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EllisD comments on Rationality Quotes March 2012 - Less Wrong
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-David Deutsch, The Beginning of Infinity.
Of course there is. A proof of a mathematical proposition is just as much itself a mathematical object as the proposition being proved; it exists just as independently of physics. The proof as written down is a physical object standing in the same relation to the real proof as the digit 2 before your eyes here bears to the real number 2.
But perhaps in the context Deutsch isn't making that confusion. What scope and limitations on mathematical knowledge, conditioned by the laws of nature, does he draw out from these considerations?
The Pythagorean theorem isn't proved or or even checked by measuring right triangles and noticing that a^2 + b^2 = c^2. Is the Pythagorean theorem not knowledge?
I don't think Deutsch means that mathematical proofs are all inductive. I think he means that proofs are constructed and checked on physical computing devices like brains or GPGPUs; and that because of that mathematical knowledge is not in a different ontological category than empirical knowledge.
I feel quite confident saying that mathematics will never undergo paradigm shifts, to use the terminology of Kuhn.
The same is not true for empirical sciences. Paradigm shifts have happened, and I expect them to happen in the future.
It believe it already has. Consider the Weierstrass revolution. Before Weierstrass, it was commonly accepted that while continuous functions may lack a derivative at a set of discrete points, it still had to have a derivative somewhere. Then Weierstrass developed a counterexample, which I think satisfies the Kuhnian "anomaly that cannot be explained within the current paradigm."
Another quick example: during the pre-War period, most differential geometry was concerned with embedded submanifolds in Euclidean space. However, this formulation made it difficult to describe or classify surfaces -- I seem to believe but don't have time to verify that even deciding whether two sets of algebraic equations determine isomorphic varieties is NP-hard. Hence, in the post-War period, intrinsic properties and descriptions.
EDIT: I was wrong, or at least imprecise. Isomorphism of varieties can be decided with Grobner bases, the reduction of which is still doubly-exponential in time, as far as I can tell. Complexity classes aren't in my domain; I shouldn't have said anything about them without looking it up. :(
Reading the wiki page, it looks like Weierstrass corrected an error in the definition or understanding of limits. But mathematicians did not abandon the concept of limit the way physicists abandoned the concept of epicycle, so I'm not sure that qualifies as a paradigm shift. But I'm not mathematician, so my understanding may be seriously incomplete.
I can't even address your other example due to my failure of mathematical understanding.
Hindsight bias. The old limit definition was not widely considered either incorrect or incomplete.
They abandoned reasoning about limits informally, which was de rigeur beforehand. For examples of this, see Weierstrass' counterexample to the Dirichlet principle. Prior to Weierstrass, some people believed that the Dirichlet principle was true because approximate solutions exist in all natural examples, and therefore the limit of approximate solutions will be a true solution.
Not true. The "old limit definition" was non-existent beyond the intuitive notion of limit, and people were fully aware that this was not a satisfactory situation.
We need to clarify what time period we're talking about. I'm not aware of anyone in the generation of Newton/Leibniz and the second generation (e.g., Daniel Bernoulli and Euler) who felt that way, but it's not as if I've read everything these people ever wrote.
The earliest criticism I'm aware of is Berkeley in 1734, but he wasn't a mathematician. As for mathematicians, the earliest I'm aware of is Lagrange in 1797.
I'm also curious about this history.
That's pretty clear, thanks. Obviously, experts aren't likely to think there is a basic error before it has been identified, but I'm not in position to have a reliable opinion on whether I'm suffering from hindsight bias.
Still, what fundamental object did mathematics abandon after Weierstrass' counter-example? How is this different from the changes to the definition of set provoked by Russell's paradox?
I don't recall where it is said that such an object is necessary for a Kuhnian revolution to have occurred. There was a crisis, in the Kuhnian sense, when the old understanding of limit (perhaps labeling it as limit1 will be clearer) could not explain the existence of e.g., continuous functions without derivatives anywhere, or counterexamples to the Dirichlet principle. Then Weierstrass developed limit2 with deltas and epsilons. Limit1 was then abandoned in favor of limit2.
Wikipedia gives the acceptance of non-Euclidean geometry as a "classical case" of a paradigm shift. I suspect that there were several other paradigm shifts involved from Euclid's math to our math: for instance, coordinate geometry, or the use of number theory applied to abstract quantities as opposed to lengths of line segments.
Would the whole Russel's paradox incident count as a mathematical paradigm shift?
Reading Wikipedia, it looks like a naive definition of a set turns out to be internally inconsistent. Does that mean the concept of set was abandoned by mathematicians the way epicyles have been abandoned by physicists? That's not my sense, so I hesitate to say redefining set in a more coherent way is a paradigm shift. But I'm no mathematician.
Its a matter of degree rather than an absolute line. However, I would say a time when even the very highest experts in a field believed something of great importance to their field with quite high confidence, and then turned out to wrong, probably counts.
I don't think "everyone in field X made an error" is that same thing as saying "Field X underwent a paradigm shift."
Why not ? That sounds like a massive shift in the core beliefs of the field in question. If that's not a paradigm shift, then what is ?
The "non-expressible in the new concept-space" thing that you think never actually happens.
What would count as one?
As I understand it, a paradigm shift would include the abandonment of a concept. That is, the concept cannot be coherently expressed using the new terminology. For example, there's no way to express coherent concepts in things like Ptolomy's epicycles or Aristole's impetus. I think Kuhn would say that these examples are evidence that empirical science is socially mediated.
I'm not aware of any formerly prominent mathematical concepts that can't even be articulated with modern concepts. Because mathematics is non-empirical and therefore non-social, I would be surprised if they existed.
A totally trivial nit pick, I admit, but there's no such thing as the Aristotelian theory of impetus. The theory of impetus was an anti-Aristotelian theory developed in the middle ages. Aristotle has no real dynamical theory.
Thanks. Did not know that.
Thanks, I did not actually know that. But I should have known.
There are perfectly fine ways to express those things. Epicycles might even be useful in some cases, since they can be used as a simple approximation of what's going on.
The reason people don't use epicycles any more isn't because they're unthinkable, in the really strong "science is totally culture-dependent" sense. It's because using them was dependent on whether we thought they reflected the structure of the universe, and now we don't. Ptolemy's claim behind using epicycles was that circles were awesome, so it was likely that the universe ran on circles. This is a fact that could be tested by looking at the complexity of describing the universe with circles vs. ellipses.
So this paradigm shift stuff doesn't look very unique to me. It just looks like the refutation of an idea that happened to be central to using a model. Then you might say that math can have no paradigm shifts because it constructs no models of the world. But this isn't quite true - there are models of the mathematical world that mathematicians construct that occasionally get shaken up.
My point was that trying to express epicycles in the new terminology is not possible. That is, modern physicists say, "Epicycles don't exist."
Obviously, it is possible to use sociological terminology to describe epicycles. You yourself said that they were useful at times. But that's not the language of physics.
Since you mentioned it, I would endorse "Science is substantially culturally dependent", NOT "Science is totally culturally dependent." So culturally dependent that there is not reason to expect correspondence between any model and reality. Better science makes better predictions, but it's not clear what a "better" model would be if there's no correspondence with reality.
I brought all this up not to advocate for the cultural dependence of science. Rather, I think it would be surprising for a discipline independent of empirical facts to have paradigm shifts. Thus, the absence of paradigm shifts is a reason to think that mathematics is independent of empirical facts.
If you don't think science is substantially culturally dependent, then there's no reason my argument should persuade you that mathematics is independent of empirical facts.
This is false in an amusing way: expressing motion in terms of epicycles is mathematically equivalent to decomposing functions into Fourier series -- a central concept in both physics and mathematics since the nineteenth century.
To be perfectly fair, AFAIK Ptolemy thought in terms of a finite (and small) number of epicycles, not an infinite series.
But it is! You simply specify the position as a function of time and you've done it! The reason why that seems so strange isn't because modern physics has erased our ability to add circles together, it's because we no longer have epicycles as a fundamental object in our model of the world.
So if you want the copernican revolution to be a paradigm shift, the idea needs to be extended a bit. I think the best way is to redefine paradigm shift as a change in the language that we describe the world in. If we used to model planets in terms of epicycles, and now we model them in terms of ellipses, that's a change of language, even though ellipses can be expressed as sums of epicycles, and vice versa.
In fact, in every case of inexpressibility that we know of, it's been because one of the ways of thinking about the world didn't give correct predictions. We have yet to find two ways of thinking about the world that let you get different experimental results if you plan the experiment two different ways. In these cases, the paradigm shift included the falsification of a key claim.
I don't think it's necessarily true (for example, you can imagine an abstract game having a revolution in how people thought about what it was doing), but it seems reasonable for math, depending on how you define "math." I think people are just giving you a hard time because you're trying to make this general definitional argument (generally not worth the effort) on pretty shaky ground.
Thanks, that's quite clear. Should I reference abandonment of fundamental objects as the major feature of a paradigm shift?
Yes, every successful paradigm shift. Proponents of failed paradigm shifts are usually called cranks. :)
My position is that the repeated pattern of false fundamental objects suggest that we should give up on the idea of fundamental objects, and simply try to make more accurate predictions without asserting anything else about the "accuracy" of our models.
I'm not seeing how the second sentence is an example of the criterion in your first sentence. That criterion seems to strict, too: in general the new paradigm subsumes the old (as in the canonical example of Newtonian vs relativistic physics).
I'm also not seeing what the attributes "empirical" and "non-social" have to do (causally) with the ability to form coherent concepts.
Maybe you should also unpack what you mean by "coherent"?
I'm not a mathematician, but from my outside perspective I would cheerfully qualify something like Wilf-Zeilberger theory as the math equivalent to a paradigm shift in the empirical sciences.
WP lists "non-euclidean geometry" as a paradigm shift, BTW.
Using modern physics, there is no way to express the concept that Ptolomy intended when he said epicycles. More casually, modern physicists would say "Epicycles don't exist" But contrast, the concept of set is still used in Cantor's sense, even though his formulation contained a paradox. So I think the move from geocentric theory to heliocentric theory is a paradigm shift, but adjusting the definition of set is not.
I'm using the word science as synonymous with "empirical studies" (as opposed to making stuff up without looking). That's not intended to be controversial in this community. What is controversial is the assertion that studying the history of science shows examples of paradigm shifts.
One possible explanation of this phenomena is that science is socially mediated (i.e. affected by social factors when the effect is not justified by empirical facts).
I'm asserting that mathematics is not based on empirical facts. Therefore, one would expect that it could avoid being socially mediated by avoiding interacting with reality (that is, I think a sufficiently intelligent Cartesian skeptic could generate all of mathematics). IF I am correct that they are caused by the socially mediated aspects of the scientific discipline and IF mathematics can avoid being socially mediated by virtue of its non-empirical nature, then I would expect that no paradigm shifts would occur.
This whole reference to paradigm shifts is an attempt to show a justification for my belief that mathematics is non-empirical, contrary to the original quote. If you don't believe in paradigm shifts (as Kuhn meant them, not as used by management gurus), then this is not a particularly persuasive argument.
If Wikipedia says that, I don't think it is using the word the way Kuhn did.
For Kuhn, the word was, if anything, a sociological term -- not something referring to the structure of reality itself. (Kuhn was not himself a postmodernist; he still believed in physical reality, as distinct from human constructs.) So it seems to me that it would be entirely consistent with his usage to talk about paradigm shifts in mathematics, since the same kind of sociological phenomena occur in the latter discipline (even if you believe that the nature of mathematical reality itself is different from that of physical reality).
As I'd mentioned elsewhere, there's actually a pretty easy way to express that, IMO: "Ptolemy thought that planets move in epicycles, and he was wrong for the following reasons, but if we had poor instruments like he did, we might have made the same mistake".
The abovementioned non-euclidean geometry is one such shift, as far as I understand (though I'm not a mathematician). I'm not sure what the difference is between the history of this concept, and what Kuhn meant.
But there were other, more powerful paradigm shifts in math, IMO. For example, the invention of (or discovery of, depending on your philosophy) zero (or, more specifically, a positional system for representing numbers). Irrational numbers. Imaginary numbers. Infinite sets. Calculus (contrast with Zeno's Paradox). The list goes on.
I should also point out that many, if not all, of these discoveries (or "inventions") either arose as a solution to a scientific problem (f.ex. Calculus), or were found to have a useful scientific application after the fact (f.ex. imaginary numbers). How can this be, if mathematics is entirely "non-empirical" ?
Hmm, I'll have to think about the derivation of zero, the irrational numbers, etc.
The motivation for derivation of mathematical facts is different from the ability to derive them. I don't why the Cartesian skeptic would want to invent calculus. I'm only saying it would be possible. It wouldn't be possible if mathematics was not independent of empirical facts (because the Cartesian skeptic is isolated from all empirical facts except the skeptic's own existence).
Hmm, "justified" generally has a social component, so I doubt that this definition is useful.
So this WP page doesn't exist? ;)
My position, FWIW, is that all of science is socially mediated (as a consequence of being a human activity), mathematics no less than any other science. Whether a mathematical proposition will be assessed as true by mathematicians is a property ultimately based on physics - currently the physics of our brains.
I disagree, as, I suspect, you already know :-)
But I have a further disagreement with your last sentence:
What do you mean, "and therefore" ? As I see it, "empirical" is the opposite of "social". Gravity exists regardless of whether I like it or not, and regardless of how many passionate essays I write about Man's inherent freedom to fly by will alone.
Yes, non-empirical is the wrong word. I mean to assert that mathematics is independent of empirical fact (and therefore non-social. A sufficiently intelligent Cartesian skeptic could derive all of mathematics in solitude).
Didn't Gödel show that nobody can derive all of mathematics in solitude because you can't have a complete and consistented mathamatical framework?
Goedel showed that no one can derive all of mathematics at all, whether in solitude or in a group, because any consistent system of axioms can't lead to all the true statements from their domain.
Anyone know whether it's proven that there are guaranteed to be non-self-referential truths which can't be derived from a given axiom system? (I'm not sure whether "self-referential" can be well-defined.)
I don't know whether this is true or not; arguments could (and have) been made that such a skeptic could not exist in a non-empirical void. But that's a bit offtopic, as I still have a problem with your previous sentence:
Are you asserting that all things which are "dependent on empirical fact" are "social" ? In this case, you must be using the word "social" in a different way than I am.
If we lived in a culture where belief in will-powered flight was the norm, and where everyone agreed that willing yourself to fly was really awesome and practically a moral imperative... then people would still plunge to their deaths upon stepping off of skyscraper roofs.
:) It is the case that the coherence of the idea of the Cartesian skeptic is basically what we are debating.
I'm specifically asserting that things that are independent of empirical facts are non-social.
I think that things that are subject to empirical fact are actually subject to social mediation, but that isn't a consequence of my previous statement.
What does rejection of the assertion "If you think you can fly, then you can" have to do with the definition of socially mediated? I don't think post-modern thinking is committed to the anti-physical realism position, even if it probably should endorse the anti-physical models position. The ability to make accurate predictions doesn't require a model that corresponds with reality.
The frequentist vs. baysian debate is a debate of computing mathematical paradigms. True mathematicians however shun statistics. They don't like the statistical pradigm ;)
Gödel's discovery ended a certain mathmatical pradigm of wanting to construct a complete mathematics from the ground up.
I could imagine a future paradigm shift way from the ideal of mathmatical proofs to more experimental math. Neural nets or quantum computers can give you answer to mathematical question that you ask that might be better than the answer s that axiom and proof based math provides.
Except, in practice mathematics still works this way.
It had damn well better be checked that way, because it rests on the assumption of flat space, which may or may not be true. The derivation from the axioms is not checked by empirical data; the axioms themselves are. If you don't check the axioms, you don't have knowledge, you have pretty equations on paper, unconnected to any fact. Pythagoras is just as much empirical knowledge as Einstein; it's just that the axioms are closer to being built-in to the human brain, so you get an illusion of Eternal Obviousness. Try explaining the flat-space axioms to squid beings from the planet Rigel, which as it happens has a gravity field twenty times that of Earth, and see how far you get. "There's only one parallel line through a given point", you say, and the squid explodes in scorn. "Of course there's more than one! Here, I'll draw them for you and you can see for yourself!"
I agree. Isn't deriving propositions from axioms what mathematics is?
A mathematician might say so, yes. I'm a physicist; I'm not really interested in what can be derived from axioms unconnected to reality.
No, that's not how you prove it, but you can check it pretty easily with right triangles. Similarly, if you believe that Pi == 3, you only need a large wheel and a piece of string to discover that you're wrong. This won't tell you the actual value of Pi, nor would it constitute a mathematical proof, but at least the experience would point you in the right direction.
If you find a right triangle with sides (2.9, 4, 5.15) rather than (3,4,5), are you ever entitled to reject the Pythagrean theorem? Doesn't measurement error and the non-Euclidean nature of the actual universe completely explain your experience?
In short, it seems like you can't empirically check the Pythagorean theorem.
That is not what I said. I said, regarding Pi == 3, "this won't tell you the actual value of Pi, nor would it constitute a mathematical proof, but at least the experience would point you in the right direction". If you believe that a^2 + b^2 = c^5, instead of c^2; and if your instruments are accurate down to 0.2 units, then you can discover very quickly that your formula is most probably wrong. You won't know which answer is right (though you could make a very good guess, by taking more measurements), but you will have enough evidence to doubt your theorem.
The words "most probably" in the above sentence are very important. No amount of empirical measurements will constitute a 100% logically consistent mathematical proof. But if your goal is to figure out how the length of the hypotenuse relates to the lengths of the two sides, then you are not limited to total ignorance or total knowledge, with nothing in between. You can make educated guesses. Yes, you could also get there by pure reason alone, and sometimes that approach works best; but that doesn't mean that you cannot, in principle, use empirical evidence to find the right path.
Peer review. If the next two hundred scientists who measure your triangle get the same measurements from other rulers by different manufacturers, you'd be completely justified in rejecting the Pythagorean theorem.
My challenge to you: go out and see if you can find a right triangle with those measurements.
Sure, how about a triangle just outside a black hole.
That was a quick trip. Which black hole was it?
You're completely justified in rejecting Euclid's axioms. You're not at all justified in rejecting the Pythagorean theorem.
Upvoted for your excellent demonstration of peer review ;) I stand corrected.
I am having trouble with this as a statement of historical fact. Isn't that how they did it?
You could call it a pradigm shift that we today don't like how they did it ;)
I'm not sure that's how it was motivated historically. Note that Euclid's proof (Edit: not Euler) doesn't require measuring anything at all.
To use a different example, how would one go about measuring whether there are more real numbers than integers? The proof is pretty easier, but it doesn't require any empirical facts as far as I can tell.
I think you mean Euclid's proof, and he was working centuries after Pythagoras, who was himself working over a thousand years after the Babylonians, who discovered Pythogorean Triples (the ones you notice by measuring).
To restate, I'm fine with saying that a proof for the Pythogorean Theorem exists that does not require measuring physical triangles, but I'm not comfortable with the statement that it cannot be proved by measuring physical triangles, which is what your original comment implied to me.
As discussed in the other subthread, I think that Deutsch's intention was to argue that any instance of a proof, as an object, has to exist in reality somewhere, which is a very different claim.
It depends on what you mean by "proved". The Pythagorean Theorem applies to all possible triangles (on a flat Euclidean plane), and the answer it gives you is infinitely precise. If you are measuring real triangles on Earth, however, the best you could do is get close to the answer, due to the uncertainty inherent in your instruments (among other factors). Still, you could very easily disprove a theorem that way, and you could also use your experimental results to zero in on the analytical solution much faster than if you were operating from pure reason alone.
It's just the problem of induction.
Other subthread? Don't see where anyone made that point. Moreover, I don't think it is a good reading of the original quote.
That's not fairly represented by saying "All actual proofs are on physical paper (or equivalent)."
I was thinking of this comment. If by "knowledge" he means "a piece of memory in reality," then by definition there is no abstract knowledge, and no abstract proofs, because he limited himself to concrete knowledge.
That knowledge can describe concepts that we don't think of as concrete- the Pythogorean Theorem doesn't have a physical manifestation somewhere- but my knowledge of it does have a physical manifestation.
There are all kinds of quantitative ways in which there are more real numbers than integers. On the other hand a tiny minority of us regard Cantor's argument (that I think you're alluding to) as misleading and maybe false.