Thursday, May 07, 2009

What is Mathematics ?

Update: Welcome Econmistsviewers ? I am amazed you are here. This is, I think, a once in a lifetime opportunity to type a joke about Hendrik Hertzberg and the Georgia State Legislature. Huh ?

Hilzoy writes

the Georgia State Senate has adopted a resolution allowing the state to nullify any federal laws it thinks are unconstitutional. Hendrik Hertzberg actually read the resolution, and wrote a post that made me want to read it as well: he described it as "a Kompletely Krazy Kocktail of militia-minded moonshine and wacko white lightning ... [and that the] resolution is written in "a mock eighteenth-century style, ornate and pompous", I thought it was an unnervingly good imitation of eighteenth-century prose.

[skip]


"That to this compact each State acceded as a State, and is an integral party, its co-States forming, as to itself, the other party:..."


[skip]

I googled a distinctive phrase, and lo! it turns out that the Georgia resolution is a lightly modified version of Thomas Jefferson's Resolutions Related To The Alien And Sedition Acts.


Odd. I always thought that the tern "co-state" had more to do with Hamiltonian than with Jeffersonian constitutional exegesis.

Oh and Mr Hertzberg, I love you, but you have to learn to use the google.



Mark Kleiman comments

the "formalist" view of mathematics, as summed up by Bertrand Russell: "Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." That remains the dominant view among philosphers of mathematics, but the disconnect between the frailty of the wand and its evident power lends some support to the alternative view of Lakoff and Núñez that mathematics starts from basic human experience.

I must confess that my own views remain primitively Platonic; that is, I belive that it was the case that a group consisting of two pairs of trees contained four trees, even before there were humans around to notice that fact.


I think that Kleiman (like Plato) confuses mathematics and physics. I have no doubt that there are laws of physics and that they existed before humans did. For example, I'm sure that two trees plus two trees made four trees back when our ancestors were climbing in those trees.

However, the claim that two plust two must always equal four in general and not just for trees can be false. Consider a taught string. Wiggling it at one point with a given force will make a wave. Wiggling it somewhere else will create a wave. Wiggling it both places may create 0 waves. It also may create a wave with power 4 times that of each of the single waves. With waves 1+1 makes somewhere from 0 to 4.

The claim that massive objects must be like trees and not like waves on a bowed string must be a claim in phyisics and not in mathematics, because it is false. Arithmetic is just the same now as it was in 1900, however, now it is clear that Kleiman's statement about trees is a falsifiable hypothesis which happens to be true and not a necessary truth.

The title of Kleiman's post is "Yet Bridges Stand," but not all bridges stand. For example, the Tacoma Narrows Bridge was blown down by the wind.

How odd, a gust of wind is very light and the bridge was very heavy and strong. One could hardly notice the effect of even a very powerful gust of wind. However, when it comes to bridges, one plus one doesn't always equal two, it can be more than 2 if the wind is in harmony with the bridge.

update: Bruce Webb has video of destructive harmony.

I agree with Russell on this one. Mathematics concerns axiom systems. Mathematical terms are defined in use. The Euclidean term "line" can be approximately illustrated by something drawn with a pencil and a straight edge. It can also be equally validly illustrated by quite different sets which would be called curves in plain English. Neither illustration is more correct than the other. The meaning of the term is abstract, there is nothing except for the axioms and that means not enough for us to imagine it. For mathematics to be valid, it is not at all necessary for any entity which can fit in our universe to correspond to the axioms.

The hypotheses that sets of axioms can be intrepreted so that they correspond to physical laws are often very fruitful. However, they belong to physics not mathematics.

No doubt Lakoff and Núñez are right that, in history, many axioms began as hypotheses. I don't doubt that Russell agrees. This does not mean that they remained hypotheses, or that mathematicians consider, or should consider, their work vulnerable to evidence. Critically not all axiom systems began as hypotheses. The key example, of course, is Lobachevskian geometry (Euclid's axioms except that given a line A and a point B there is more than one line through B parallel to A). This began as an effort to prove by contradiction that no such axiom system was possible.

What is the basis in human experience for p-adic analysis ?

15 comments:

Bruce Webb said...

Hi Robert. I stuck up an awesome color video of the collapse of 'Galloping Gertie' at AB last december
http://angrybear.blogspot.com/2008/12/now-for-something-completely-different.html

As to the math question. Given that some animals have a rudimentary ability to count there is a sense in which arithmetic is Platonic but in general I find it more useful to see the laws of Physics and the higher maths as simply being part of Popper's World Three and so allowing inhabitants of World Two to try to explain observed events of World One.
http://en.wikipedia.org/wiki/Popperian_cosmology

If you try to explain Laws of Physics as being inherent in an undifferentiated world you risk casting all past progress in science into the dustbin of psuedo-science. To the extent that Maxwell's and then Einstein's equations are incomplete they are not in fact Platonic Ideals, that would seem to be a contradiction. But that does not make them non-scientific and so not 'laws' at all.

Robert said...

Ooops I will add a link to the film.

I think the worse problem is that current theories are probably false. So how why is a good theory which is falsified better than pseudoscience ? It is a useful approximation. Newton's equations are still used all of the time.

More importantly, it is a stylized fact. One can remember a whole lot of data with the shortcut that Newton's law of gravity works to within measurement error for the solar system except for the precession of the epihelion of the orbit of Mercury.

This is not just a convenience. Many data have been collected and not reported, because they fit Newtonian predictions. I think there is no other way to fit the facts than to make sure your theory gives almost exactly the same predictions as Newton's for, say, the location of planets.

Thus the theory is immortal. Now it lives on in Einstein's theory (which I was confidently told the first day of a course on general relativity is just an approximation).

Statements can be just plain true and also useful so long as they include qualifiers like "approximately" and "for speeds under 1,000 km/second" and "for densities less than 1000 that of lead at standard temperature and pressure." Such claims are, as yet, unrefuted. I personally can't doubt that they will never be refuted.

Actually this problem only arises for physics. No one pretends that hypotheses of chemists or biologists work fine at the center of the Sun.

Anonymous said...

I've read, enjoyed, and learned from a large number of your posts here and elsewhere. This is the only one which has been simply silly. I suggest you delete the thread and try to forget the whole thing.

Anonymous said...

symbolic information analysis

Anonymous said...

Some of this seems to be nothing more than playing with words. "With waves 1+1 makes somewhere from 0 to 4." Sure, if you mean by "+" something very different than what is generally assumed by people when they see the sign, "+". Likewise, in addition modulo 1, 3/4+3/4=1/2. But Platonists aren't arguing about different systems, where "+" has a wholly different sense.

In any case, when I hear someone support the idea that mathematics is merely a system of axioms (or systems, I guess), the first question I want to hear answered is how that can be squared with Godel. Godel considered his result, particularly the demonstration of truths that no axiomatic system can prove, a vindication of his own Platonism. I'm not sure I accept it entirely, but I can't say I've heard a better explanation.

Robert said...

Anonymous 1 Glad you've enjoyed and learned from other of my posts.

Anonymous 3. Godel first. If I understood correctly, the claim pre-Godel is that the only things we can say about axiom systems is that they don't lead to logical contradictions and they are interesting (an aesthetic judgement or maybe a claim that they are useful to physicists). Godel showed that we can't prove that axiom systems are consistent.

In general even if one has been used for centuries and no contradictions proven, one might be. So not that it can't be true that an axiom system is consistent but that we can't know that an axiom system is consistent.

One can still define mathematics as the set of all consistent axiom systems if one pleases (and I do).

The weasle phrase "in general" is because there are very simple axiom systems in "first order logic" such that Godel described a procedure which would find an inconsistency if there was one and which took a finite amount of figuring.

On addition, the problem really is that particles are sort of like waves. This means that it isn't easy to keep track of cases when 1+1=2 or, as you put it, when "+" is to be used or some other symbol.

It is simply a fact that there appear to be perfectly sound proofs that something can't happen and yet it does (I am thinking of the EPR experiment).

The rules for trees don't work for photons or electrons. This means that the fact that they work for trees is an empirical discovery, that is physics and not mathematics. The history of predicting what will happen given the pre-existing necessary truths of mathematics is littered with false predictions.

Another example, the original one, Euclidean geometry. Eintstein asserts that the Euclidian entity a line does not fit in our universe. If he was right, then something which was perceived to be a necessary truth which pre-existed Euclid is not a true statement about anything in this universe but rather an abstract concept which doesn't fit in here with us which was created by mathematicians.

I think that there is no statement about the real world which is necessarily true. Within axiom systems there are necessarily true statements. About reality there are only hypotheses which haven't been rejected yet.

Will Hardy said...

There are plenty of bridges in Japan that have stood for over a thousand years. But Japan is a very windy country.

Surely some of those winds must be in the harmonic for some of those bridges. And yet the bridges stand.

So therefore it cannot be true that when winds match the harmonic of bridges that 2 + 2 > 4.

Kaleberg said...

Every physicist will tell you that the Standard Theory is wrong. It doesn't deal with gravity, and that omission is what is driving a big chunk of theoretical physics today.

As for mathematics, some mathematicians consider axiomatic systems evidence about actual mathematical objects. That's why they tend to prove the same things over and over again in different ways, to confirm the evidence and extend their understanding.

(Mathematics is actually evidence based. If someone had produced a two dimensional map that could not have been colored with four colors it would have disproved the four color hypothesis.)

gromit said...

> What is the basis in human experience for p-adic analysis ?

as a recovering mathematician i would say that the basis in human experience for much of mathematics is the experience of thought, which is just as real (for a human) as the experience of the world outside the head.

as for 'there is no statement about the real world which is necessarily true' i think you are overstating the case. probably there are true statements which are imprecise and for which the level of imprecision can be bounded, so you can actually devise a true statement!

Anonymous said...

Godel says more than that. The self-consistency is one part of it, but the first part, the meat of the paper, is the demonstration that PM (or any axiomatic system for the naturals) is incomplete. There are statements such that neither the statement nor its negation are provable by the axioms. And the nature of the statement given by Godel is such that it is necessarily true, though unprovable. (It boils down to, roughly, 'This statement is not provable' -- proof of either this statement or its negation would lead to a contradiction.) Godel was a Platonist, and took this to be an important defense of his philosophical leanings -- a mathematical truth exists, but doesn't derive (in fact, can't derive) from axioms.

As for the counting, I find that empirical outlook hard to believe. If I had an apple in each hand, and putting them together and counting found that I had 3, I'd believe that my eyes or mind were failing sooner than I'd believe that 1+1=3. I think this would be true no matter how many times I repeated the experiment. Similarly, when quantum mechanics comes along and things don't seem to behave as they 'should', researchers don't conclude that we've been wrong about counting all along. They presume that the models must be more complicated, but "+" stays the same underneath all the crazy spaces and probability distributions.

And counting is about as far as I'd go with this. I'm not convinced that many of the derivative systems (like geometry) are anything more than axioms + symbols + rules. Iirc, there were even (pre-Godel) some proofs of relative consistency and completeness for some of these systems (ie, if arithmetic is consistent/complete, so is system X). But the naturals seem somehow special. (Like the Kronecker -- I think? -- quote, something like, 'God made the naturals, the rest is the work of man.')

Bruce Wilder said...

When Einstein was conducting the "thought experiment" of imagining himself sitting on the cowcatcher of a speed-of-light locomotive, was he testing the world-as-it-is, or the limits of geometric concepts? Was he discovering something about geometry, or the world?

"there is no statement about the real world which is necessarily true" seems maddeningly ambiguous to me.

Do (all) statements about the real world contain assertions of logical relation? Can even a possibly true statement about the real world be logically invalid?

The central puzzle of Platonic shadows runs both ways: the being of concepts does not cast shadows in the world; the being of the world does not cast shadows in the dream world of mathematics. Yet, somehow, we need the concept world to "see" relations in the real world: we need names and categories to distinguish one thing from another; we need preconceptions to recognize the facts of "cause and effect" in action.

We need logic, because we can not directly observe in the world the useful relations that enable a bridge to stand.

With respect to Newton's Law of Gravity, what is an empirical test meant to accomplish? Is it to confirm the algebra by arithmetical mimicry? Or, is it to establish a value for the gravitational constant?

What is the difference between General Relativity and Newton's Law of Gravity? Is it to be expressed only in the realist-pragmatist's probability and precision? Isn't General Relativity also conceptually more satisfactory? General Relativity resolves Newton's concerns about action at a distance and the mysterious coincidence of gravitational and inertial mass.

Just some random musings. Back to Mom.

Anonymous said...

"Anonymous 3. Godel first. If I understood correctly, the claim pre-Godel is that the only things we can say about axiom systems is that they don't lead to logical contradictions and they are interesting (an aesthetic judgement or maybe a claim that they are useful to physicists). Godel showed that we can't prove that axiom systems are consistent."

Nonsense! We KNOW many axiomatic systems are consistent. Godel proved that no system complex enough to include arithmetic can be both consistent and complete.

"In general even if one has been used for centuries and no contradictions proven, one might be. So not that it can't be true that an axiom system is consistent but that we can't know that an axiom system is consistent."

That not anything like what he Godel showed! And, we DO know many axiom systems are consistent, provably so.

"One can still define mathematics as the set of all consistent axiom systems if one pleases (and I do)."

Well, if you don't care that what you're defining ha nothing to do with actual mathematics, you can define it that way!

Do you take the least trouble to see if you have any clue what you are talking about?

Robert said...

Anonymous three, your vigorous criticism of my reply to your comment completely ignores "The weasle phrase "in general" is because there are very simple axiom systems in "first order logic" such that Godel described a procedure which would find an inconsistency if there was one and which took a finite amount of figuring."

My reply to your first comment explicitely noted that it is possible to show that some axiom systems are consistent, and that Godel had personally done so.

The claim to which you object was vague. "axiom systems" doesn't mean "any axiom sysem" (so my claim would be false" or "every axiom system" (so it would be true). I added the sentence which you ignore to clarify.

Sloppy writing (on a topic where sloppy writing is veyr inapprorpriate). Not, as bad, I think, as making a claim about what I said which contradicts a sentence in my reply.

Anonymous said...

Robert, as this is your blog, I assume you have the IP addresses of particular commenters, even if we sign as "anonymous". If so, you should know that I am the one who wrote the initial comment about Godel, and that I am NOT the person who wrote, "Nonsense! &c." I'm disappointed that you would try to discredit a poster by associating him with a completely separate, over-the-top reply.

My follow-up was the one dealing with completeness, not consistency. I understand that the first-order predicate calculus was shown to be complete and consistent by Godel (in his first dissertation). I have no objection to your characterization of consistency. My point centers around completeness. If you want to address that point, fine. If you don't, I hope you will at least publish this comment to make clear what I am actually saying. That much is a matter of basic honesty.

Robert said...

Dear anonymous not 3. Sorry. My mistake. I apologize.

Also you vastly over estimate my computer literacy. I have no idea how to find the IP addresses of commenters. I know it can be done, but I can't do it.