Math is Subjunctively Objective
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 your 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.
Part of The Metaethics Sequence
Next post: “Does Your Morality Care What You Think?”
Previous post: “Can Counterfactuals Be True?”
- Morality as Fixed Computation by (8 Aug 2008 1:00 UTC; 72 points)
- Changing Your Metaethics by (27 Jul 2008 12:36 UTC; 62 points)
- The Meaning of Right by (29 Jul 2008 1:28 UTC; 61 points)
- Can Counterfactuals Be True? by (24 Jul 2008 4:40 UTC; 33 points)
- Does Your Morality Care What You Think? by (26 Jul 2008 0:25 UTC; 21 points)
- The role of mathematical truths by (24 Apr 2010 16:59 UTC; 15 points)
- 's comment on What is Eliezer Yudkowsky’s meta-ethical theory? by (30 Jan 2011 0:10 UTC; 14 points)
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- [SEQ RERUN] Math is Subjunctively Objective by (12 Jul 2012 2:07 UTC; 6 points)
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Hmm, Eliezer likes Magic the Gathering (all five basic terrains?)...
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.
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...).
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.
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.
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.
“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.
Come on, everyone knows 2 + 3 = 11!
Well, in base 4, 2+3=11. I don’t think there’s a base where 2+3=(11!), though. It looks like there’a numerical approximation for base 1.8523...
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.
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?
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 felt that 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 why these 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.
It seems to me that when I say “every Hilbert space is convex”, I’m not saying something in math; I’m saying something about math, in English. 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.
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.
WWPD? What Would Plato Do?
Thomas: which set theory? There are lots of them.
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.
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.
It seems to me that math is a set of symbolic tools for clarifying the tautological nature of non-transparently tautological assertions.
″...then where is it?”
Same place all the other true counterfactuals are.
That was me at 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.
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.
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?
“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 understanding of the correlated material processes that 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.
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.
Where is the location of “dead”?
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”?
“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.
If you had a Turing machine that perfectly simulated the physical laws of our universe, could an external person use that machine’s source code to derive the laws of arithmetic as they are within our universe, even if the laws of arithmetic for the external person’s universe were different?
Suppose we think about it the opposite way: what if we built a machine that simulated the physical laws of a universe where 2+3 = 6, where if you stick 2 whatsits by 3 whatsits you get 6 whatsits total. What would that universe be like? Could it even be built?
It helps to differentiate between “real” and “existant”. Mathematics is as real as the laws of logic—neither, however, exists.
What is “real” is that which proscriptively constraints that which exists. That which exists is that which interacts directly with other phenomena which also exist (that also interact).
When we say “2+3=5” what we are doing is engaging in the definition of real patterns of that which exists. So while, yes, the patterns themselves are external to us; the terms we assign them are subjective. In mathematics we ‘understand’ what happens when you add 2 and 3 together. But these are just symbols; just representations by which we predict outcomes based upon our understanding of those constraining patterns. In other words; we define 0 as none of a thing, and 1 as a single thing. We therefore define 2 as 1+1, 3 as 1+1+1, etc., etc.. (The dots).
However, if we were to encounter a person who defined 6 as 1+1+1+1+1, and we were to continue definining 1+1+1+1+1 as five, then neither would agree with one another, and both would be correct. This is, given the understanding involved of the difference between “real” and “existent”, neither exceptional nor inscrutable. It’s a trick of definition—much as is the Law of Identity and the Law of the Excluded Middle. (That which I define as “only-A” cannot ever simultaneously be “not-A”.)
I’m quite unconfident about this whole line of argument, and concerned that we’re heading for some moral conclusions based on appeals to this argument. If you have to get into odd discussions about the truth and meaning of mathematical entities to make a metaethical l argument, I doubt you have a good metaethical argument.
The funny thing is I consider morality subjective objective, just like yummyness. What is subjectively yummy to you is an objective fact about you, just as what is moral to you is an objective fact about you. If we run the You algorithm to evaluate yummyness of a root beer float, we’ll get an answer, depending on a lot of state variables about you. Similarly, we can run your moral algorithm to evaluate a person’s actions, and depending on a lot of state variables, we’ll get an answer there too. Both answers are objective facts about the subjective you.
I don’t know about that. People continually make metaethical mistakes by assuming that “morality is defined by your brain” is the same as “morality is about your brain”, and draw all kinds of faulty conclusions, like that unless there’s a stone tablet somewhere out in space with the Thousand Commandments of Morality written on it, it must be okay for people—in different societies, with different beliefs—to torture children if they want to (because hey, if morality isn’t objective it must be relative, right?). That’s exactly the error being talked about here, collapsing levels, and I think it’s kind of an important one, metaethically.
Not arguing against what you said, but on your view, what, if anything, distinguishes morality from yumminess? Aren’t they, as you describe them, just “what I like”, applied to different classes of things, morality being about dealings with other people and yumminess being about food and drink?
I want to get to the end of the metaethics sequence before pontificating too much, but I’ll say a little.
What is yummy? A sense of taste, an evaluation of a particular sensory modality which impels action—eat more.
What is scary? Well, also an evaluation, but not confined to a particular sensory modality. It’s an evaluation of threat, a fairly complex evaluation, and it impels action too—fight or flight.
What is moral? The moral sense also evaluates—it evaluates the actions and attitudes of people, and it also impels action—attitudes and reward or punishment, for actions, for attitudes about actions, for properly rewarding/punishing actions, for properly rewarding/punishing attitudes about actions, …
Personally I believe that mathematics is little else but text with rules. One of these rules is that when a certain rule is satisfied, we are allowed to write that something is true. But when do we know that a rule is satisfied? What does that even mean? Well, I believe that in the end we have to trust our intuition. That is, when we have a strong enough, honest feeling that something satisfies our rule of “being true”, we say that it is true. This definition makes mathematical truths very vaque and even subjective, which is unfortunate, but so far no other philosophy has satisfied me.
Checking whether mathematical rules are satisfied does not require intuition; it can be done by a computer program (and often is).
Someone doing that still puts faith on the computer, and the person who made the computer program to check the rules. Essentially, he has strong feeling that A holds because the computer program said so. He still has to rely on his “intuition” or “belief” that the computer program gives true statements.
Some people (mostly young children, though some adults as well) believe that the ratio of a circle’s circumference to its diameter should be an integer, or at worst a rational fraction. Most other people, however, do not believe this to be the case.
If mathematical truths are subjective as you claim, then a person who believes that pi == 3 should be able to build himself a 5-foot wide hula-hoop using exactly 15 feet of plastic tubing. Do you think this is actually the case ?
Maybe he is able to construct some sort of an abstract hula-hoop in his mind, which he believes to have those properties, but of course he isn’t able to do it in the physical reality. Strong intuition suggests that it isn’t possible.
However, we should not forgot that mathematical models of physical reality and mathematics itself are separate things. We can use mathematics to understand nature, but nature cares very little about anyones mathematical truths. Well, I think it’s safe to say so anyway.
Ok, so consider what happens when this person does indeed attempt to construct a physical hula-hoop. After failing a few times, assuming he doesn’t give up altogether, he’ll be forced to accept (however provisinally) that pi is not 3, but approximately 3.14159265 (in our current physical reality, at least). He now has two conflicting models in his mind: one of an abstract hula-hoop made with pi == 3, and another one made with pi ~= 3.14159265. Which one will he “have a strong feeling / intuition / belief” about, do you think ?
I think he will have a strong feeling that pi is about 3.141… . Like I said, in my definition truth is subjective and may chance since it’s tied to the person’s beliefs / feelings. This may not seem beatiful to everyone, but I can live with that.
Why ?
Hmm, well, if you truly believe that truth is subjective, then there’s nothing I can do to dissuade you, by definition—since my subjective opinion is as good as yours. Now if you’ll excuse me, I’ve got to go build some hula-hoops, and then maybe take to the skies by will alone.
Doesn’t seem to apply here, because Randolf admits that reality doesn’t care what nonsense he believes. The only problem is he seems intent on describing that nonsense as ‘truth’ and refusing to label what it is that reality is doing, which is what everyone else is calling ‘truth’.
Hehe, I knew someone would pick up on my reference, I just didn’t realize how fast it would happen :-)
But my point was this: if Randolf really does believe that truth is subjective, and that it is arrived at mostly through feelings and intuitions, then he has effectively removed himself from rational debate. There’s nothing I can say that will persuade him one way or another, because there’s no useful mechanism by which my subjective beliefs can influence his subjective beliefs. So, there’s little point in arguing with him on this (or any other) topic.
Randolf, my apologies if I seem to be putting words in your mouth; the above paragraph is simply my personal interpretation of your claim, taken to its logical conclusion.
No, I think you understood pretty well what I meant. However, even though I may not be a rationalist myself, I think I can still take part in rational debate by embracing the definition of rational truth during that debate. Same way a true Christian can take part in a scientific debate about evolution, even if he doesn’t actually believe that evolution is true. Rational talk, just like any talking, can also change my feelings and intuitions and hence persuade me to change my subjective beliefs.
However, I now realise this wasn’t exactly the right place to tell about my idea of subjective truth. Sorry about that.
I don’t think it will work in this case, because we’re debating the very notion of rational truth.
I personally didn’t mean to give you that impression at all, I apologize if I did. Just because I happen to think that using reason to debate with someone who does not value reason is futile, doesn’t mean that I want to actively discourage such debate. After all, I could be wrong !
Yes, I agree, it doesn’t work on this case. It was an interesting talk though, thank you for that. Now I must sleep over this..
Oh, you probably could. I’m not so fond on this definition. It’s just something I have found most satisfying so far but it’s still subject to chance (How ironic!).
That’s the key issue. Reality is doing something here. And you know, in advance what his model will move to. You don’t think he will succeed at his event. At the end of the day, you are pretty sure that there’s something objective going on.
More starkly, I can give you mathematical examples where your intuition will be wildly at odds with the correct math. Some of those make fun games to play for money. I suspect that you won’t be willing to play them with me even if your intuition says that you should win and I shouldn’t.
That’s a bit differend from what I’m trying to say. My word choosing of intuition was clearly bad, I should have talked about mental experiences. My point is that when I do the mathematics, when I, for example, use the axioms and theorems of natural numbers to proof that 1+1 is 2, I have to rely on my memories and feelings at some point. If I use a theorem proven before, I must rely on my memories that I have proven that theorem before and correctly, but remembering is just another type of vaque mental experience. I could also remember axioms of natural numbers wrong, even if it would seem clear to me that I remember them correctly. I have to rely on the feeling of remembering correctly. This is why I define truth as what you truly believe. Once you have carefully checked that you used all the axioms and theorems correctly, you will truly believe that you made no mistake. Then you can truly believe that 1 + 1 is 2, and it’s safe to say its the truth.
FWIW: I agree with you that:
my beliefs are always the outputs of real-world embodied algorithms (for example, those associated with remembering previously proven axioms) and therefore not completely reliable.
there exists a non-empty set S1 of assertions that merit a sufficiently high degree of confidence that it is safe to call them “true” (while keeping in mind when it’s relevant that we mean “with probability 1-epsilon” rather than “with probability 1”).
I would also say that:
there exists a non-empty set S2 of assertions that don’t merit a high degree of confidence, and that it is not safe to call them true.
the embodied algorithms we use to determine our confidence in assertions are sufficiently unreliable that we sometimes possess a high degree of confidence in S2 assertions. This confidence is not merited, but we sometimes possess it nevertheless.
Would you agree with both of those statements?
Assuming you do, then it seems to follow that by “what I truly believe” you mean to exclude statements in S2. (Since otherwise, I could have a statement in S2 that I truly believe, and is therefore definitionally true, which is at the same time not safe to call true, which seems paradoxical.)
Assuming you do, then sure: if I accept that “what I truly believe” refers to S1 and not S2, then I agree that truth is what I truly believe, although that doesn’t seem like a terribly useful thing to know.
Yes, I think you managed to put my thoughts into words very well here. Probably a lot more clearly than I.
May I recommend “Godel, Escher, Bach” to you? It discusses the issue of what proof is at a rigorous but accessible level, including that a proof is just a well-formed finite string.
Yes, I believe that proof is just a well-formed finite string, but I take that a little bit futher because one can always ask that “what a well formed finite string is?”. Basically, I tell that person to use his honest intuition to check which things are “well-formed finite strings”.
These questions have simple answers. Please explain what part of carrying out a proof-checking procedure—which can be by hand if need be—requires intuition.
I am not Randolf, but I’ve met people who would answer this question thusly:
I don’t think it’s possible to use logic to convince someone of the importance of logic, unless he happens to be convinced already.
You can use naive logic to convince people of the importance of more rigorous logic, though, and I suspect that most of the people decrying logic, axiomatic systems, etc. aren’t objecting to reasoning in general so much as certain levels of formality, or certain attitudes surrounding them. I’ve met a lot of people claiming to put more stock in gut feelings than clever reasoning, but I’ve never met one such that didn’t have a handy store of justifications for their beliefs—which seems to point to a certain trust even if it’s unacknowledged.
Or, in my experience, specific topics. For example, such a person would say that reasoning does apply to topics such as deciding which car to buy, or which stock to invest to, or what the sum of the angles in a triangle is. Reasoning does not, however, apply to other topics such as deciding what to eat for lunch, which deity to worship (if any), whom to date, and which topics are subject to reason in the first place.
The above is a real example, BTW (assuming I understood the person’s position correctly).
(nods) I generally summarize this as “reason is useful only for those topics where I’m confident I’m right or am willing to be corrected if wrong.” To which my response is typically “how very convenient for you that it works out that way.”
Yes, I wouldn’t have bothered if he had said something like that; the thing is from the above that didn’t seem to be the objection he was making. Since he now says it essentially is, I think I’ll step out of this argument. (Well, the first two sentences are easily answerable, but I’ll let someone else do that if they really want.) Also apparently by “intuition is required”, he means “brains cannot carry out an algorithm 100% reliably, and 100% reliability is required (or something like that)”. Which would I suppose make him the first person I’ve heard to actually (effectively) endorse “ordinary person reasoning”, where only chains of reasoning of a bounded (and very short!) length are valid! (I seem to recall this being discussed somewhere here before… can’t find it right now, though.) Anyway, I won’t bother commenting on this any further.
Oh, I found it. It wasn’t a discussion here, it was a post on Scott Aaronson’s blog: http://www.scottaaronson.com/blog/?p=232
Yes, that’s pretty much what I would say. Also, a simple answer to the question would also be:
My world view used to be differend until I read the following pharse somewhere. That moment I realised I can only be as sure as my feelings let me.
I still have a great interest in mathematics and am hoping my studies and everything goes well so I can bear the title of mathematican one day. Maybe my beliefs change when I get less green.
That’s a much weaker statement than the one you originally stated. This new statement says, basically, “you can never be 100% sure of anything”, whereas before you seemed to be saying, “there exist no objective standards of truth at all, any story is as good as any other”.
Whetever it is a weaker statement or not isn’t the point. I only brought it up because it made me change the way I think about mathematics and the world. While I don’t know what you mean by “any story is as good as any other”, I do not believe that it is possible to give truth a honest definition which would leave no open questions about the very nature of truth, while still being entirely objective.
Well, let’s say I believe that I can fly by will alone. You, on the other hand, believe that I cannot fly by will alone. Which one of us is right ? If truth is entirely subjective, then we’re both “right”, in the sense that we both have some sort of a story in our heads regarding flight, and in our respective worldviews this story makes perfect sense, and since there’s no objective standard for truth (at least, none that we can access in any way), the stories are all that matters. Thus, all stories are equally true, just by the virtue of being stories.
According to a weaker interpretation of your statements, however, one of us is probably closer to the truth than the other. More specifically, it is very likely that my belief about my ability to fly by will alone is false. It’s still not 100% likely, of course—there’s always that chance that we live in the Matrix, or that I’m a superhero, or that by “flight” I really mean “pretending to fly without physically moving”, etc. -- but such chances quite small. Thus, for all practical purposes, we can say, “Bugmaster’s belief about flight is false”, with the understanding that we can never be 100% sure.
There could be other interpretations of your claims, of course; these are just the two I could come up with. I could support the second interpretation, though whether it applies to math or not is highly debatable. However, if you support the first interpretation, or if you don’t place any significant value on reason, then any further discussion on the topic is pointless—because, by definition, there’s nothing I can say that will make any difference to you.
It doesn’t really work this way. And to demonstrate, I bring up the prime numbers.
What many people don’t quite understand is that mathematics, like the sciences does not invent things, it discovers them. The structures are already there. We did not invent cells, electricity, or gravity. They were already there. All Mathematics does is name them, categorize them, and show properties that they have. There is nothing human about the prime numbers, for instance. There really is nothing human about mathematics.
Counting is essentially the building block of all of mathematics. 1 2 3 etc… There is no other way to count than the way we count. Is this because of our definition of counting? Well of course, but it is nonetheless true. If Aliens were to count, they would have to count this way. Can I construct systems where 1+1=1? Of course. Consider clouds. If you add two clouds together, you just get a cloud. However, counting is still not changed. In order to even ask the question, I need to be able to discretely differentiate clouds, which means that counting is still there. You simply have a bizarre algebra on top of it.
To even consider a universe where counting goes by different rules is mind-boggling, because it would require the impossibility of discrete objects. Even waves would have peaks and valleys they would be able to be counted. Time generates rhythm and beats that would be counted. And there is only one way to count.
And once you realize there is only one way to count. You realize that addition gives us multiplication and that gives us the prime numbers. We didn’t invent prime numbers. We discovered them.
That is not such an unquestionable truth, there are many different schools of thoughts. None overly useful.
I’m not quite sure what you mean by that, but Platonism has been useful for inspiring Tegmark’s Ultimate Ensemble, which has been useful for inspiring UDT.
Can’t say that I find either very useful (in the instrumentalist sense, anyway), but I suppose if you count inspiration for a couple of rather speculative ideas as useful, I agree.
This is only true to a point. In some sense, yes, the real numbers are the only complete & [canonically] totally-ordered field, up to isomorphism; but this last part is a bit of a snag for the language being used here, since the tools used to develop the real numbers in those different ways are certainly created as much as language & software are created.
You could cling to the idea that even these things are merely “discovered,” but eventually you’d find yourself talking about the Platonic ideal of the wobbly, scratched up table in the neighbors’ house, and how the carpenter originally discovered the Form of this particular table.
This is more a criticism of the English words for invention, creation, discovery, & the like; but then, philosophy of math that gets too far afield from actually doing logic is basically just philosophy of language.
I suggest this may be a map/territory problem. Math is part of the map, but it has no physical analog with the territory. Rather, it tells us (some of) what to expect about the way the territory behaves under certain specific conditions (like when two sheep and then three sheep leave the pen).
Another way to look at it is that quantity (on which math operates) is a quality, akin to redness or sourness, but operating only on groups. That is to say, there is something there that causes fiveness to appear in my brain, but that thing is not an inherent part of the sheep any more than fluffiness or whiteness. Thus ‘2+3=5’ has the same truth value as ‘black + white = gray’.
It seems that Mathematics as we know it (Russel’s axioms) is both an emergent phenomenon as well as the most basic law of them all. In macroscopic physics we observe that two rocks next to three rocks is five rocks, two hydrogen atoms next to three hydrogen atoms is five hydrogen atoms, two oscilliations of a cyclic system followed by three more is five such, and so on and so forth… But the Schrodinger equation contains addition of complex numbers, which we know to be a superset of the naturals.
Man, I really need to write a top level article on the Tegmark IV Hypothesis.
I still stand by my belief that 2 + 3 = 5 does not in fact exist, and yet it is still true that adding two things with three things will always result in five things.
I don’t think that what you just said means anything.
I think he is trying to say he is a fictionalist.
Oh, well if it has a name...
Why not call the set of all sets of actual objects with cardinality 3, “three”, the set of all sets of physical objects with cardinality 2, “two”, and the set of all sets of physical objects with cardinality 5, “five”? Then when I said that 2+3=5, all I would mean is that for any x in two and any y in three, the union of x and y is in five. If you allow sets of physical objects, and sets of sets of physical objects, into your ontology, then you got this; 2+3=5 no matter what anyone thinks, and two and three are real objects existing out there.
Everywhere two and three things exist, “2 + 3 = 5” exists. Much like there is only one electron, there is one “2 + 3 = 5″. Electrons and mathematics are described by their behaviors. “If the behavior of electrons exists outside your brain… then where is it?”
Everywhere.
Some day EY will learn to taboo “exist”, and that will be his awakening as an instrumentalist.
Odd, EY never seemed to me as particularly opposed or holding views going against / away from instrumentalism, when I was reading the sequences.
I’m curious to see where that comment comes from.
I cannot speak for him, but my understanding is that he identifies instrumentalism with “traditional rationality”, which is but a small step toward Bayesianism.
Isn’t the territory and the map an explicit distinction between what exists and what we theorize?
As I said many times before on this forum, the instrumental approach is that the map-territory distinction is a model, i.e. territory is in the map, not in the territory :)
I think I see where you are coming from with that now.
It seems to me that the territory assumption is necessary for morality, and not much else (because we want to care about things that “exist”, but otherwise probability theory is defined over possible observations only).
Of course a great number of unnecessary things have been called “necessary for morality”...
I’m going to read your comments a bit more and see if I can settle my mind on this instrumentalism thing. Do you reccommend anything I should check out?
I think morality is a red herring here. “Wanting to care” about something is a confused state. I care about what I care about. If it so happens that what I care about is an element of a model rather than being something else, I don’t necessarily stop caring about it solely because of that fact.
That said, personally my response to instrumentalism is to take a step back and talk about expectations regarding consistency.
If we can agree that some models support predictions of future experiences better than others, I’m content to either refer to the model that best supports those predictions as a reality that actually exists, as a territory that maps describe, or as my preferred model, depending on what language makes communication easier. I suppose you could say I’m a compatibilist with respect to instrumentalism.
If we can’t agree on that, I’m not sure where to go from there.
I used to feel the same way, but then it is easy to start arguing about the imagined parts of the territory for which no map can ever exist, because “the territory is out there”, and about which of the many identical maps is “more right” (as opposed to “more useful for a given task”). And, given that there can be no experimental evidence to resolve such an argument, it can go on forever. Examples of this futile argument are How many angels can dance on the head of a pin?, QM interpretations, Tegmark’s mathematical universe, statements like “every imaginable world exists” and other untestable nonsense.
As an engineer, I don’t enjoy unproductive futile debates, so expending effort arguing about interpretations seems silly to me. Instrumentalism avoids worrying about “objective reality” and whether it has some yet-undiscovered “true laws” of which our theories are only an approximation. Life is easier that way. Or would be, were it not for the “realists”, who keep insisting that their meta-model is the One True Path. That is not to say that I reject the map-territory distinction, I just place both parts of it inside the [meta]map.
Agreed that futile debates are silly. (I do sometimes enjoy them, but only when they’re fun.)
That said, I find it works for me, in order to avoid them, to accept that questions about the persistent thing (be it reality or a model) are only useful insofar as they lead us to a clearer understanding of the persistent thing. It’s certainly possible to construct and argue about questions that don’t do this, but it’s not a useful thing to do, and I try to avoid it.
I haven’t yet found it necessary to assert a firm position on the ontological nature of reality beyond “the persistent thing” in order to do that. Whether reality is “in the map” or “in the territory” or “doesn’t exist at all” seems to me just another futile debate.
I largely agree. I assert that the territory is in the map mostly as a Schelling fence of sorts, beyond which there is a slippery slope into philosophizing about untestables.
I don’t see how. Feel free to explicate.
Sorry. I wish I could say “Popper”, since he basically , but he argued against Bohr’s instrumentalism on some grounds I don’t fully understand. quote from Wikipedia:
Usually when I read a critique of instrumentalism, it is straw-manned first (I think of it as InSTRAWmentalism). I am quite well aware that this could be a problem with my, admittedly patchy, understanding of the issue, and am happy to change my mind when a good argument comes along.
Do you think the limit of the map as its error goes to zero exists? Do you think we will ever be able to determine whether or not the limit exists? What name would you give that limit if it existed?
I’m just trying to get a better idea of what you believe about instrumentalism. Personally, I think that every map is a territory (mathematical realism) because among all the vacuous explanations for why we experience something instead of nothing it seems to be a simpler model. Instrumentalism, in this case, means trying to figure out the probability distribution of the territories/maps you are a member of, or in other words which map is most likely to predict the measurements I make?
I can see how mathematical realism is obviated by Occam’s Razor since it’s not necessary to explain any measurement, but it’s probably the best metaphysical idea I’ve ran into and it does lend some insight into the question of what to simulate (it doesn’t matter; every simulation already exists just as much as we do), what to care about (everything happens in some universe, so just try to optimize your own), immortality (some universes have infinite time and energy, and some of those universes will simulate us), and god/Omega (there exist beings in other universes that simulate our universe, but it doesn’t matter since our existence is independent of being simulated).
The equivalent language I prefer is more lay-person: will science ever explain everything we observe and predict everything we may ever observe? And my answer is: there is no way to tell at this point, and the answer[ability] is not relevant to anything we do. After a moment of thought you can see that this might not even be the right question to ask: some day we might be powerful enough and smart enough to create new physical laws, so even defining such a limit will be meaningless.
Even if the Universe’s fundamental nature can be changed without limit there would still be a current territory that hasn’t changed yet. The future territory would be different, but if we knew how to create new laws we could also probably predict what the new territory would be like.
If the fundamental nature of the universe just changes over time on its own, then your argument is a lot stronger.
But should my map mark territory as being in the map, or in the territory?
It helps if you start by tabooing the words “territory”, “real”, “exist” and explaining what you mean by them.
He explicitly identifies as a realist somewhere. Saying things along the lines of “once you have all these theories describing things, why postulate the additional fact that they don’t exist?” (that’s not an exact quote)
There
I already thought that Yudkowsky was a Platonist given his position on Everett’s interpretation and Tegmark’s multiverses, but that’s can be cosidered conclusive evidence.
Because that’s how naive class theory works, not how consistent formal mathematics works.
The closest thing to a canonical approach these days is to start from what you have, nothing, and call that the first set. Then you make sets from those sets in a very restrictive, axiomatic way. Variants get as exotic as the surreal numbers, but the running theme is to avoid defining sets by intension unless you’re quantifying over a known domain.
For the record, I don’t think any of these things “exist” in any meaningful sense. We can do mathematics with inconsistent systems just as well, if less usefully. The law of non-contradiction is something I don’t see how to get past (ie I can’t comprehend such a thing), and there is nothing much else distinguishing the consistent systems as being anything other than collections of statements to the effect that this & that follow if we grant these or those axioms. (Fortunately, it’s more interesting than that at the higher levels.)
You’ve misunderstood me. It’s really not at all conspicuous to allow a none-empty “set” into your ontology, but if you’d prefer we can talk about heaps; they serve for my purposes here (of course, by “heap”, I mean any random pile of stuff). Every heap has parts: you’re a heap of cells, decks are heaps of cards, masses are heaps of atoms, etc. Now if you apply a level filter to the parts of a heap, you can count them. For instance, I can count the organs in your body, count the organ cells in your body, and end up with two different values, though I counted the same object. The same object can constitute many heaps, as long as there are several ways of dividing the object into parts. So what we can do, is just talk about the laws of heap combination, rather than the laws of numbers. We don’t require any further generality in our mathematics to do all our counting, and yet, the only objects I’ve had to adopt into my ontology are heaps (rather inconspicuous material fellows in IMHO).
I should mention that this is not my real suggestion for a foundation of mathematics, but when it comes to the challenge of interpreting the theory of natural numbers without adopting any ghostly quantities, heaps work just fine.
(edit): I should mention that while heaps, requiring only for you to accept a whole with parts, and a level test on any gven part, are much more ontologically inconspicuous than pure sets. Where exactly is the null set? Where is any pure set? I’ve never seen any of them. Of course, i see heaps all over the place.
In college, I made the observation that math majors tended to think that math itself was something real, while physcis majors, studying the exact same math in the same classes at the same time, tended to think that math was just a conceptual tool that was sometimes useful when trying to discover things about reality, but that math wasn’t itself real. I’m not sure which view is more valid then the other, or how you even distinguish the two views.
Taboo “real” (I find this a uniquely useless word in general) and I don’t think the typical physics major and the typical math major are actually disagreeing about anything. Most of what I’ve heard physics and math people say on this subject is signaling tribe affiliation, e.g. “physics is to math as sex is to masturbation.”
It’s possible that they are not, but it seems like there’s more to the question that that.
I guess what I would say is “is math a fundamental property of the universe, like the laws of physcis, or is it a useful and consistent tool that only exists in our mind, like morality?”
You would probably have to break it down further then that. Pythagorean theorem clearly seems to be a property of the real world, as does pi, and geometry in general. Once you get to more abstract math, though, it becomes less clear to me if you are describing something fundamental or merely manipulating symbols.
How would the world look different on each of those hypotheses? (Can you please taboo “fundamental” and “exists,” too?)
You’re confusing the map and the territory here. The Pythagorean theorem and pi are both mathematical features that fall out of a particular model of the world, namely Euclidean geometry, which is an inaccurate model for at least two historically major reasons (the Earth not being flat and relativity).
How does “the Earth not being flat” make Euclidean geometry inaccurate?
If you draw a big enough right triangle on the Earth, it will visibly fail to satisfy the Pythagorean theorem. The geometry of the Earth is approximately spherical geometry, not Euclidean geometry.
Euclidean geometry is a set of principles and conclusions for flat space. That Earth is not flat in no way makes Euclidean geometry inaccurate.
The Pythagorean theorem and pi could both accurately be described as predictive scientific hypothesis of observed phenomenon. “In 3 dimensional space, if I measure two sides of a right triangle, the third side will be the square root of the sum of the squares of the other two sides.” That is a scientific hypothesis, and it can be tested; not only that, but you lose none of the meaning of Pythagorean theorem by putting it in those terms. (Yes, if you bring relativity into it, it turns out to be a slightly inaccurate hypothesis because of the curvature of space in a gravitational field, so I suppose that puts it in the same catagory of hypothesis as Newtonian physcis.) It still seems to be an attempt to describe a feature that exists in nature, though.
(minor edits for clarification)
Ah. I wouldn’t call that claim the Pythagorean theorem. To me, the Pythagorean theorem is a mathematical statement about mathematical objects called Euclidean triangles (or if we want to get really fancy, it’s a statement about vectors in inner product spaces), and there is a separate claim, which is not mathematical, which asserts that a certain model which includes things like Euclidean triangles describes some part of the real world in some way.
In other words, I think it’s sensible to enforce a strong separation between talking about the mathematical details of a mathematical model and the relation of that mathematical model to reality. To me this dissolves what I think your original question is (although I am not sure I have correctly understood what your original question is).
Maybe your question is secretly a question about the unreasonable effectiveness of mathematics in the natural sciences?
Actually, the pythagorean theorem and pi still apply regardless of what dimension of geometry the world obeys (3-dimensional newtonian physics, 4-dimensional relativistic spacetime, 11-dimensional string theory, etc).
The Pythagorean theorem doesn’t apply to curved space, only to flat space (regardless of number of dimension). And pi is the number 3.14159..., which can be defined in ways that have nothing to do with geometry, so I’d put it as “in convex (concave) space, the ratio of a circumference to its diameter is less (greater) than pi”, not as “in convex (concave) space, pi is less (greater) than 3.14159...)”.
I don’t know what you mean by that.
There is nothing clear about either of those. Both can be proven without empirical investigation. P’s T is not true in curved space.
The theorem (as ISTM is understood nowadays) is a statement about flat space, so I’d put it as “it doesn’t apply to curved space”; saying that it’s false in curved space sounds to me like saying that “in the US, people drive on the right side of the road” is false in the UK.
Ultimately, while there is a useful distinction to be made here, I’m not sure this is the way to make it; most (all?) of what we call “laws of physics” are actually surface behaviors of more complex systems; models, ultimately, of the emergent behavior of a mathematically simpler yet chaotic and computationally dense reality. Which, when investigated, may prove less fundamental rules but rather guidelines to understanding the results of a truth you don’t yet fully comprehend, and perhaps couldn’t model if you did.
Then again, which field the phrase “the field” refers to does depend on who is asking the question where and when.
Among all possible judgements, sure; but among all those judgements that a real person will have to make in the real world...
It makes no sense to call something “true” without specifying prior information. That would imply that we could never update on evidence, which we know not to be the case for statements like “2 + 3 = 5.” Much of the confusion comes from different people meaning different things by the proposition “2 + 3 = 5,” which we can resolve as usual by tabooing the symbols.
Consider the propositions ” A =“The next time I put two sheep and three sheep in a pen, I will end up with five sheep in the pen.”
B = “The universe works as if in all cases, combining two of something with three of something results in five of that thing.” C = “the symbolic expression 2 + 3 = 5 is consistent with mathematical formalism”
These are a few examples of what we might mean when we ask “Is ‘2+3=5’ true?” In all cases, we can in principle perform the computation of P(A|Q), or P(B|Q), etc, where Q represents prior information including what I know about sheep and mathematical formalism.
The map is not the territory. There’s no little XML tag attached to helium atoms with the wave equation written on it. Math was created by humans to describe our observations—we didn’t arrive at it by pure thought. The reason 2 + 3 = 5 is a theorem of Peano arithmetic and moving three large, distinct objects next to two large, distinct objects makes a group of five large, distinct objects is the correspondence of the Peano axioms and inference rules to reality.
So I think Eliezer’s error here was a fallacy of compression. “2 + 3 = 5” refers to two distinct propositions—that “2 + 3 = 5″ is a theorem of Peano arithmetic, and that moving a group of 3 apples (or one of many other types of large, distinct objects) next to a group of 2 apples will result in a group of 5 apples.
That’s interesting… Did you actually count sheep and rocks when writing this article? Did the character you give voice to count sheep and rocks?
Usually, when I make this kind of arguments, what I really say is “If I counted 2 sheep and 3 sheep, I would find 5 sheep” which means that it actually is what I expect, but that’s not evidence if my cognition process is put into question.
Yet, I don’t think it is necessary to actually count sheep and rocks when making this argument… But if I was discussing with someone who thought that 2 + 3 = 6 (or someone who thinks that either answer is meaningless), then it would be necessary to make the experiment, because we would expect different results.