Monday, April 28, 2008

Is Mathematics Discovered or Invented?

Nicholas Knisely has an interesting post up today that plays off two recent articles that discuss the nature of mathematics. The real issue here is whether there is an underlying order in the world that we discover or do we invent that order ourselves? In other words was Plato right:

A post on Slashdot (h/t) points to an article in Science News about an ongoing debate about the connection between mathematics and the nature of reality. (We were just discussing that question here earlier this week.)

The article in Science News begins:

"[A]re new mathematical truths discovered or invented? Seems like a simple enough question, but for millennia, it has provided fodder for arguments among mathematicians and philosophers.

Those who espouse discovery note that mathematical statements are true or false regardless of personal beliefs, suggesting that they have some external reality. But this leads to some odd notions. Where, exactly, do these mathematical truths exist? Can a mathematical truth really exist before anyone has ever imagined it?
On the other hand, if math is invented, then why can’t a mathematician legitimately invent that 2 + 2 = 5?

...Plato is the standard-bearer for the believers in discovery. The Platonic notion is that mathematics is the imperturbable structure that underlies the very architecture of the universe. By following the internal logic of mathematics, a mathematician discovers timeless truths independent of human observation and free of the transient nature of physical reality. ‘The abstract realm in which a mathematician works is by dint of prolonged intimacy more concrete to him than the chair he happens to sit on,’ says Ulf Persson of Chalmers University of Technology in Sweden, a self-described Platonist.

The Platonic perspective fits well with an aspect of the experience of doing mathematics, says Barry Mazur, a mathematician at Harvard University, though he doesn’t go so far as to describe himself as a Platonist. The sensation of working on a theorem, he says, can be like being ‘a hunter and gatherer of mathematical concepts.’"

Read the rest here.

The article on Slashdot about the piece above also includes a link to a paper recently published entitled "Let Platonism Die" which includes a claim that Platonism "has more in common with mystical religions than with modern science".

The point of which seems to be that there's a fundamental question about reality. Does it reflect a designers intent, and is that intent mirrored at all levels of reality, or is creation essentially a result of random processes and any attempt to find an purpose or meaning is simply a human desire for order projected onto the Cosmos.


Father Knisely, who was a theoretical physicist and astronomer before becoming a priest comes down on the side of a ordered reality:

I'm such a thoroughgoing Platonist (neo actually) that it seems obvious to me that mathematics (and science) is about uncovering the underlying order. But I'm also religious and believe I've encountered in one way or another the orderer, so perhaps my sense that mathematics is discovered isn't all that surprising.




Read it all here.

4 comments:

Anonymous said...

It's an interesting question, one that has occupied both amateur and professional philosophers for a long time. I think it's probably fair to say that no one accepts a fully Platonistic view anymore (in the strict sense of accepting exactly what Plato said about mathematics), but there are lots of full-blooded realists around. For an interesting (and extremely influential) discussion of the debate, see Paul Benacerraf's 1973 paper "Mathematical Truth".

Lately, the major viewpoints are the following:
1) Structuralism, the view that mathematics is the science of structures. This does well at explaining the applicability of mathematics, for those of us who believe in a structured universe. Getting the details right has involved some pretty sophisticated logic, though.

2) Neo-logicisim, one of the current views most directly inspired by Plato. Holds that numbers, sets, etc really do exist, and that we know about them by certain 'abstraction principles', the classic example of which is Hume's Principle. This view is primarily associated with Crispin Wright, Bob Hale, and their students.

3) Nominalism, the view that mathematical entities don't exist at all, but are a useful (eg, for science) fiction.

Overall, the trend seems to be leaning towards the 'discovered' side of the debate, but nominalism is certainly a very active and real option. If you're interested in any one of these views in particular (or all of them) I'll happily point you to some good introductory reading.

Nicholas+ said...

I'd be delighted for some pointers to some introductory reading. Thanks!

Chuck Blanchard said...

anonymous:

This was very useful and interesting. Thanks! The references would be most welcome!

Anonymous said...

For an introductory level work there's really none better than Stewart Shapiro's book "Thinking About Mathematics". It assumes very little in terms of mathematical and philosophical background. It is aimed at undergraduate students, so in some parts it might seem a bit slow, but the overview is still quite good. Shapiro covers the three major theories of the early 20th century (logicism, formalism, and intuitionism) before moving on to more contemporary ideas. One point to be warned about, however: Shapiro is an advocate of mathematical structuralism, so his presentation of that view as a solution to all the problems is not entirely unbiased.

For more advanced overviews, the entries in the Oxford Handbook of Philosophy of Mathematics and Logic (also edited by Shapiro, and perhaps the largest handbook I've ever seen) are generally quite good, and most of them cite good references.

For the specific topic of mathematical truth the starting point remains Paul Benacerraf's paper "Mathematical Truth". His papers "What Mathematical Truth Could Not Be" are also quite good, though in parts they expect some knowledge of mathematics and logic. A lot of the contemporary issues regarding the nature of mathematical truth focus on what kinds of philosophical conclusions can be drawn from metamathematical theorems.

Finally, there's still a large influence of the idea, due to W.V.O. Quine, amongst others, that the only kinds of mathematics we should even believe are true are the things used in the natural sciences. Quine's classic paper on this issue, and on ontology more generally, is "On What There Is".

Hopefully this is helpful/interesting. Not knowing the precise audience of this blog, I've tried to steer away from work that requires any substantial philosophical or mathematical background. Start with Shapiro's text, and see where you go from there.