as:
“what we normally think of as whole numbers, exist, in the same sense that what we normally think of as sets exist”.

Well, OK, in what sense DO sets exist?

It doesn’t really seem to advance the argument.

28. 28 28 Alan Wexelblat

Steve: OK thank you that’s much clearer. I think I was misinterpreting some of the ways you used terms like “prove” and “applying” in the original post.

29. 29 29 Roger Schlafly

Steve, I don’t know why you are picking on Silas, because I don’t accept your argument either. Your reason (1) is wrong because all statements imply the consistency of Peano arithmetic, whether true or false. Implying consistency is no evidence at all for the truth of a statement.

Your reason (2) is false because it only implies the existence of some model of the Peano axioms, and that model may not be the natural numbers.

Furthermore, I don’t see how your skeptic would be convinced by an argument of this type. He would say that you are assuming more than you are proving, and I would agree.

30. 30 30 Silas Barta

@Steve_Landsburg: I apologize for misquoting you; I should not have attempted any quote until I had the book in front of me, and I should have held off on the accusation until I could point to the exact passage.

That said, I do understand the difference between an argument and a definition; my point was that (I believe) there is a passage in TBQ in which you ground your _definition_ of (not argument for) existence in consistency. I have never said that any attempt, on your part, to argue for “A implies B” mean you’re defining B as A (or vice versa); every such inference was based, at least, on my belief that there was a specific point where you invoked an unhelpful definition, and I’m sorry for not providing more substantiation earlier.

However, in your latest reply, you are still making (what I regard as) a typical mistake, which is that you’ve switched senses in which you are using a term, and I can show it more clearly than before. You say:

(we know Peano arithmetic exists, because it’s a series of marks on paper that we can write down and look at)

But previously, you said Peano arithmetic was an immaterial, platonic object with a specific conceptual structure and boundary. How did it now become transcribable series of marks on paper for which looking at constitutes proof? If making the marks and looking at them suffices to prove their existence, this is a very difference concept than that immaterial-but-existent thing you had established before. I submit you are equivocating again.

(And on a side note, before you write me off as “too dumb” to understand this, keep in mind that, a) you already called me an expert in information theory, and b) my arguments in the thread from last October matched those of the commenter Snorri_Ghodi, whom you already regarded as an intelligent commenter. And let’s not forget the numerous commenters on this thread who are as unimpressed as I am by your deductions and definition choices here.)

31. 31 31 Steve Landsburg

Silas: I appreciate the apology. Part of my snit was due to the fact that you’ve continued to misquote me on this point, in many times and many places, despite numerous corrections.

my point was that (I believe) there is a passage in TBQ in which you ground your _definition_ of (not argument for) existence in consistency

I will be very surprised if there is a passage that seems to say this. It certainly is not what was intended.

Now—-here you go making the exact same mistake AGAIN:

But previously, you said Peano arithmetic was an immaterial, platonic object with a specific conceptual structure and boundary.

No, I said that the natural numbers are an immaterial, platonic object (not in this post, but elsewhere). Once again, you’ve confused Peano arithmetic with the natural numbers.

As Snorri Godhi — a) Snorri was a useful, intelligent commenter, and I miss him. b) I did get mighty pissed at him when he (baselessly) accused me of trying to lie to him about standard textbook material. c) I am aware that you thought that you and Snorri were saying the same thing. I take this as further evidence for how confused you are/were, since you were in fact not saying the same thing at all. d) Intelligent people disagree with me all the time, and they do so intelligently. That doesn’t make you any less dumb.

As for calling you an expert in information theory, I apologize for the error.

32. 32 32 Silas Barta

No, I said that the natural numbers are an immaterial, platonic object (not in this post, but elsewhere). Once again, you’ve confused Peano arithmetic with the natural numbers.

Even if I have, that still wouldn’t allow you to assert such a difference between the two. If Peano arithmetic _isn’t_ a immaterial, platonic object, then where is it, so that I can touch it? And if you use a classification of Peano arithmetic that isn’t refuted by that test, then what relevant difference do you think exists between that class and the natural numbers?

I am aware that you thought that you and Snorri were saying the same thing. I take this as further evidence for how confused you are/were, since you were in fact not saying the same thing at all.

Of course we weren’t saying the _exact_ same thing, and I wasn’t claiming otherwise. But you can’t deny that there was significant overlap (e.g. in the relevance of the fact that computers work). If an intelligent commenter and I independently come up with many of the same inferences, why are you using those inferences — made *long* before you crowned me information theory expert! — as evidence of my stupidity?

Intelligent people disagree with me all the time, and they do so intelligently. That doesn’t make you any less dumb.

I’m not the only one making the criticisms of the choices of definitions and inferences thereon that you’ve made here — are all of those other commenters dumb too?

As for calling you an expert in information theory, I apologize for the error.

Fine, just let me know which information-theoretical statement I made that changed your mind.

33. 33 33 Super-Fly

“Even if I have, that still wouldn’t allow you to assert such a difference between the two. If Peano arithmetic _isn’t_ a immaterial, platonic object, then where is it, so that I can touch it?”

Hey Silas, sorry about that, Prof. Landsburg let me borrow Peano arithmetic and I still haven’t given it back. You can stop by if you want to see it, but my dog chewed it up a bit.

I’m a bit confused as to what’s being argued at this point.

34. 34 34 XiXiDu

@Silas Barta

If Peano arithmetic is a description of a timeless pattern that are the natural numbers then the description is itself a subset of what it describes and therefore having the same ontological status. That is, there is no difference between the theory and the model that can be adequately differentiated by the concept of existence. Given that we assume a mathematical universe, the map and the territory both exist as timeless patterns. Is that what you are trying to argue?

35. 35 35 XiXiDu

@Steve Landsburg

The stones (atoms, natural numbers) exist independently of human thought-action while the labels (Peano axioms) describing the stones are human dependent. That is, the labels are emergent qualities of the stones but differentiated from human thought-action dependent subsets. Existence is defined by differentiating what is ‘human dependent subsidiary output by thought-action’ from the top-level foundations of the stones that make up humans and subsequently the labels. Therefore stones exist as models independently of the theory of labels by which humans are describing them. Is that what you are arguing?

36. 36 36 Steve Landsburg

Silas:

And if you use a classification of Peano arithmetic that isn’t refuted by that test, then what relevant difference do you think exists between that class and the natural numbers?

Part of the reason you’re so frustrating to talk to is that this is yet another question I’ve answered for you several times in several forums, without any apparent success in penetrating your skull. You can describe Peano arithmetic with a finite string of symbols. You can’t describe the natural numbers finitely, or even recursively.

The difference between you and Snorri Godhi — and for that matter between you and pretty much anybody else who’s ever commented here — is the way stuff just bounces right off you. You cling stubbornly to fundamental conceptual errors (e.g. your confusion between theories and structures) no matter how many times and no matter how carefully they’re explained to you. Your reading comprehension is practically nil. And now I think we can add another — your bizarre inference that if you agree with Snorri Godhi (even if you *don’t actually understand what he’s saying*) that this somehow you makes you less dumb.

37. 37 37 Silas Barta

@Steve_Landsburg:

You can describe Peano arithmetic with a finite string of symbols. You can’t describe the natural numbers finitely, or even recursively.

What does that have to do with the issue of whether Peano arithmetic is an immaterial, platonic object? That would speak to the _separate_ issue (that you also fumbled) of whether PA has finite complexity. It has nothing to do with whether it’s an immaterial/platonic object.

Just out of the blue you said that PA is a set of marks on paper that we can look at, and those marks … somehow constitute their existence. I’m pretty sure that *nobody* would agree with you that those marks *are* Peano arithmetic (as opposed to simply representing it). Maybe that’s not what you really meant, but then, why say it, right?

I’m not going to reply to your future mudslinging — the brilliant entourage of (other) posters here will certainly be around shortly to substantiate your claim to superior insight on this issue, and that should be more than sufficient to make whatever point you intended on that matter, and more to your standards.

38. 38 38 Super-Fly

Silas:

Does Beethoven’s 9th Symphony exist? Where is it? Surely, it’s not the dots and squiggles on a piece of paper. Is it the vibrations in the air when a bunch of musicians get together? Where does it go when those vibrations dissipate? Does it still exist? Of course! Beethoven said it did. We know it exists because musicians still play it and we have the musical scores. Does Beethoven’s 23rd symphony exist? No, because Beethoven never composed it (or if he did, he should have written it down).

So, the \$64,000 question: Does Peano arithmetic exist? … ¡¡¡Yes!!! This should not be too exciting. Plus, this post is about the existence of the natural numbers, NOT Peano arithmetic.

The important thing is whether or not the Peano axioms are consistent. Aristotelian physics, for example exists as a theory (just like PA!!!) but it doesn’t describe empirical evidence so… it’s not that helpful. The Peano axioms certainly appear to be consistent. If you find a contradiction, that would really be great. Now, because PA is consistent, our old buddy Godel tells us that there exists a mathematical structure that obeys PA. The natural numbers is the smallest such structure (this is all from the post).

And one last thing I’m still confused as to what the actual argument is!! I’m not trying to be snarky. Are you suggesting that the natural numbers don’t exist? Are you unhappy with the definition of ‘Existence’? Are you denying that Godel is wrong or that logic doesn’t exist? The only thing you seem to have claimed is that Prof. Landsburg is wrong. Not about anything particular… just wrong. I realize he has the home-blog advantage, but still, some specifics would be nice.

P.S. Prof. Landsburg, feel free to correct me anywhere if I made a mistake.

39. 39 39 Steve Landsburg

Superfly:

P.S. Prof. Landsburg, feel free to correct me anywhere if I made a mistake.

This was quite accurate and well put. Nothing in this post was intended to be controversial. It was meant as an extremely elementary post covering some standard textbook notions, as a precursor to future posts where I’ll want to reference these notions. You summed it up nicely.

40. 40 40 Josh

A more generalized comment: the last two days of this blog have been extremely interesting, on many levels. Steven debating with a Princeton professor over a fundamental economic issue, then today shows that Steven is not afraid to use intense rhetoric (and has towards me before on race issues). What a great blog!

41. 41 41 Silas Barta

@Super-Fly:

Which of my points is your last comment intended to be responsive to, but not answered in the link I gave? (Looks like Steve_Landsburg is agreeing with you mainly because he regards you as basically on his side, irrespective of why you are.)

In my last comment, I wasn’t criticizing the original post, just (what I regard) as the latest instance of equivocation on Steve_Landsburg’s part. He said, quite clearly,

“(we know Peano arithmetic exists, because it’s a series of marks on paper that we can write down and look at)”

Now, to the extent that PA exists, it exists as a non-material entity. There is no object you can point to to say — “Oh, that thur’s our Peano arithmetic”. No, it is a non-material object (I’d prefer “pattern”) that can be instantiated in places, but those instantiations are not PA itself. So that comment of his is just wrong.

Unable to admit a contradiction, correct his mistake, and move on, Steve_Landsburg tried to actually *defend* that statement by saying:

“You can describe Peano arithmetic with a finite string of symbols. You can’t describe the natural numbers finitely, or even recursively.”

which is clearly non-responsive, as it speaks to the issue of PA’s complexity or describability, not its status as a material vs. immaterial object. And then Steve_Landsburg tries to ground crucial parts of his argument with “well, hey, you can see PA right on the paper there!” — which is asserting that PA is material, which you rightly mocked when you said:

“Hey Silas, sorry about that, Prof. Landsburg let me borrow Peano arithmetic and I still haven’t given it back. You can stop by if you want to see it, but my dog chewed it up a bit.”

So, whatever the merit of Steve_Landsburg’s original post, clearly he’s erred or equivocated thereafter, and he can’t just shrug his shoulders and say, “Well, Super-Fly’s on my side and rephrased my point, so I get to ignore all that.”

As for the problems with his original post, did you understand the points first comment?

And as for my general position on the issues you raised: Yes, things like Beethoven’s symphony exist, for the appropriate definition of “exist”. You have to be clear about which sense you mean “exist”, and make sure you don’t try to use the properties of one when you’ve only prove the other. Nor should you dictate the meanings of “exist” that others are using, just to turn what they said into a contradiction.

Hence, my persistent disagreement with Steve_Landsburg.

42. 42 42 Silas Barta

Okay, I’ve found the passage in TBQ I was referring to with this statement:

“”

and which Steve_Landsburg took so much offense at. Here it is, on page 90 (turns out you said “is real” instead of “exists” … oh, how I butchered your words!):

“Admittedly, the word real is a little vague. If it makes you feel better, let’s give it a precise definition: ‘The natural numbers are real’ means that the laws of arithmetic are consistent. [italics original]”

So, you didn’t define “X exists” as “X is consistent”, but you certainly defined “X exists” as “something related to X is consistent”, which isn’t much better. It would be one thing if you were saying that “something related to X is consistent” *implies* “X exists”. And indeed, that’s what you do here. Still, you did use definitional fiat to do your work, and thereby reduce the claim about realness of natural numbers (in whatever sense you want it to mean) to an easy claim about arithmetic.

(It’s a bit disingenuous to say you didn’t have any idea what passage I was referring to, by the way. Surely a man with your tremendous brain saw at least a similarity … you did write the book, after all.)

43. 43 43 Silas Barta

Sorry, the empty quotes above should contain:

“This is a completely different approach than you take in The Big Questions, in which your definition of exist is only that (from memory) “It exists if it’s consistent” (at least when the concept is applied to math), and then you port over whatever implications you want from other definitions of existence, such as when you criticize others, who may be using different definitions — see the link above.”

44. 44 44 Super-Fly

OK Silas, I need to get some sleep. I can’t keep playing this game, as much as I would like to. I also don’t have time to peruse the bottomless links on the blogo-tubes. The stuff that Prof. Landsburg is saying in this post is truly not groundbreaking or controversial. Over-analyzing his word use does not make you right. We can also nitpick our respective definitions of existence all night long, but we won’t get anywhere. On the stuff about existence, he said he will talk about it later. Have patience.

45. 45 45 Steve Landsburg

Silas:

“Admittedly, the word real is a little vague. If it makes you feel better, let’s give it a precise definition: ‘The natural numbers are real’ means that the laws of arithmetic are consistent. [italics original]”

So, you didn’t define “X exists” as “X is consistent”, but you certainly defined “X exists” as “something related to X is consistent”, which isn’t much better. It would be one thing if you were saying that “something related to X is consistent” *implies* “X exists”. And indeed, that’s what you do here. Still, you did use definitional fiat to do your work, and thereby reduce the claim about realness of natural numbers (in whatever sense you want it to mean) to an easy claim about arithmetic.

Except for this:

1) “The natural numbers” and “the laws of arithmetic” are two entirely different things, with “the laws of arithmetic” in this case referring to the manipulations allowable by the Peano axioms.

2) I have told you 11,000 times, most recently about 6 times in *THIS THREAD*, that they are two different things.

3) Therefore the passage does not relate “X exists” to “something about X is consistent” but rather relates “X exists” to “something about Y is consistent”.

4) I stand by my conclusion that you are thoroughly ineducable.

46. 46 46 Skinner

Steven, I don’t know what this ‘extreme sceptic’ guy thinks, he seem to be a straw-man to me. But if he really exists, he surely
uses the word ‘exist’ in a different sense than you, so he will not be very interested in playing the rest of this game by your rules.

The way you expropriate the word ‘exist’ is almost rude. I don’t even think your informal notion can be made logically consistent.

47. 47 47 Steve Landsburg

Skinner: The extreme skeptic is indeed a straw man. The point of the post was to clarify some simple points that most people don’t often think about, and for that purpose it seemed useful to introduce the straw man and explain how one might respond to him. I use this technique all the time when I teach mathematics. To understand, say, continuity, it is useful to say “Suppose you met an extreme skeptic who didn’t believe that the function f(x)=x is continuous. How might you respond to him?”

48. 48 48 Silas Barta

@Steve_Landsburg:

Therefore the passage does not relate “X exists” to “something about X is consistent” but rather relates “X exists” to “something about Y is consistent”.

It doesn’t _relate_ them, it _defines_ one as the other. Your 1-4 are non-responsive. The quote shows that you did, in fact, definte “X exists (well, is real, but whatever)” as “something related to X is consistent” (despite your pretense that you said nothing like that, or were unable to remember that passage). That’s not showing implication, that’s *defining* the two as equivalent.

Anyone want to try their hand at defending this sleight-of-mind?

49. 49 49 Steve Landsburg

Silas: Obviously you are never ever ever going to get it through your monumentally thick skull that “the natural numbers” and “the laws of arithmetic” are two different things.

That would be the same thick skull that resists every attempt to grasp that this is not an arbitrary definition; it’s the one definition that’s naturally motivated by Godel’s Theorem.

And, come to think of it, the same thick skull that houses the bizarre belief that the natural numbers can be defined by axioms, which you’ve also repeated umpteen gazillion times, including in responses to posts (on other forums) where it had JUST BEEN EXPLAINED TO YOU that this was an error.

You are much much much too stupid to be thinking about this, and I’m certainly through trying to re-explain thest things to you.

50. 50 50 Silas Barta

Oh, okay, thanks for clearing that up. Now I understand why you get to prove something’s existence by defining it as being equivalent to something trivial, as do all the other commenters here.

51. 51 51 Super-Fly

“I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.”

Charles Babbage

52. 52 52 Steve Landsburg

Super-Fly: Thank you for this quote. I am sure I will have reason to use it in the future.

53. 53 53 Benkyou Burito

Wait a minute. I thought Godel’s second incompleteness theorem concluded that consistency cannot be proven within PA arithmetic. And proofs that show that it can rely on some very subjective interpretations of what “finite” means.

But I think I found where you and I fundamentally disagree, something I was trying to get at last week.

“Unicorns themselves do not exist and therefore it makes no sense to study their properties.”

How can you be so sure? This is not a matter of opinion but one of fact, the strongest argument you can make on unicorns is that there is no credible reason at all to believe that they exist.

We have no reason to believe that the element Ununoctium exists, but we study its properties and will know it when we find it. And if we wish, we can produce it in a lab. I’m pretty sure we could produce a unicorn too.

I’m not sure if it is your reality or your theory that depends so strongly on definitive-absolute statements. You are certain that zero is a number, others are not so sure. Early Russian computer languages had a version of boolean with three states, yes-no-maybe.

54. 54 54 watercott

Hopefully it isn’t too late and you’re still replying here. I am missing one piece of the puzzle in your argument I think.

Steve:” For example, write down the Peano Axioms. Then the Completeness Theorem tells you that as long as those axioms are consistent, there must be some mathematical structure that obeys those axioms. (Note that “be” is a synonym for “exist”.) The smallest of those structures (known as “models”) is our good old friend the natural numbers.

In other words, Godel’s Theorem tells you that if the Peano axioms are consistent, then the natural numbers must exist.”

It seems to me that you can’t actually make this statement without somehow showing that the natural numbers are one of the models that obey the Peano axioms. i.e. how do we know all of the statements that the Peano axioms make are true about the natural numbers.

I had always thought that the Natural Numbers, as formally defined, were actually derived from axioms, and that there relationship with the natural numbers, as defined by, say, counting sheep, were just taken on faith to correspond with the formal Natural Numbers (albeit faith very well informed by experience). Is there something more to it?

55. 55 55 Steve Landsburg

watercott:

It seems to me that you can’t actually make this statement without somehow showing that the natural numbers are one of the models that obey the Peano axioms. i.e. how do we know all of the statements that the Peano axioms make are true about the natural numbers.

If the Peano Axioms are consistent, then Godel’s Theorem guarantees the existence of a model (in fact of many models). The smallest of these models is in fact exactly the same as what most people think of as the natural numbers. (Though I suppose you’re right that this does require a bit of extra argument.)

I had always thought that the Natural Numbers, as formally defined, were actually derived from axioms,

This is certainly false. Godel’s incompleteness theorem guarantees that there is no way to define the natural numbers via axioms.

56. 56 56 watercott

I’m a bit removed from my math degree, so bear with me.

Is there some proof that the natural numbers are a model described by the Peano Axioms, though (and in addition, that they are the smallest such model)? This is probably out of the scope of this simple explanation – I’m just curious whether a proof exists, or if it is agreed upon in some other fashion.

Like I mentioned, I had thought that it was almost by definition – but if that’s incorrect and the Peano Axioms don’t define the natural numbers, how ARE the natural numbers defined?

57. 57 57 Steve Landsburg

Watercott:

how ARE the natural numbers defined?

You have to start somewhere with undefined terms. A perfectly good place to start is with the natural numbers.

Or, if you prefer, you can start by taking sets and membership as undefined terms and then construct the natural numbers in any of several (essentially equivalent) ways as sets. For example, you can define 0 to be the empty set, 1 to be the set {0}, 2, to be the set {0,1}, 3 to be the set {0,1,2}, etc.

58. 58 58 dave

The small number of mathematicians who advocate ultrafinitism reject Peano’s axioms because the axioms require an infinite set of natural numbers.

-from the peano wiki

its true. people have been thinking about numbers for millions of years. peano does seem to be the second, third, or fourth guy to have written about the unification of math and logic.

i fully agree with you doc, we do need to start somewhere and having some common language with which to talk about our new ideas is a fundamental need.
my problem with the peano model first surfaced in high school algebra. why on earth does dividing by zero not compute?
if youre going to arbitrarily include that rogue zero in your set of naturals…you better damm well be able to work with it.

i am one of those skeptics. if a number is so big, that a person could not compute it in a lifetime, does it exist?
also. pi…infinite? cmon…its just a simple circle…there must be some simple, concrete, easy, natural and finite solution to the ratio between its radius and its circumference.
in nature, a gas bubble in solution would be perfectly spherical? or not? a quantum function that most closely resembles a sphere? it seems to me that if circles form in nature, there must be some easy and simple reason for it. 22/7 works for me. =]

Comments are currently closed.