Not everyone likes to go and read the full text of references given as backup. I thought it important to have the term out there for anyone reading the thread, and not just those who bother to read the article.
To me as a physicist, mathematics is the language of science rather than a science in it’s own right.
Like Derleth said,
Using maths (sorry USAers I’m a Brit), we can communicate the ides of science universally, well at least anywhere on earth. Regardless of different spoken languages, the tools of maths are the same the world over.
Yes, I do see where this is coming from. But I also asked a (tangental to this thread) question and I hope someone can give me an answer.
Sometimes a cigar is just an extremely oblate spheroid. 
In this aspect, it seems to me, mathematics is sort of like a language. There have to be certain terms that are just intuitive or instinctive because words are defined in terms of other words and so there is always just a touch of circularity about the basics.
Yes, the term does originate from the title of Wigner’s article.
Yes, this is precisely the case. Mathematics is a language, and if you attempted to define everything used you would indeed run into a problem of circularity or infinite recursion, which is the same thing.
bonzer: Thank you for the answer. I suspected as much, but I didn’t know that the title of that paper had become a general term.
That’s just my point, no pun intended. For any given interpretation of ‘point’, etc, you have to decide whether the axioms hold true.
For instance, the Peano axioms are supposed to capture the behavior of the integers. How do we know if they really do? We apply the axioms to derive theorems (“predictions”) about the integers. We then use those theorems to work with the integers (“experiments”). If the experiments work, then we believe that the axioms have captured the essence of the integers.
For an example of a failed experiment, look at the Russell paradox in set theory. Using the axioms of the time, Russell wanted to consider the “set of all sets that don’t contain themselves.” Does this set contain itself? Whether you answer “yes” or “no”, you obtain a contradiction. (I’m sure you know this, Mathochist, I’m just recapping it for others.) Anyway, Russell discovered that the then-current axioms weren’t sufficiently restrictive to capture the behavior of sets. Mathematicians had to revise their theory of sets in order to reflect what they found “experimentally”. That’s just like a scientist revising a theory to reflect the demands of a laboratory experiment.
Math (or maths–whatever) isn’t a science. In the sciences, the hypotheses whose truth or falsehood you wish to determine are falsifiable. (I find Popper’s notion of falsifiability very useful in determining which disciplines are and are not sciences.) In mathematics, you can logically prove theorems you propose. That makes math a very different kind of discipline from anything in the sciences.
The fact that math provides uncannily good descriptions of various physical phenomena is beside the point.
You’re confusing the natural numbers with our real-world experiences which motivated the definition of natural numbers. What is being tested is not the N itself, but the model which uses N as an analogue of counting. That’s not math any more than a gravitometer testing the model which uses differential geometry as an analogue of gravity is.
What Russell showed was that the axioms as they stood were inconsistent, meaning not that they “failed to capture the behavior of sets” , but that they didn’t capture the behavior of anything. The naïve set theory of the time is still a perfectly valid piece of mathematics, but it’s almost useless since it says anything is true.
A good analogy is the way a universe with a fine structure constant significantly different from 1/137 is – at least within the standard model – a perfectly valid universe. It just doesn’t have much of anything in it.
Others have mentioned that this activity can and should be carried out according to the scientific method, so I won’t bother with that. I just wanted to know if the pun was intended.
Actually, I don’t think anyone has explained how that would happen. If you’re doing nothing but counting an animal population, unless you’re verifying an earlier hypothesis, where is the scientific method in that? The results of the count can be falsified, but does it predict anything? I think not. That’s why some purists would say that it “isn’t science.”
It was.
Now I see what you were trying to get at, but that’s not what you said originally:
Of course, it’s quite possible to wander around in the marsh for weeks counting fecal pellets in order to test hypotheses about distribution in relation to environmental variables, population models, or other things.
It is true that many scientists don’t consider purely observational or descriptive (non-hypothesis-testing) studies as “real” science. However, such data collection does often form the foundation for testing hypotheses in the future. For example, descriptive studies in natural history and taxonomy provides the fodder on which evolutionary analyses can be based. In any case, such studies may be considered more “scientific” (in the strict sense) than mathematics because they are empirical, depending on observation of the real world for their validity. As others have pointed out, mathematics has no such constraint.
OK, so suppose the Peano postulates are found to be inconsistent. After all, you can’t prove them consistent using just the postulates themselves - Goedel showed that. So, at some point in the future, they may be found to be inconsistent. Or, suppose you were able to prove something in N that turns out not to be true for the counting numbers. (Suppose, for instance, that we had a perfectly rigorous proof of Fermat’s last theorem and an explicit counterexample.) Are you going to keep proving things using them, or are you going to throw them out and replace them with other axioms? In the end, it’s the “real world experiences” that count. The definitions are only useful to the extent that they capture some aspect of those experiences.
To put it another way, if naive set theory “is still a perfectly valid piece of mathematics”, why aren’t people still using it to prove theorems? You answer the question yourself: it “didn’t capture the behavior of anything.”
That’s just what I’m saying: that a formal system is only interesting when it captures the behavior of something: points, numbers, functions, matrices.
It’s rather obvious that you haven’t spent much time around mathematicians. Most of us don’t much care whether or not our work applies to the “real world”. As a case in point, Cohen forcing is a technique developed to show the existence of (at least) two different models of ZFC – two different topoi. Only one’s natural-numbers object can “be” the one which obtains in the “real world”. Would you say that this result is uninteresting because at most one of the models it talks about can reflect the real world?
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That’s just what I’m saying: that a formal system is only interesting when it captures the behavior of something: points, numbers, functions, matrices.
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Now you’re getting into the deep ontological question of mathematics. Do points, numbers, functions, matrices, and so on really “exist”?
First, let’s throw out this antiquated “formal system” verbiage. A consistent structure is only interesting if it captures the behavior of something, yes. Any consistent structure does capture the behavior of any of its instantiations, though.
Projective geometry captures the behavior of its “points” and “lines”, but those can be what we normally think of as “lines” and “points”, respectively in the plane (suitably extended at infinity). A “point” in the dual instantiation of the structure of projective geometry does have an extension, while a line has none. Our real-world experience of “points” includes a lack of extension (as reflected in Euclid’s definitions), but the axioms of projective geometry don’t refer to such a concept and so asking about “extension” (or any other real-world concept not included in the axioms) is meaningless.
As a further example, all evidence from cosmology and physics seems to indicate that our observable universe is essentially finite in content: there are only so many fundamental particles, for instance. The natural numbers cannot, then, be interesting under your view since most of them are much larger than any counting numbers which arise in the “real world”. There are many people who seriously question the existence of any instantiation of the structure of natural numbers (defined by the Peano axioms), and by extension of any instantiation of any mathematical structure which depends on the natural numbers (most of them). Do they say that mathematics is uninteresting? Hardly. They just say that as currently formulated it doesn’t apply to the “real world”.
OK, I confess that I don’t hang out with mathematicians very much these days. But I don’t think that what I mean by the “real world” is the same as what you’re thinking. See below.
Um, sorry for using “antiquated verbiage”. I promise to use “structure” instead from now on, if it makes you happy. :rolleyes:
I think this is what I’m getting at. What’s the difference between a “structure” and an “instantiation”? (is this the same as a “model”? what about “topos”?)
If I understand you correctly, you’re saying that the word “point” means one thing if it’s a point in the plane and a different thing if it’s a point in the dual instantiation of projective geometry. Fine, I understand that. I’m trying to make a deeper observation. When you talk of “what we normally think of as “lines” and “points”, respectively in the plane”, what are you talking about? How do you define the plane, for example? I’m trying to say that it’s based on an intuitive idea, abstracted from our experience. In the dual instantiation the link is different, but it’s still based on an abstraction from our experience. The ZFC axioms are based on abstractions from our experience.
You obviously don’t spend much time around physicists.
All the current evidence from cosmology indicates that the universe is spatially infinite.
Let me try to make this clearer. I’m not trying to say that math is only interesting if it has applications, or if it models something that actually exists in the physical world, or anything like that. So, whether the universe is actually infinite or finite is irrelevant.
Now, above you admitted that of the two different models of ZFC, only one (at most) can be true of the “real world” natural numbers. This is precisely why I say math is experimental. Which of those models gives a correct view of the natural numbers? The question only makes sense if the natural numbers exist independently of ZFC. So you’ve admitted that there are experimental questions in math!
As an example: the Peano axioms define a structure. Within ZFC, we can construct two distinct instantiations of this structure. The object filling the “zero slot” in each one is the empty set {}, but in one the successor function is S(n) = {n} and in the other it is S’(n) = n U {n}. 0 to 3 are respectively
{}, {{}}, {{{}}}, {{{{}}}}
{}, {{}}, {{},{{}}}, {{},{{}},{{},{{}}}}
in the two instantiations. Of course, for either to exist we need to assume the existance of a model of set theory: an instantiation of the structure defined by the ZFC axioms.
A topos is a much more technical concept tying into logical structures, but among other things it provides a model for set theory.
The original motivations of Euclidean geometry are abstractions from experience, yes. By now, however, the various branches have been logically wrested from that weak foundation. Differential geometry rests on looking “locally” like an affine space over R or C, which are themselves defined completely abstractly. Algebraic geometry was redefined practically from the ground up by Grothendieck in the middle of last century in terms of schemes and representable functors, both of which are motivated not as abstractions of experience from the physical world but as abstractions of experience from mathematics itself.
But there’s only a finite amount of stuff in it. It’s the limited ontology that I was using.
You miss the point. Nobody working in mathematics cares which model works. They’re both logically sound (or at least as sound as each other), so you just go ahead and work in one or the other and note when you need to use a property of one that the other doesn’t have. Maybe a physicist or philosopher cares which model obtains, but I’m not even sure that’s possible (even theoretically) to determine. The difference between the two that Cohen forcing distinguishes is a technical point about large transfinite numbers, which I’ve already pointed out the physical world may not have the ontology to support.
Even if it is managed, though, one of two large groups of mathematicians will have their work suddenly provably having absolutely nothing to do with the real world. Does that make it wrong?
I was about to comment on this, too, but Mathochist referred to the observable Universe, which is almost certainly finite, not the whole shebang, which may or may not be. It’s a philosophical question as to whether and to what extent physicists ought to concern ourselves with the parts which are not observable.
If you can’t measure and experiment on it, it’s not in an experimental ontology, which is what FriendRob is trying to push on mathematicians.
I’d like to put that on a t-shirt. May I? 