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The two meanings of mathematical terms
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In the latest Rationality Quotes thread, CronoDAS quoted Paul Graham:
It would not be a bad definition of math to call it the study of terms that have precise meanings.
Sort of. I started writing a this as a reply to that comment, but it grew into a post.
We've all heard of the story of epicycles and how before Copernicus came along the movement of the stars and planets were explained by the idea of them being attached to rotating epicycles, some of which were embedded within other larger, rotating epicycles (I'm simplifying the terminology a little here).
As we now know, the Epicycles theory was completely wrong. The stars and planets were not at the distances from earth posited by the theory, or of the size presumed by it, nor were they moving about on some giant clockwork structure of rings.
In the theory of Epicycles the terms had precise mathematical meanings. The problem was that what the terms were meant to represent in reality were wrong. The theory involved applied mathematical statements, and in any such statements the terms don’t just have their mathematical meaning -- what the equations say about them -- they also have an ‘external’ meaning concerning what they’re supposed to represent in or about reality.
Lets consider these two types of meanings. The mathematical, or ‘internal’, meaning of a statement like ‘1 + 1 = 2’ is very precise. ‘1 + 1’ is defined as ‘2’, so ‘1 + 1 = 2’ is pretty much the pre-eminent fact or truth. This is why mathematical truth is usually given such an exhaulted place. So far so good with saying that mathematics is the study of terms with precise meanings.
But what if ‘1 + 1 = 2’ happens to be used to describe something in reality? Each of the terms will then take on a second meaning -- concerning what they are meant to be representing in reality. This meaning lies outside the mathematical theory, and there is no guarantee that it is accurate.
The problem with saying that mathematics is the study of terms with precise meanings is that it’s all to easy to take this as trivially true, because the terms obviously have a precise mathematical sense. It’s easy to overlook the other type of meaning, to think there is just the meaning of the term, and that there is just the question of the precision of their meanings. This is why we get people saying "numbers don’t lie".
‘Precise’ is a synonym for "accurate" and "exact" and it is characterized by "perfect conformity to fact or truth" (according to WordNet). So when someone says that mathematics is the study of terms with precise meanings, we have a tendancy to take it as meaning it’s the study of things that are accurate and true. The problem with that is, mathematical precision clearly does not guarantee the precision -- the accuracy or truth -- of applied mathematical statements, which need to conform with reality.
There are quite subtle ways of falling into this trap of confusing the two meanings. A believer in epicycles would likely have thought that it must have been correct because it gave mathematically correct answers. And it actually did . Epicycles actually did precisely calculate the positions of the stars and planets (not absolutely perfectly, but in principle the theory could have been adjusted to give perfectly precise results). If the mathematics was right, how could it be wrong?
But what the theory was actually calcualting was not the movement of galactic clockwork machinery and stars and planets embedded within it, but the movement of points of light (corresponding to the real stars and planets) as those points of light moved across the sky. Those positions were right but they had it conceptualised all wrong.
Which begs the question of whether it really matters if the conceptualisation is wrong, as long as the numbers are right? Isn’t instrumental correctness all that really matters? We might think so, but this is not true. How would Pluto’s existence been predicted under an epicycles conceptualisation? How would we have thought about space travel under such a conceptualisation?
The moral is, when we're looking at mathematical statements, numbers are representations, and representations can lie.
If you're interested in knowing more about epicycles and how that theory was overthrown by the Copernican one, Thomas Kuhn's quite readable The Copernican Revolution is an excellent resource.
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Comments (78)
Or: "Physics is not Math"
This seems to be a common response - Tyrrell_McAllister said something similar:
I take that distinction as meaning that a precise maths statement isn't necessarily reflecting reality like physics does. That is not really my point.
For one thing, my point is about any applied maths, regardless of domain. That maths could be used in physics, biology, economics, engineering, computer science, or even the humanities.
But more importantly, my point concerns what you think the equations are about, and how you can be mistaken about that, even in physics.
The following might help clarify.
A successful test of a mathematical theory against reality means that it accurately describes some aspect of reality. But a successful test doesn't necessarily mean it accurately describes what you think it does.
People successfully tested the epicycles theory's predictions about the movement of the planets and the stars. They tended to think that this showed that the planets and stars were carried around on the specified configuration of rotating circles, but all it actually showed was that the points of light in the sky followed the paths the theory predicted.
They were committing a mind projection 'fallacy' - their eyes were looking at points of light but they were 'seeing' planets and stars embedded in spheres.
The way people interpreted those successful predictions made it very hard to criticise the epicycles theory.
The issue people are having is, that you start out with "sort of" as your response to the statement that math is the study of precisely-defined terms. In doing so, you decide to throw away that insightful and useful perspective by confusing math with attempts to use math to describe phenomena.
The pitfalls of "mathematical modelling" are interesting and worth discussing, but it actually doesn't help clarify the issue by jumbling it all together yourself, then trying to unjumble what was clear before you started.
I've never gotten that impression. Proponents of epicycles were working from the assumption that celestial motion must be perfect, and therefore circular, and so were making the math line up with that. Aside from trying to fit an elliptical peg into a circular hole, they seemed to merely believe that the points of light in the sky follow the paths the theory predicts.
But then, it's been a few years since I've read any of the relevant sources.
You seem to be trying to, somewhat independently of the way it is done in science and mathematics typically, arrive at distinct concepts of accurate, precise, and predictive. At least how I'm using them, these terms can each describe a theory.
Precise - well defined and repeatable, with little or no error.
Accurate - descriptive of reality, as far as we know.
Predictive - we can extend this theory to describe new results accurately.
A theory can be any of these things, on different domains, to different degrees. A mathematical theory is precise*, but that need not make it accurate or predictive. It's of course a dangerous mistake to conflate these properties, or even to extend them beyond the domains on which they apply, when using a theory. Which is why these form part of the foundation of scientific method.
*There's a caveat here, but that would be an entire different discussion.
And this is not just an abstract issue. The fact that predictiveness has almost nothing to do with accuracy, in the sense of correspondence is one of the outstanding problems with physicalism.
"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."
-- Albert Einstein
As far as I can see, that's just an acknowledgement that we can't know anything for certain -- so we can't be certain of any 'laws', and any claim of certainty is invalid.
I was arguing that any applied maths term has two types of meanings -- one 'internal to' the equations and an 'external' ontological one, concerning what it represents -- and that a precise 'internal' meaning does not imply a precise 'external' meaning, even though 'precision' is often only thought of in terms of the first type of meaning.
I don't see how that relates in any way to the question of absolute certainty. Is there some relationship I'm missing here?
The quote is getting at a distinction similar to yours. It's from the essay Geometry and Experience, published as one chapter in Sidelights on Relativity (pdf here).
A different quote from the same essay goes:
As the other commenters have indicated, I think that your distinction is really just the distinction between physics and mathematics.
I agree that mathematical assertions have different meanings in different contexts, though. Here's my attempt at a definition of mathematics:
Mathematics is the study of very precise concepts, especially of how they behave under very precise operations.
I prefer to say that mathematics is about concepts, not terms. There seems to me to be a gap between, on the one hand, having a precise concept in one's mind and, on the other hand, having the ability to articulate that concept in words.
It is more difficult to say what mathematical statements mean.
Often, a mathematical statement will just be about the mathematical concepts it mentions. A statement of geometry, for example, is often just about the concepts of triangle, circle, or whatever geometrical concepts it mentions.
However, in a mathematician's working practice, these statements often stop being about the original mathematical concepts. Rather, they might become about isomorphism classes of concepts (which are themselves concepts), where "isomorphism" is taken in the sense of some mathematical theory. Or the statements might become mathematical concepts in their own right. When the mathematician thinks of them this way, they no longer refer to anything outside of themselves. They are themselves the precise concepts being studied mathematically.
Thus, different utterances of a mathematical statement can mean different things. That is, the things that it refers to, and what it asserts about those things, can change with different utterances (while never being about actual physical things, which is the domain of physics). Usually, mathematicians try to arrange it so that a given statement has (provably) the same truth value in all cases, but not always.
I have not read Kuhn's work, but I have read some Ptolemy, and if I recall correctly he is pretty careful not to claim that the circles in his astronomy are present in some mechanical sense. (Copernicus, on the other hand, literally claims that the planets are moved by giant transparent spheres centered around the sun!)
In his discussion of his hypothesis that the planets' motions are simple, Ptolemy emphasizes that what seems simple to us may be complex to the gods, and vice versa. (This seems to me to be very similar to the distinction between concepts that are verbally simple and concepts that are mathematically simple, which EY and others have referenced repeatedly here and at OB.) And while the device of the Equant* is fairly simple mathematically, it would raise so many mechanical complications that Copernicus rejected it, not because it's inaccurate, but because he considered it too mechanically complicated.
Ptolemy also tended to demonstrate equivalence between two ways of accounting for observations, which again suggests that he was not trying to describe the mechanics of the planets' motion, but rather only the mathematics of their motion.
I am not familiar with later Geocentric astronomy, and it may for all I know be the case that later thinkers thought that the epicycles had their own existence and moved the planets mechanically, but the history of astronomy is a little more complex than the popular account of the Copernican revolution would suggest.
If anything, Copernicus's insight was to permit the mathematics to inform his physical judgment rather than the other way around. Ptolemy rejected the heliocentric hypotheses intentionally and explicitly on common-sense grounds. Copernicus permitted the famous Ptolemaic coincidences (e.g. the fact that all the planets' epicycles tend to follow the mean motion of the sun) to suggest the sun as a more simple and natural center.
*The Equant is a device for describing planets whose uneven motion relative to the earth cannot be adequately accounted for by eccentricity and epicycles. In essence, Ptolemy assigns three centers to the planet's motion. One is the earth (the point of observation), another is the center of the circle the planet describes in space (the eccenter), and the third is the point with respect to which the angular motion of the planet is constant (i.e. the planet would appear to be revolving at a uniform rate if observed from this third point).
From what I've heard and read, Ptolemy was a believer in the "shut up and calculate" interpretation of astronomical mechanics. If the equations make accurate predictions, the rest doesn't matter, right?
Bohr took a similar attitude toward quantum mechanics when Einstein complained about it not making any sense: the "meaning" or "underlying reality" simply isn't important - the only thing that matters is whether or not the equations work.
Considering that, in the end, the Earth does go around the Sun, there are some fascinating lessons to be derived from all this.
In particular - yes, the Gods may have a different notion of simplicity, as 'twere, but unless you can exhibit that alternative notion of simplicity, it seems we should still penalize hypotheses that sure look complicated.
And the Sun does go around the Earth.
Of course, most of the observation that led to people thinking that the Sun goes around the Earth in the first place was based on the Earth's rotation on its axis, so that's a whole different issue.
I offer this link not as any sort of pedantic correction, but simply as a resource for those interested in learning exactly what modern physics has to say about this question. (Not difficult; highly recommended.)
(An ulterior motive for posting this is that I always have a terrible time tracking down that particular post.)
Would it have been better for Ptolemy to forego the epicycles and suggest that the planets describe simple circles around the earth? Not only would that have been less accurate, but it would have obscured the coincidences that enabled later astronomers like Copernicus to take a god's eye view and notice that a heliocentric framework was a much simpler interpretation of the data.
My point is that complexity was not the problem. If Ptolemy had tried on purpose to make his model less complex, it would likely have come at the expense of accuracy. The problem was that Ptolemy had too much common sense, and was not willing to let the math dictate his physics rather than the other way around.
To add to what others have already commented...
It is theoretically possible to accurately describe the motions of celestial bodies using epicycles, though one might need infinite epicycles, and epicycles would themselves need to be on epicycles. If you think there's something wrong with the math, it won't be in its inability to describe the motion of celestial bodies. Rather, feasibility, simplicity, usefulness, and other such concerns will likely be factors in it.
While 'accurate' and 'precise' are used as synonyms in ordinary language, please never use them that way when talking technically about the meanings of words. They are very useful jargon.
Similarly, please never use 'begs the question' or any form of it when not referring to the logical fallacy.
But I don't think there's anything "wrong with the math" - I even said precisely that:
I was trying to talk about how people actually use them, and one of the things I was suggesting is that people do actually tend to treat them as synonymous.
Isn't this a little picky? The way I used 'begs the question', in the sense of 'raises the question', is fairly common usage. Language is constantly evolving and if you wanted to claim that people only should use terms and phrases in line with their original meanings you'd have throw away most language.
Language is always evolving, but more recently, and especially currently, evolving usages are still pretty sloppy. If you want to be less wrong you need to use language more precisely. That is, don't use new usages when an older usage is more precise or accurate, unless there is a real need, especially don't use technical terms in sloppy common usages.
Formatting point: please use the "summary break" button when you have a long post.
Please see this previous comment of mine.
The point here is that it "1+1=2" should not be taken as a statement about physical reality, unless and until we have agreed (explicitly!) on a specific model of the world -- that is, a specific physical interpretation of those mathematical terms. If that model later turns out not to correspond to reality, that's what we say; we don't say that the mathematics was incorrect.
Thus, examples of things not to say:
"Relativity disproves Euclidean geometry."
"Quantum mechanics disproves classical logic"
"I am an infinite set atheist - have you ever actually seen an infinite set?"
I fully agree, and this is completely in line with the points I was trying to make.
Re the last quote: I didn't expect Eliezer to say something like that. Has he actually ever seen a finite set?
Perhaps he meant "seen" in the sense of "visualized." What happens when we try to introspect on our visualization of some mathematical terms?
Well I can't visualize an infinite set, but neither can I imagine a finite set, nor the number 5 for that matter. I can imagine five dots, or five apples, but not 5. In terms of my visualization, "5" seems to be an unfinished utterance. My mind wants to know, "5 what?" before it will visualize anything, or else it just puts up 5 black circles or whatever.
I interpreted that to mean that Eliezer doubts that a model that requires infinite sets will correspond to reality, not that the mathematics are incorrect. The figurative use of the word "atheist" makes the statement ambiguous, but his use of the phrase "actually seen" indicates that his concern is with modeling reality, not the math per se.
That was my (charitable) interpretation too, until, to my dismay, Eliezer confirmed (at a meetup) that he had "leanings" in the direction of constructivism/intuitionism -- apparently not quite aware of the discredited status of such views in mathematics.
And indeed, when I asked Eliezer where he thinks the standard proof of infinite sets goes wrong, he pointed to the law of the excluded middle.
His idol E.T. Jaynes may be to blame, who in PTLS explicitly allied himself with Kronecker, Brouwer, and Poincaré as opposed to Cantor, Hilbert, and Bourbaki -- once again apparently not understanding the settled status of that debate on the side of Cantor et al. One is inclined to suspect this is where Eliezer picked such attitudes up.
Huh. Good to know.
Can you elaborate on constructivism, intuitionism, and their discrediting? And what that has to do with the law of the excluded middle? I thought constructivism and intuitionism were epistemological theories, and it isn't immediately obvious how they apply to mathematics. Does a constructivist mathematician not believe in proof by contradiction?
Also, I don't know what you mean by "the standard proof of infinite sets".
Yes, a constructivist mathematician does not believe in proof by contradiction.
I think komponisto is a little confused about the discredited status of intuitionism, and you're a little confused about math vs epistemology. Here's a short sweet introduction to intuitionist math and when it's useful, much in the spirit of Eliezer's intuitive explanation of Bayes. Scroll down for the connection between intuitionism and infinitesimals - that's the most exciting bit.
PS: that whole blog is pretty awesome - I got turned on to it by the post "Seemingly impossible functional programs" which demonstrates e.g. how the problem of determining equality of two black-box functions from reals in [0, 1] to booleans turns out to be computationally decidable in finite time (complete with comparison algorithm in Haskell).
Not at all. Precious few are the mathematicians who take the views of Kronecker or Brouwer seriously today. I mean, sure, some historically knowledgeable mathematicians will gladly engage in bull sessions about the traditional "three views" in the philosophy of mathematics (Platonism, intuitionism, and formalism), during which they treat them as if on par with each other. But then they get up the next day and write papers that depend on the Axiom of Choice without batting an eye.
The philosophical parts of intuitionism are mostly useless, but it contains useful mathematical parts like Martin-Löf type theory used in e.g. the Coq proof assistant. Not sure if this is relevant to Eliezer's "leanings" which started the discussion, but still.
Right, but in this context I wouldn't label such "mathematical parts" as part of intuitionism per se. What I'm talking about here is a certain school of thought that holds that mainstream (infinitary, nonconstructive) mathematics is in some important sense erroneous. This is a belief that Eliezer has been hitherto unwilling to disclaim -- for no reason that I can fathom other than a sense of warm glow around E.T. Jaynes.
(Needless to say, Eliezer is welcome to set the record straight on this any time he wishes...)
I do not understand what the word "erroneous" is supposed to mean in this context.
For the sake of argument, I will go ahead and ask what sort of nonconstructive entities you think an AI needs to reason about, in order to function properly.
Some senses of "erroneous" that might be involved here include (this list is not necessarily intended to be exhaustive):
Mathematically incorrect -- i.e. the proofs contain actual logical inconsistencies. This was argued by some early skeptics (such as Kronecker) but is basically indefensible ever since the formulation of axiomatic set theory and results such as Gödel's on the consistency of the Axiom of Choice. Such a person would have to actually believe the ZF axioms are inconsistent, and I am aware of no plausible argument for this.
Making claims that are epistemologically indefensible, even if possibly true. E.g., maybe there does exist a well-ordering of the reals, but mere mortals are in no position to assert that such a thing exists. Again, axiomatic formalization should have meant the end of this as a plausible stance.
Irrelevant or uninteresing as an area of research because of a "lack of correspondence" with "reality" or "the physical world". In order to be consistent, a person subscribing to this view would have to repudiate the whole of pure mathematics as an enterprise. If, as is more common, the person is selectively criticizing certain parts of mathematics, then they are almost certainly suffering from map-territory confusion. Mathematics is not physics; the map is not the territory. It is not ordained or programmed into the universe that positive integers must refer specifically to numbers of elementary particles, or some such, any more than the symbolic conventions of your atlas are programmed into the Earth. Hence one cannot make a leap e.g. from the existence of a finite number of elementary particles to the theoretical adequacy of finitely many numbers. To do so would be to prematurely circumscribe the nature of mathematical models of the physical world. Any criticism of a particular area of mathematics as "unconnected to reality" necessarily has to be made from the standpoint of a particular model of reality. But part (perhaps a large part) of the point of doing pure mathematics (besides the fact that it's fun, of course), is to prepare for the necessity, encountered time and time again in the history of our species, of upgrading -- and thus changing --our very model. Not just the model itself but the ways in which mathematical ideas are used in the model. This has often happened in ways that (at least at the time) would have seemed very surprising.
Well, if the AI is doing mathematics, then it needs to reason about the very same entities that human mathematicians reason about.
Maybe that sounds like begging the question, because you could ask why humans themselves need to reason about those entities (which is kind of the whole point here). But in that case I'm not sure what you're getting at by switching from humans to AIs.
Do you perhaps mean to ask something like: "What kind of mathematical entities will be needed in order to formulate the most fundamental physical laws?"
Aaaand this makes me curious. Eliezer, for the sake of argument, do you really think we'd do good by prohibiting the AI from using reductio ad absurdum?
Oi, that's not right. The domain of these functions is not the set of reals in [0, 1] but the set of infinite sequences of bits; while there is a bijection between these two sets, it's not the obvious one of binary expansion, because in binary, 0.0111... and 0.1000... represent the same real number. There is no topology-preserving bijection between the two sets. Also, the functions have to be continuous; it's easy to come up with a function (e.g. equality to a certain sequence) for which the given functions don't work.
Of course, it happens that the usual way of handing "real numbers" in languages like Haskell actually handles things that are effectively the same as bit sequences, and that there's no way to write a total non-continuous function in a language like Haskell, making my point somewhat moot. So, carry on, then.
Your comment is basically correct. This paper deals with the representation issue somewhat. But I think those results are applicable to computation in general, and the choice of Haskell is irrelevant to the discussion. You're welcome to prove me wrong by exhibiting a representation of exact reals that allows decidable equality, in any programming language.
Relativity teaches us that "the earth goes around the sun" and "the sun goes around the earth, and the other planets move in complicated curves" are both right. So to say, "Those positions [calculated by epicycles] were right but they had it conceptualised all wrong," makes no sense.
Hence, when you say the epicycles are wrong, all you can mean that they are more complicated and harder to work with. This is a radical redefinition of the word wrong.
So, basically, I disagree completely with your conclusion. You can't say that a representation gives the right answers, but lies.
You're technically right about general relativity (so far as I grok it), but the hypothesis of geocentrism as understood pre-GR still fails hard compared to that of heliocentrism understood pre-GR.
Geocentrism doesn't logically imply that the earth doesn't rotate, but that hypothesis was never taken seriously except by heliocentrists, who then found experimental evidence of that rotation. Not to mention that epicycles were capable of explaining practically any regular pattern, and thus incapable of making the novel predictions of Newtonian gravity, which gives almost the right predictions assuming heliocentrism but gives nonsense assuming geocentrism.
It's far worse than just being more complicated; most epicycle-type hypotheses fail harder than neural networks once they leave their training set.
Grahams point s straightforward if expressed as 'pure maths is the study of terms with precise meannigs'.
I'm not in the business of telling people what values to have, but if you are a physcalist, you are comited to more than instrumental.
The fact that predictiveness has almost nothing to do with accuracy, in the sense of correspondence is one of the outstanding problems with physicalism