What are Numbers?

Tuesday, March 14, 2006

What Is It

Plato claimed that numbers exist in some mind-independent abstract heaven. Nominalists claim that there is no such heaven. Clearly, we can't see, hear, taste or feel numbers. But if there are no numbers what is mathematics all about? John and Ken count on a great discussion with Gideon Rosen from Princeton University, co-author of A Subject With No Object: Strategies for Nominalistic Interpretation of Mathematics.

Listening Notes

What are mathematical objects? What kind of things are they? John claims that he has never seen a "three" or a "million". John mentions the notion of "ontology". The ontology of a particular thing refers to what it is made of, what its existence really entails.

Ken introduces Professor Gideon Rosen of Princeton University. John explains that school children are taught that numbers are just numerals. Plato on the other hand thought that numbers were non-physical, non-sensible things in a realm beyond time and space. Ken challenges the Platonist view. If numbers belong to this Platonic realm, they can not be causally active. If numbers are not causally active, how can be aware of them? Rosen explains that there are two answers to this question. One asserts that knowledge regarding numbers is acquired in a mysterious, para-psychological way. Rosen's view is that mathematical knowledge is required through doing mathematics.

Another view is that numbers are ideas in our minds. Rosen challenges this view. If "six" is just an idea, when I refer to "six", do I refer to the six in your mind or in my mind? There are infinitely many mathematical objects and most are not even thought of. Can there be "ideas" that no one has ever thought of?

Ken explains a third view: there are no numbers in the same way that there is no Sherlock Holmes. Mathematics is simply fiction with certain constraints. Rosen points out that according to this view, statements such as "there is an even number between 1 and 3" are all false, since there are no even numbers, which isn't that intuitive.

Yet another view is that numbers are just abstractions from physical magnitudes. John points out that different units lead to assignment of different numbers to the same measurement. Rosen argues that it is doubtful to think that numbers are part of the real world. There are infinitely many numbers but there isn't an infinite amount of any quantity in the real world.

Do mathematicians need to care about what numbers really are? Does the ontology of mathematics matter to the purposes of math and science? Rosen explains that mathematicians don't need to answer philosophical questions about mathematics. They postulate the existence of numbers, in the fictional or Platonist sense, as a theorem, just as "2+2=4" is a theorem.

Comments (1)


John_CaliforniaCentralValley's picture

John_California...

Friday, October 16, 2020 -- 8:51 AM

Many years ago, when my math

Many years ago, when my math teacher told me about imaginary numbers ("the square root of negative one" times "a real number") I was happy to see the new idea. He drew the complex plane for me. He used a dashed line for the imaginary axis, which seems appropriate. To me it felt like a world of possibilities was opening up.

Later, however, there was something that bothered me about it. I read that imaginary numbers are useful in solving some real-world problems, with practical applications in engineering. It took a long time to get used to that idea: that something "not real" ("imaginary") could be a valid number, similarly as "real numbers" or "natural numbers" (positive integers) are valid numbers; and moreover, that it could be useful when modeling the real world.

Finally today the thought came to me that any number has some similarities to a real thing in the world, and some differences from that thing. There is no need to feel uncomfortable about what is "real" or "not real" about numbers; just like everything else, they have similarities to, and differences from, other things. Also we can say that any number (even made-up fictional numbers like square roots of negative numbers) might have some useful relationship with something in the real world (such as when solving problems in the above-mentioned engineering applications). And, even the most "natural" of numbers, such as the number 6 or the number 7, are abstractions, not entirely identical to the things in the "real" world that they are describing.

Aside from numbers based on the square root of negative one, there's another kind of "unreal" or "imaginary" number that we learned about even earlier in school: negative numbers. To me, the positive rational numbers seem "real" and all other numbers seem to be useful fictions; that's just how _I_ feel about it. A negative number such as "negative 3" ("-3") cannot be visualized as easily as just "3". I can have 3 things to look at, but it's a little harder to imagine a negative 3 things to look at. But I can imagine that the level of water in a pool can be expressed as a positive number when it is above a certain level, and a negative number when it is below that level. So, "-3" has some relationship with reality, but it does not coincide with all reality in all ways. So it is with all numbers, even made-up numbers, and even the square root of a negative number.