Basic Arithmetic, Part III: The Map is Not the Territory

Today let’s talk about consistency.

Suppose I show you a map of Nebraska, with as-the-crow-flies distances marked between the major cities. Omaha to Lincoln, 100 miles. Lincoln to Grand Island, 100 miles. Omaha to Grand Island, 400 miles.

You are entitled to say “Hey, wait a minute! This map is inconsistent. The numbers don’t add up. If it’s 400 miles straight from Omaha to Grand Island, then there can’t be a 200 mile route that goes through Lincoln!”

So a map can be inconsistent. (It can also be consistent but wrong.) Nebraska itself, however, can no more be inconsistent than the color red can be made of terrycloth. (Red things can be made of terrycloth, but the color red certainly can’t.)

With that in mind, suppose I give you a theory of the natural numbers — that is, a list of axioms about them. You might examine my axioms and say “Hey! These axioms are inconsistent. I can use them to prove that 0 equals 1 and I can also use them to prove that 0 does not equal 1!” And, depending on the theory I gave you, you might be right. So a theory can be inconsistent. But the intended model of that theory — the natural numbers themselves — can no more be inconsistent than Nebraska can. Inconsistency in this context applies to theories, like the Peano axioms for arithmetic, not to structures, like the natural numbers themselves.

(See yesterday’s post for more on theories and models.)

Philosopher Alfred Korzybski admonishes us to remember that the map is not the territory. The theory is the map. The model is the territory. The hallmark of Internet crankery in this area is the refusal to distinguish them.

Whenever someone drones on at length about “the consistency (or inconsistency, or possible consistency, or possible inconsistency) of the natural numbers”, you’ll know he’s blathering. The concept simply doesn’t apply. Nebraska can’t be inconsistent. Only a description of it can be inconsistent.

(This brief snarky detour is brought to you by the small but determined band of commenters who consistently and vocally ignore this distinction in order to spout nonsense both here and on other blogs. I am not talking about anyone who’s commented here lately.)

It can be easy — and therefore entirely excusable — to get confused about this issue because in informal discussions of this subject — as in every other informal discussion of every subject in the English language — a single word can have multiple meanings. That sometimes happens with the word “arithmetic”. The phrase “Peano arithmetic” is the name of a theory — a list of axioms. On the other hand, some of us (me, for example) sometimes use the word “arithmetic” (a bit sloppily) to refer to a structure, namely the natural numbers themselves, which form a model of Peano arithmetic. Fortunately, the meaning is usually clear from context. If someone talks about “the consistency of arithmetic” you know that he’s talking about the theory (unless of course you have reason to suspect that he’s badly confused).

Now then. Let’s start with a theory. There are (at least) two sorts of questions you could ask about this theory. First: Is this theory consistent? In other words, is the theory free of self-contradiction? Second: Does this theory have a model? In other words, is there actually some structure that this theory describes?

If you’re given a map, the first question is like asking whether all the distances add up. The second question asks whether this is a map of someplace that actually exists or just a figment of the mapmaker’s imagination.

Inconsistent theories, obviously, have no models. A map that makes no sense cannot be a map of anything.

What about consistent theories? A consistent theory might a priori have either no models, or just one model, or many models.

The first possiblity is ruled out by Godel’s Completeness Theorem, not to be confused with the far more famous Godel’s Incompleteness Theorem. According to Godel’s Completeness Theorem, every consistent theory has at least one model. This is like saying that if you draw a map, and if nothing about the map is self-contradictory, then somewhere there is a territory that corresponds to the map. You should find this at least mildly surprising, but there it is.

If your theory is a theory of the natural numbers — in other words, if the natural numbers constitute a model for your theory — then the Lowenheim/Skolem Theorem says that your theory has a jillion other models as well. In other words, your map applies equally well to a jillion different territories. And there is no way, just by looking at the map, to tell those territories apart.

In other words, no theory — no list of axioms — can be a complete description of the natural numbers. It will always be a partial description, which applies equally well to a jillion other mathematical structures that look a little bit like the natural numbers but mostly a whole lot different.

Today’s moral: The map is not the territory. The map — the set of axioms — is either consistent or it’s not. If it’s inconsistent, there’s no corresponding territory. If it’s consistent, there are many corresponding territories and the map can’t tell you which one you’re in. That’s a fundamental limiitation on the power of the axiomatic method to describe a mathematical structure such as the natural numbers. It means there’s more to the natural numbers than any set of axioms can possibly know about.

Still to come: Are the Peano axioms consistent? Do the natural numbers really exist? And just how much about the natural numbers is any axiomatic system doomed not to know?

And finally: Thanks to those of you who encouraged me to continue this series. Let me know if you want still more.

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35 Responses to “Basic Arithmetic, Part III: The Map is Not the Territory”


  1. 1 1 Phil Maymin

    Great, great, keep it coming!

    BTW why do you call it a “model” instead of “reality”? Nebraska isn’t a model. Wouldn’t it make more sense to say that theories can apply to particular aspects of reality?

  2. 2 2 Phil Maymin

    Also if you could comment on “alternatives” to theories, if there are any. After all, you and I understand each other when we talk about natural numbers, so if our method of formally describing them with axioms fails to be both consistent and complete, are there other methods that do a better job?

  3. 3 3 Tobias Schmidt

    I’ve been enjoying this series of entries immensely so far so please keep them coming!

  4. 4 4 Coupon_Clipper

    Again, this is a great explanation. Keep it going.

    I understand what a non-standard model of PA is, but what about a non-standard model of ZFC? And what if we’re allowed to use 2nd order logic? What do the non-standard models look like then?

    Finally, I understand that you can’t prove Goodstein’s Theorem or HvH under PA w/ 1st order logic even though we know they’re true. Are there similar results that you can’t prove under PA (or ZFC) using 2nd order logic? (I imagine that this is much harder to answer for reasons you detailed in the post about SOL.)

  5. 5 5 loveactuary

    I cannot get enough of this please keep it up.

  6. 6 6 Harold

    Theorem: statement proved by previously accepted theorems or axioms. Godels completeness theorem is therefore based on the list of axioms in the same way that a theory is. How can we use this to tell us anything about the “trueness” of the initial axioms? Are we not in a circular argument? If we wished to say the natural numbers exist, can we use Godels completeness theorem to say that if the theory is self consistent, then there must be a model – i.e. the natural numbers? Surely, Godels theorem is based on axioms, one of which is perhaps that natural numbers exist?

  7. 7 7 Adam Chlipala

    This discussion seems to cover only first-order logic. In higher-order logic with inductive definitions, you can _construct_ the natural numbers, such that the “axioms” can then be proved in terms of the constructed “model.” Most real mathematical proofs will benefit from using higher-order logic, anyway, so why not just ditch the hang-ups that come from the first-order restriction?

    Here you do run into problems regarding models for your higher-order logic itself, but most users of such logics ignore these issues in practice, while reaping the benefits of being able to reason about domain-specific models directly.

  8. 8 8 John Lynch

    Yes. More please.

  9. 9 9 Marcelo

    Keep ‘em coming, these are great.

  10. 10 10 Dave

    Apologies to the rest but you’re losing me on these….

  11. 11 11 Matthew

    Keep them coming.

  12. 12 12 Don

    Steve,
    Would you consider extending this discussion to economic theories and models? Or does your earlier post that economics is not mathematics apply?

  13. 13 13 Jonathan Kariv

    My 2 cents. I’m enjoying these, keep going

  14. 14 14 Alan Wexelblat

    if you draw a map, and if nothing about the map is self-contradictory, then somewhere there is a territory that corresponds to the map. You should find this at least mildly surprising

    I do find this mildly surprising and so require some clarification. Let’s say that I, like many a gamer, invent a map of my fictional world for my D&D players. Let’s further say it doesn’t contain self-contradictions. Does this theory say that I have now drawn a map of someplace on Earth? That can’t be true, as I could draw a map of a world that is completely a desert.

    Have I drawn a map of some other world that exists in the universe? I suppose that’s possible, but let’s further say that my map violates rules that we know exist for desert planets (such as surface smoothness versus cratering or the presence of dunes). I have violated some constraints of geophysics but not constraints of consistency in map-making.

    How, then, can my map describe any territory anywhere?

  15. 15 15 Neil

    The map is not the territory because there is no territory. Its maps, all the way down. Yesterday, when I gazed over the Cascade mountains from a perch high on Mt. Rainier, I was simply looking at a map–a map cast on the retinas of my eyes. I can achieve finer and finer maps by looking closer and closer or by employing other sensory apparatus, but all I know are maps.

    A territory is the “a coastline”. It appears to be a definitive object only according to the scale of the map you are using.

    I suspect this is true in mathematics as well.

  16. 16 16 tkw

    i like this series a lot. but do i like it more than what you would be writing about otherwise?

  17. 17 17 Clifford Nelson

    First, thank you for discussing this, but I have these questions:

    “A map that makes no sense cannot be a map of anything.”

    So if something fails to make sense it does not exist? Or is it that something that does not make sense just can’t be mapped?

    Can numbers be used to map everything including emotional and physical in the universe? If not, is there alternate model that can?

  18. 18 18 Steve Landsburg

    Alan Wexelblat: Godel’s Completeness Theorem applies to axiomatic systems, not literal maps.

    The theorem says that if you write down a list of axioms, and if those axioms are not self-contradictory, then there is some mathematical structure that they describe.

    I was offering a metaphor: An axiomatic system is like a map, a mathematical structure is like a territory, and Godel’s theorem is like saying that *if* your map is consistent, then there must be some corresponding territory.

    The fact that this is not true of literal maps and literal territories is part of why it’s at least mildly surprising.

  19. 19 19 Harold

    I though about it this way. I draw a map of an imaginary treasure island, with coastline, palm trees and treasure chest. If this is not contradictory, then it represents a model somewhere. This does not mean that the actual island exists, but in a field, any field, there is this patch of grass that represents the trees, that patch of grass that represents the coastline etc. Therefore, there is a territory that corresponds to my map.

  20. 20 20 Steve Landsburg

    Harold: Your is a pretty good analogy. A slightly better one would be to say that somewhere there is a place that corresponds to your map, after assigning possibly non-standard meanings to the symbols on your map. So maybe all of the “rest stop” symbols on your map represent trees, and all of the rivers on your map represent highways, and all of the desert areas on your map represent oceans.

    Of course, there might in fact *not* be such a place. But the Godel theorem tells us that in the mathematical situation, there *is* such a place (in fact many such).

  21. 21 21 Super-Fly

    I’m a fan of the math posts as well.

  22. 22 22 Seth

    Stephen Wright has a map of the United States that is actual size.

  23. 23 23 Phil Maymin

    Seth is right. Last summer Stephen folded it.

  24. 24 24 Benkyou Burito

    I accept that you have no responsibility or disposition towards indulging me on this, but maybe this topic is interesting enough from a different perspective. I think I’m with Neil on this one.

    You state that the map is the theory and that reality is the model. You then suppose that the blathering of people who seek to hedge at this position just are not differentiating between the two. But yesterday you made a very clear point to name “the real reality” as the thing that interests you when considering models to choose from.

    But models are created aren’t they? To explain or to establish a ‘reality’ that can not otherwise be measured. Are not economic models formulated the same way?

    I am deeply suspicious of your conclusions in many cases not because I don’t know the difference between what your theory is and what reality is, but because of the pre-supposition that reality can be modeled.

    As it happens there are scientists of the most prestigious caliber contemplating the possibility of a rout through Lincoln that is half the distance of the natural distance between the two cities.

    As it stands we have little (not zero) reason to believe that space can folded to allow for a smaller travelling distance than the natural distance between two points. But if your model IS reality and not just a creation of human ability, then it seems you should have to allow for the possibility.

  25. 25 25 nobody.really

    This brief snarky detour is brought to you by the small but determined band of commenters who consistently and vocally ignore this distinction in order to spout nonsense both here and on other blogs. I am not talking about anyone who’s commented here lately.

    Sorry, I’ve gotten preoccupied elsewhere. But I’ll try to make up for lost time.

    Going (continuing?) a bit off-topic:

    Neil:

    The map is not the territory because there is no territory. Its maps, all the way down. Yesterday, when I gazed over the Cascade Mountains from a perch high on Mt. Rainier, I was simply looking at a map–a map cast on the retinas of my eyes. I can achieve finer and finer maps by looking closer and closer or by employing other sensory apparatus, but all I know are maps.

    A territory is the “a coastline”. It appears to be a definitive object only according to the scale of the map you are using.

    I suspect this is true in mathematics as well.

    Benkyou Burito:

    I think I’m with Neil on this one.

    You state that the map is the theory and that reality is the model. You then suppose that the blathering of people who seek to hedge at this position just are not differentiating between the two. But yesterday you made a very clear point to name “the real reality” as the thing that interests you when considering models to choose from.

    But models are created aren’t they? To explain or to establish a ‘reality’ that can not otherwise be measured. Are not economic models formulated the same way?

    I sense there’s a cadre of us who have a greater affinity for epistemology than math theory. And, who knows, we’ve been discussing ideas of proof and knowledge, so maybe the subjects overlap more than I imagine.

    Anyway, my defenses go up when I hear people discussing “reality” absent a discussion of empiricism. I share Neil’s view that humans do not know “reality.” We have experiences from which we create “maps” in our heads to navigate our world. We can pursue new experiences with which to refine our maps, but it seems unlikely that we’ll ever create a map that matches the world perfectly.

    That said, I regard math as fundamentally different than the world of experience. The world of mathematical proofs may have undiscovered patterns and unexplored assumptions, but it has no error terms and darn few hidden caves. Metaphorically, the straight coastline of mathematics appears the same regardless of scale. As a result, I suspect that we may mislead ourselves when we take the limits of our ability to know the physical world and apply those limits to the study of mathematics.

    And that creates an irony. This blog attracts me because of Landsburg ’s skill at making esoteric ideas concrete. Yet I suspect that aspects of mathematics do not lend themselves to concrete explanation. (Indeed, I regard mathematics as the study of principles abstracted from the physical world; I sense Landsburg has a somewhat different metaphysical view on that.) Like others, I’ve had difficulty recognizing when Landsburg means to speak metaphorically – with all the limitations that metaphors entail – and when he means to speak literally. I appreciate his subsequent clarification of Godel’s Completeness Theorem:

    I was offering a metaphor: An axiomatic system is like a map, a mathematical structure is like a territory, and Godel’s theorem is like saying that *if* your map is consistent, then there must be some corresponding territory.
    The fact that this is not true of literal maps and literal territories is part of why it’s at least mildly surprising.

    (Ironically, some cosmology theories suggest that every possible combination of matter will almost certain occur somewhere, and probably occurs multiple times over. In other words, Landsburg’s remarks may apply literally, although I doubted that Godel – and therefore Landsburg – intended this interpretation of the Completeness Theorem.)

  26. 26 26 Jeffrey

    Keep ‘em coming. I’ve learned more about mathematical logic from this blog this year than I did while earning a Masters degree in theoretical math.

  27. 27 27 Neil

    [quote]This blog attracts me because of Landsburg ’s skill at making esoteric ideas concrete.[/quote]

    Steve is a brilliant expositor of a Platonic philosophy. Too bad it is the wrong philosophy. He should have studied more Hume and John Stuart Mill.

  28. 28 28 Neil

    I, on the other hand, am not particularly brilliant at formatting this stuff.

  29. 29 29 Steve Landsburg

    Neil: What I’m expositing here is not a philosophy; it’s standard mathematics, as it appears in all the textbooks.

  30. 30 30 Steve Landsburg

    Benkyou: I will address your points at greater length next week, but in brief:

    1) If the Peano axioms are consistent, the natural numbers exist. This is Godel’s Completeness Theorem and it is pretty much irrefutable; it rests only on simple combinatorial reasoning of the sort we need in order to think at all. This is a matter of math, not philosophy, and it is not controversial.

    2) Almost everyone believes The Peano axioms are consistent. There is a proof of this, though the proof relies on slightly iffier principles, so an extremist might want to reject it. Nevertheless, even among that small population that rejects the proof, almost everyone believes the axioms are consistent. So this consistency is just barely controversial.

    3) If you believe the axioms are consistent per 2), you must admit the natural numbers exist per 1). There is still room for disagreement about the *way* they exist. Have they existed since the beginning of time? Do they exist outside of time? Did people invent them? This is philosophy, not math, and there’s plenty of controversy here.

    We can disagree about point 3), but it’s important to distinguish it from point 1), which is all I’ve been addressing in these posts.

  31. 31 31 Neil

    The naturals must logically exist if the Peano axioms are consistent and *true*. I guess that means, as an empiricist, I must doubt the truth of one or more of the axioms. Not being a mathematician, I’ll choose the successor axiom, putting me in the crazy company of Zeilberger.

  32. 32 32 dave

    ill choose neil as ive never heard of zeilberger.

    maps all the way down.

  33. 33 33 Harold

    I think the axioms on which the completeness theory are based are generally accepted because, as Steve says, if they are not accepted it becomes difficult think at all. We find it hard to conceive of a world without them – how can a number not have an immediate succesor?However, these are still arbitrarily accepted without proof. We find it hard to conceive of a cause without an effect, and vice versa, but causality is not “proved” – what is the cause of one nucleus of Uranium decaying?. It may be that the whole universe will blink out of existence tomorrow without any cause. To question these axioms is not mathematics, but philosophical speculation. Nonetheless, is not discussing whether the natural numbers “exist”, also philosphical speculation rather than mathematics? The basis of the axioms perhaps then becomes relevant. I seem to be going down the “how do we know anything” line here, which is probably not very useful to anything.

  34. 34 34 Steve Landsburg

    Neil:

    The naturals must logically exist if the Peano axioms are consistent and *true*.

    No, you’ve missed the point entirely.

  35. 35 35 nobody.really

    For what it’s worth, I also stumble on the idea that certain mathematical ideas “exist.”

    Somewhere I’d acquired the view that appeals to “reality” or “existence” violate the norms of mathematical exposition. The capacity to entertain competing sets of assumptions facilitates the study of mathematics (among other fields). Consequently asking people to believe one set of assumptions over another would seem to ask them to place an impediment in their own intellectual capacities. We’re supposed to restrict ourselves to attributes such as consistency, completeness, logical necessity within the framework, etc. — and NOT attributes such as truth or realty or existence.

    Consequently, I’ve understood references to existence to mean, “If you adopt assumptions A, B, and C — assumptions that seem quite intuitive to minds adapted to thrive in this world, etc. — then you can fashion tool D [the natural numbers, or a theorem, or whatever]. And with A, B, C AND D, we can do E, F and G….” But perhaps I’m missing something here.

    Landsburg promises further elaboration; I’m looking forward to it.

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