**Followup to:** Probability is Subjectively Objective, Can Counterfactuals Be True?

I am quite confident that the statement 2 + 3 = 5 is *true*; I am far less confident of what it *means *for a mathematical statement to be true.

In "The Simple Truth" I defined a pebble-and-bucket system for tracking sheep, and defined a condition for whether a bucket's pebble level is "true" in terms of the sheep. The bucket is the belief, the sheep are the reality. I believe 2 + 3 = 5. Not just that two sheep plus three sheep equal five sheep, but that 2 + 3 = 5. That is my belief, but where is the reality?

So now the one comes to me and says: "Yes, two sheep plus three sheep equals five sheep, and two stars plus three stars equals five stars. I won't deny that. But this notion that 2 + 3 = 5, *exists only in your imagination, and is purely subjective.*"

So I say: Excuse me, *what?*

And the one says: "Well, I know what it means to observe two sheep and three sheep leave the fold, and five sheep come back. I know what it means to press '2' and '+' and '3' on a calculator, and see the screen flash '5'. I even know what it means to ask someone 'What is two plus three?' and hear them say 'Five.' But you insist that there is some fact *beyond* this. You insist that 2 + 3 = 5."

Well, it kinda *is.*

"Perhaps you just mean that when you *mentally visualize* adding two dots and three dots, you end up visualizing five dots. Perhaps this is the content of what you mean by saying, 2 + 3 = 5. I have no trouble with that, for brains are as real as sheep."

No, for it seems to me that 2 + 3 equaled 5 *before* there were any humans around to do addition. When humans showed up on the scene, they did not *make* 2 + 3 equal 5 by virtue of thinking it. Rather, they thought that '2 + 3 = 5' *because* 2 + 3 did in fact equal 5.

"Prove it."

I'd love to, but I'm busy; I've got to, um, eat a salad.

"The *reason you believe *that 2 + 3 = 5, is your mental visualization of two dots plus three dots yielding five dots. Does this not imply that this physical event in your physical brain is the *meaning* of the statement '2 + 3 = 5'?"

But I honestly don't think that *is* what I mean. Suppose that by an amazing cosmic coincidence, a flurry of neutrinos struck my neurons, causing me to imagine two dots colliding with three dots and visualize six dots. I would then say, '2 + 3 = 6'. But this wouldn't mean that 2 + 3 actually *had *become equal to 6. Now, if what I mean by '2 + 3' consists entirely of what my mere physical brain merely *happens to output*, then a neutrino *could* make 2 + 3 = 6. But you can't change arithmetic by tampering with a calculator.

"Aha! I have you now!"

Is that so?

"Yes, you've given your whole game away!"

Do tell.

"You visualize a subjunctive world, a counterfactual, where your brain is struck by neutrinos, and says, '2 + 3 = 6'. So you know that in this case, your future self will *say* that '2 + 3 = 6'. But then you add up dots in your *own, current brain,* and your *current* self gets five dots. So you say: 'Even if I believed "2 + 3 = 6", then 2 + 3 would still equal 5.' You say: '2 + 3 = 5 regardless of what anyone thinks of it.' So your *current* brain, computing the same question while it *imagines* being different but is not *actually* different, finds that the answer *seems to be the same*. Thus your brain creates the *illusion* of an additional reality that exists outside it, independent of any brain."

Now hold on! You've *explained* my belief that 2 + 3 = 5 regardless of what anyone thinks, but that's not the same as explaining away my belief. Since 2 + 3 = 5 does not, *in fact*, depend on what any human being thinks of it, therefore it is *right and proper* that when I imagine counterfactual worlds in which people (including myself) *think* '2 + 3 = 6', and I ask what 2 + 3 *actually* equals in this counterfactual world, it still comes out as 5.

"Don't you see, that's just like trying to visualize motion stopping everywhere in the universe, by imagining yourself as an observer outside the universe who experiences time passing while nothing moves. But really there is no time without motion."

I see the analogy, but I'm not sure it's a deep analogy. Not everything you can imagine seeing, doesn't exist. It seems to me that a brain can *easily* compute quantities that don't depend on the brain.

"*What?* Of *course* everything that the brain computes depends on the brain! Everything that the brain computes, is computed inside the brain!"

That's not what I mean! I just mean that the brain can perform computations that *refer to* quantities outside the brain. You can set up a question, like 'How many sheep are in the field?', that isn't *about* any particular person's brain, and whose *actual* answer doesn't *depend on* any particular person's brain. And then a brain can faithfully compute that answer.

If I count two sheep and three sheep returning from the field, and Autrey's brain gets hit by neutrinos so that Autrey thinks there are six sheep in the fold, then that's not going to *cause* there to be six sheep in the fold - right? The whole question here is just *not about* what Autrey thinks, it's *about* how many sheep are in the fold.

Why should I care what *my* subjunctive future self thinks is the sum of 2 + 3, any more than I care what *Autrey* thinks is the sum of 2 + 3, when it comes to asking what is *really* the sum of 2 + 3?

"Okay... I'll take another tack. Suppose you're a psychiatrist, right? And you're an expert witness in court cases - basically a hired gun, but you try to deceive yourself about it. Now wouldn't it be a bit suspicious, to find yourself saying: 'Well, the only reason *that I in fact believe*
that the defendant is insane, is because I was paid to be an expert
psychiatric witness for the defense. And if I had been paid to witness
for the prosecution, I undoubtedly would have come to the conclusion
that the defendant is sane. But my belief that the defendant is insane, is *perfectly justified;* it is justified by my observation that the defendant used his own blood to paint an Elder Sign on the wall of
his jail cell.'"

Yes, that *does* sound suspicious, but I don't see the point.

"My point is that the *physical cause* of your belief that 2 + 3 = 5, is the physical event of your brain visualizing two dots and three dots and coming up with five dots. If your brain came up six dots, due to a neutrino storm or whatever, you'd think '2 + 3 = 6'. How can you possibly say that your belief *means* anything other than the number of dots your brain came up with?"

Now hold on just a second. Let's say that the psychiatrist is paid by the judge, and when he's paid by the judge, he renders an honest and neutral evaluation, and his evaluation is that the defendant is sane, just played a bit too much Mythos. So it is true to say that if the psychiatrist had been paid by the defense, then the psychiatrist would have found the defendant to be insane. But that doesn't mean that when the psychiatrist is paid by the *judge,* you should dismiss his evaluation as telling you *nothing more than* 'the psychiatrist was paid by the judge'. On those occasions where the psychiatrist *is* paid by the judge, his opinion varies with the defendant, and conveys real evidence about the defendant.

"Okay, so now what's y*our* point?"

That when my brain is *not* being hit by a neutrino storm, it yields honest and informative evidence that 2 + 3 = 5.

"And if your brain *was* hit by a neutrino storm, you'd be saying, '2 + 3 = 6 regardless of what anyone thinks of it'. Which shows how reliable *that* line of reasoning is."

I'm not claiming that my saying '2 + 3 = 5 no matter what anyone thinks' represents stronger *numerical* evidence than my saying '2 + 3 = 5'. My saying the former just tells you something extra about my epistemology, not numbers.

"And you don't think your epistemology is, oh, a little... *incoherent?"*

No! I think it is perfectly coherent to simultaneously hold all of the following:

- 2 + 3 = 5.
- If neutrinos make me believe "2 + 3 = 6", then 2 + 3 = 5.
- If neutrinos make me believe "2 + 3 = 6", then I will say "2 + 3 = 6".
- If neutrinos make me believe that "2 + 3 = 6", then I will thereafter assert that "If neutrinos make me believe '2 + 3 = 5', then 2 + 3 = 6".
- The cause of my thinking that "2 + 3 = 5
independently of what anyone thinks" is that my
*current*mind, when it subjunctively recomputes the value of 2 + 3 under the assumption that my*imagined*self is hit by neutrinos, does not see the*imagined*self's beliefs as changing the dots, and my*current*brain just visualizes two dots plus three dots, as before, so that the imagination of my*current*brain shows the same result. - If I were
*actually*hit by neutrinos, my brain would compute a different result, and I would assert "2 + 3 = 6 independently of what anyone thinks." - 2 + 3 = 5 independently of what anyone thinks.
- Since 2 + 3 will
*in fact*go on equaling 5*regardless*of what I imagine about it or how my brain visualizes cases where my future self has different beliefs, it's a*good thing*that my imagination doesn't visualize the result as depending on my beliefs.

"Now that's just crazy talk!"

No, *you're* the crazy one! You're *collapsing your levels*; you think that just because my brain asks a question, it should start mixing up queries about the state of my brain *into* the question. Not every question my brain asks is *about* my brain!

Just because something is computed *in* my brain, doesn't mean that my computation has to depend on my brain's *representation of* my brain. It certainly doesn't mean that the *actual quantity* depends on my brain! It's my brain that computes my beliefs about gravity, and if neutrinos hit me I will come to a different conclusion; but that doesn't mean that I can think different and fly. And I don't *think* I can think different and fly, either!

I am
not a calculator who, when someone presses my "2" and "+" and "3"
buttons, computes, "What do I output when someone presses 2 + 3?" I am
a calculator who computes "What is 2 + 3?" The former is a circular question that can consistently return any answer - which makes it not very *helpful.*

Shouldn't we expect non-circular questions to be the *normal* case? The brain
evolved to guess at the state of the environment, not guess at
'what the brain will think is the state of the environment'. Even when
the brain models itself, it is trying to *know itself*, not trying to
know *what it will think about itself*.

Judgments that depend on our representations of *anyone's* state of mind, like "It's okay to kiss someone only if they want to be kissed", are the exception rather than the rule.

*Most* quantities we bother to think about at all, will appear to be 'the same regardless of what anyone thinks of them'. When we imagine thinking differently about the quantity, we will imagine the quantity coming out the same; it will feel "subjunctively objective".

And there's nothing wrong with that! If something *appears* to be the same regardless of what anyone thinks, then maybe that's because it *actually is* the same regardless of what anyone thinks.

Even if you explain that the quantity *appears* to stay the same in my imagination, *merely* because my current brain computes it the same way - well, how *else* would I imagine something, *except* with my current brain? Should I imagine it using a rock?

"Okay, so it's possible for something that appears thought-independent, to actually be thought-independent. But why do you think that 2 + 3 = 5, in particular, has some kind of existence independently of the dots you imagine?"

Because two sheep plus three sheep equals five sheep, and this appears to be true in every mountain and every island, every swamp and every plain and every forest.

And moreover, it is also true of two rocks plus three rocks.

And further, when I press buttons upon a calculator and activate a network of transistors, it *successfully predicts* how many sheep or rocks I will find.

Since all these quantities, correlate with each other and successfully predict each other, surely they must have something *like* a common cause, a similarity that factors out? Something that is true beyond and before the concrete observations? Something that the concrete observations hold in common? And this commonality is then also the sponsor of my answer, 'five', that I find in my own brain.

"But my dear sir, if the fact of 2 + 3 = 5 exists somewhere outside your brain... *then where is it?"*

Damned if I know.

Hmm, Eliezer likes Magic the Gathering (all five basic terrains?)...

Posted by: Pyramid Head | July 25, 2008 at 07:24 AM

Math is just a language. I say "just" not to discount its power, but because it really doesn't exist outside of our conception of it, just as English doesn't exist outside of our conception of it. It's a convention.

The key difference between math and spoken language is that it's unambiguous enough to extrapolate on fairly consistently. If English were that precise we might be able to find truth in the far reaches of the language, just like greek philosophers tried to do. With math, such a thing is actually possible.

So, 2+3=5 corresponds to your dots or sheep, and that's the whole fact of the matter. Cats are called cats because that's what we feel like calling them and calling them dogs won't change their cat-ness.

It FEELS like there should be more because of the way we are accustomed to extrapolating math. There is no additional fact to account for, though.

The only time this isn't really the case is with exotic math which corresponds to a basically "counterfactual" world like "What if the world were made of city blocks?" (Taxi Cab Geometry). It's true that we can imagine false worlds and invent precise language to describe those worlds, but such a description does not make them less false, just more vivid fiction.

Posted by: Ken Sharpe | July 25, 2008 at 09:13 AM

Why do you have to say the math is "outside" the brain? I do understand that the model of the natural numbers is particularly useful in making elegant predictions about our physical universe, but why does that say something about the numbers or the math? The integers are an example of a formal system, but we can construct other formal systems where the formula 2+3=6 holds (I don't know of any *interesting* such formal systems, though). I can easily see that we have these formal systems, and we also have inductive arguments that they describe the world well. I get the sense Eliezer that you posit a third thing "exists". But, wouldn't this be a case of the "mind-projection fallacy"? Why do we need a third thing exist when the formal system and the inductive argument account for everything (or, perhaps they don't, and I'm missing the point...).

Posted by: Student Commenter | July 25, 2008 at 09:43 AM

This might be stupid, but it's probably more intelligent than the 'subjunctive mood' grammar-joke I was going to tell.

Suppose I say, "Even if my mother were kidnapped by terrorists, I would still consider all terrorists freedom-fighters."

Suppose I believe that with such conviction that I'm unable to imagine a reality in which, regardless of whether the physical state of my brain changes, it would not still be true that terrorists+mom=freedom fighters. (The "terms" of this "equation" don't necessarily correspond with anything in the OP. The analogy is still functional).

In other words, I can dream up a scenario where terrorists are just terrorists, but I cannot fathom such a state of affairs actually coming to be.

So would this be subjunctively objective like your numerical epistemology or would it simply mean that my imagination is defective?

BTW, I don't truly believe anything I just wrote.

Posted by: Stephen | July 25, 2008 at 09:43 AM

Math isn't a language,

mathematical notationis a language. Math is a subject matter that you can talk about in mathematical notation, or in English, etc.Posted by: steven | July 25, 2008 at 09:54 AM

2+3=5 is an outcome of a set of artificial laws we can imagine. In that sense, it does exist "purely in your imagination", just as any number of hypothetical systems could exist. "2+3=5" doesn't stand alone without defining what it means - ie. the concept of a number, addition etc. It corresponds to the statement that IF addition is defined like so, numbers like this, and such-and-such rules of inference, then 2+2=5 is a true property of the system.

In a counterfactual world where people believe 2+3=6, in asking about addition you're still talking about the same system with the same rules, not the rules that describe whatever goes on in the minds of the people. (Otherwise you would be making a different claim about a different system.)

So yes, 2+3=5 is clearly true and has always been true even before humans because its a statement about a system defined in terms of its own rules. Any claims about it already include the system's presumptions because those are part of the question, and part of what it means to be "true".

2 rocks + 3 rocks is a different matter - you're talking about the observable world rather than a system where you get to define all the rules in advance. To apply mathematical reasoning to the real world, you have to make the additional claim "combining physical items is isomorphic to the rules of addition", and you're now in the realm of justifying this with empirical evidence. (Of which there is plenty)

I think a better phrasing of your final question then is to ask why do physical systems seem to correspond to the rules of *this* particular system, but there is a degree of circularity there - obviously we haven't just made up the rules of mathematics arbitrarily - we've based the lowest levels on recognised concepts, and then found that the same laws seem to apply at very deep levels with very high degrees of congruence with the world. If the universe were somehow different and nothing ever acted in any way corresponding to our model of "addition" or "numbers" though, then we'd not attach any special significance to it. Our "mathematics" would be quite different, and we'd be asking the same question about *that* system.

Posted by: Brian | July 25, 2008 at 10:03 AM

"But my dear sir, if the fact of 2 + 3 = 5 exists somewhere outside your brain... then where is it?"A mathematical truth can be formalized as output of a proof checking algorithm, and output of an algorithm can be verified to an arbitrary level of certainty (by running it again and again, on redundant substrate). When you say that something is mathematically true, it can be considered an estimation of counterfactual that includes building of such a machine.

Posted by: Vladimir Nesov | July 25, 2008 at 10:10 AM

Come on, everyone knows 2 + 3 = 11!

Posted by: Jay | July 25, 2008 at 10:17 AM

I am quite confident that the statement 2 + 3 = 5 is true; I am far less confident of what it means for a mathematical statement to be true.There are two complementary answers to this question that seem right to me: Quine's Two Dogmas of Empiricism and Lakoff and Núñez's Where Mathematics Comes From. As Quine says, first you have to get rid of the false distinction between analytic and synthetic truth. What you have instead is a web or network of mutually reinforcing beliefs. Parts of this web touch the world relatively closely (beliefs about counting sheep) and parts touch the world less closely (beliefs about Peano's axioms for arithmetic). But the degree of confidence we have in a belief does not necessarily correspond to how closely it is connected to the world; it depends more on how the belief is embedded in our web of beliefs and how much support the belief gets from surrounding beliefs. Thus "2 + 3 = 5" can be strongly supported in our web of beliefs, more so than some beliefs that are more directly connected to the world, yet ultimately "2 + 3 = 5" is anchored in our daily experience of the world. Lakoff and Núñez go into more detail about the nature of this web and its anchoring, but what they say is largely consistent with Quine's general view.

Posted by: Peter Turney | July 25, 2008 at 10:18 AM

Math isn't a language, mathematical notation is a language. Math is a subject matter that you can talk about in mathematical notation, or in English, etc.

What is the useful distinction here? Are you claiming that Math has a reality outside the notation? If Math isn't defined by the notation we use, then what is it?

Posted by: Ken Sharpe | July 25, 2008 at 10:27 AM

I think it doesn't make sense to suggest that 2 + 3 = 5 is a belief. It is the result of a set of definitions. As long as we agree on what 2, +, 3, =, and 5 mean, we have to agree on what 2 + 3 = 5 means. I think that if your brain were subject to a neutrino storm and you somehow

feltthat 2 + 3 = 6, you would still be able to verify that 2 + 3 = 6 by other means, such as counting on your fingers.I think once you start asking

whythese things are the way they are, don't you have to start asking why anything exists at all, and what it means for anything to exist? And I'm pretty sure at that point, we are firmly in the province of philosophy and there are no equations to be written, because the existence of the equations themselves is part of the question we're asking.But I mean, this question has been in my mind since the beginning of the quantum series. I've written a lot of useful software since then, though, without entertaining it much. Do you think maybe it's just better to get on with our lives? It's not a rhetorical question, I really don't know.

Posted by: ME | July 25, 2008 at 11:15 AM

It seems to me that when I say "every Hilbert space is convex", I'm not saying something

inmath; I'm saying somethingaboutmath,inEnglish. Yes, I might talk about the world by saying "the world has the structure of a Hilbert space". But then I might talk about blog commenters (not the ones here at OB) by saying they are like a horde of poo-throwing chimpanzees, and yet that doesn't make primatology a language.Posted by: steven | July 25, 2008 at 11:38 AM

I would encourage Peter's route related to Quine. A formalist in Phil of Math would say that a mathematical statement is true if it can be derived from axiomatic set theory. That is, the truth of the statement is then grounded in formal logic.

This does, of course, beg the question of what grounds our formal logic, but at least it puts basic arithmetic on more firm footing ... in Peter's words, even more deeply imbedded in our belief system.

Posted by: Thomas Schminke | July 25, 2008 at 12:15 PM

WWPD?

What Would Plato Do?

Posted by: david | July 25, 2008 at 01:05 PM

Thomas: which set theory? There are lots of them.

Posted by: Peter de Blanc | July 25, 2008 at 01:06 PM

Math isn't supposed to be some sort of universal truth, but I also don't think it's quite accurate so say it's just a language. It just happens to be a useful abstraction. Granted, an apparently universally useful abstraction, but it's still an invention of humans, the same as boolean logic or physical models.

Posted by: Steve Downing | July 25, 2008 at 01:25 PM

I'm not convinced that it makes sense to talk about visualizing two dots and three dots that are six dots. I would say that the physical event of visualizing two dots and of visualizing three more dots IS the event "visualizing five dots". There is then a separate event, lets call it "describing what you have visualized", that can be mistaken. You can visualize five dots and as a result of interference in the information flow to your mouth end up saying "I see six dots". For that matter, you can visualize five dots, and as a result of either noise or the fact that it is hot out and other parts of your brain are competing to control your vocal apparatus, end up saying "it sure is hot out" instead of "I see five dots". In either of these cases you would say that the vocalization is not about the visualization, but rather, about the other mental and physical processes that caused it. In that case, why not say that "I see six dots" is about your visualization AND the neutrinos, not about the visualization.

Posted by: michael vassar | July 25, 2008 at 02:08 PM

It seems to me that math is a set of symbolic tools for clarifying the tautological nature of non-transparently tautological assertions.

Posted by: michael vassar | July 25, 2008 at 02:09 PM

"...then where is it?"

Same place all the other true counterfactuals are.

Posted by: | July 25, 2008 at 02:15 PM

That was me at July 25, 2008 at 02:15 PM.

Posted by: Q the Enchanter | July 25, 2008 at 02:15 PM

Can we taboo the words "math", "maths", and "mathematics"? I think there are mathematical facts and then there is the study of mathematical facts, and these two things are as different in the same sense that the universe isn't cosmology, crops aren't agronomy, minds aren't psychology, and so on.

Posted by: steven | July 25, 2008 at 02:28 PM

3 + 2 = 6 for me if I choose to define 6 to signify five. 3 + 2 = 5 only for common mathematical definitions of 2, 3, 5, + and =. Otherwise everything is fine, your opponent agreed somewhere at the beginning, that a group of three objects (such as sheep) and two objects will make five objects for our definitions of two, three and five weather we exist or not.

Posted by: Dmitriy Kropivnitskiy | July 25, 2008 at 03:01 PM

Is it useful to say that "2+3=5" is our shorthand for referring to the infinite number of statements of this form:

2 sheep and 3 sheep make 5 sheep

2 rocks and 3 rocks make 5 rocks

2 dinis and 3 dinis make 5 dinis

and so forth? And that the external truth of the statement depends in principle on all these various testable sub-statements?

Posted by: Rolf Andreassen | July 25, 2008 at 06:13 PM

"But my dear sir, if the fact of 2 + 3 = 5 exists somewhere outside your brain... then where is it?"The truth-condition for "There are five sheep in the meadow" concerns the state of the meadow.

My guess is that the truth condition for "2 + 3 = 5" concerns the (more complex, but unproblematically material) set of facts you present: the facts that e.g.:

*It's easy to find sheep for which two sheep and three sheep make five sheep

*It's fairly easy to build calculators that model what happens with the sheep

*It's fairly easy to evolve brains that model what happens with the calculators and the sheep.

*It's fairly easy to find "formal mathematical models" that can run on these evolved brains, that model what's going on with the sheep *and* the rocks *and* various other systems, with axioms and rules of inference that can be briefly described in English.

We have good reason to claim that "2+3 = 5" has an existence outside your mind. We have such reason because, as you point out, we see many different material systems that "correlate with each other and successfully predict one another".

But... do we have any reason to claim that "2+3=5" has an existence outside of these correlations between material systems? My guess is "no". My guess is that we should say that "2+3=5" is about these correlations. Once we say this, we can go ahead and investigate these correlations the way we'd investigate other aspects of material systems: we can try to spell out just what systems do "correlate with each other and successfully predict one another" in the ways we summarize with addition, and then what systems correlate with one another in the ways we summarize with Euclidean geometry, and then look for the meta-level pattern that unites the two sets of correlations.

These questions about the correlations are interesting and partially unsolved. But my guess is that they aren't gaps in our understanding of what math is

about, just gaps in our understandingof the correlated material processesthat math is about. The lines of questioning needed to explain these correlations are different from the lines of questioning that tend to be invoked when someone asks "where" math is.Posted by: Anna Salamon | July 25, 2008 at 06:30 PM

I've been wondering. The conventional wisdom says that it's a problem for mathematical realism to explain how we can come to understand mathematical facts without causally interacting with them. But surely you could build causal diagrams with logical uncertainty in them and they would show that mathematical facts do indeed causally influence your brain?

Also, I would say the problem (if any) is the location of 2, 3, and 5, not the location of 2+3=5, unless the location of "Napoleon is dead" is also a problem.

Posted by: steven | July 25, 2008 at 08:45 PM

Isn't this George Berkeley's issue? Isn't math just the structural part of another sort of language? Isn't 2 + 3 = 5 the same as red and blue make purple in the sense that each observer has a sense of red, blue, purple, 2, 3 and 5 all his/her/its own?

If space aliens find Voyager and read 1 *, 2 **, 3 ***, 4 ****, 5 *****, etc do they see those ***s in any context other than the three tentacles on their second heads?

How then is "2" in any sense different than "red"? How then is "2" any more independently real than "red"?

Posted by: Thanatos Savehn | July 27, 2008 at 01:32 AM

"But my dear sir, if the fact of 2 + 3 = 5 exists somewhere outside your brain... then where is it?"

For some reason most mathematicians don't seem to feel this sort of ontological angst about what math really means or what it means for a mathematical statement to be true. I can't seem to articulate a single reason why this is, but let me say a few things that tend to wash away the angst.

* it doesn't matter "where it is", it is a proven consequence of our axioms.

* it is in every structure in the universe capable of representing integers and performing arithmetic on them.

* there are many ways you can define the real numbers, but they're all isomorphic. When making statements like "2 + 3 = 5" we don't need to worry about which version of the reals we're talking about; it's true for all of them.

* there's a hierarchy of types of mathematical questions. At the bottom are recursive ones: questions we could answer with a big enough computer and enough time. Then there are R.E. questions: questions that if-the-answer-is-yes, we can confirm with a big enough computer and enough time (also, co-R.E., for if-the-answer-is-no). R.E. + co-R.E. is exactly the questions you can write in first-order logic (with the variables taking on integer values) with symbols for all recursive functions and only one quantifier. More quantifiers move you further up the hierarchy. Past that there are questions like the continuum hypothesis that aren't even *about* numbers, and don't seem to be constrained by anything physical. So even if you feel quite uneasy about what some mathematics means, remember that the stuff low on the hierarchy can be on solid ground even if the higher stuff isn't.

Posted by: Larry D'Anna | July 27, 2008 at 09:19 PM