### The Field or the Object of Study?

So I want to say biology is the study of life. But I also want to say mathematics is the study of mathematics. In other words, it seems as if mathematics is an independent thing that's out there in the objective reality of the universe, and we just discover its truths. In contrast, it seems as if biology is a human invention, a fairly crude approach to grappling with the complexities of life that are "out there."

Discuss.

1. Anonymous12:02 AM

Someone has to link to this: http://xkcd.com/435/

2. Sean Coughlin9:22 AM

From http://en.wikipedia.org/wiki/Mathematics#Mathematics_as_science

Karl Popper concluded that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures..."

3. Well mathematics is sort of different from things like biology or even physics in that it seems more closely related to logic than to any attempt to systematically understand existence. There's a sense in which the finding of mathematics are true simply because of the definitions of the terms used, whereas in the natural or social sciences people generally take a further step of attempting to connect conceptual definitions to things that exist in the real world (sort of like how Mises' economics was supposed to be a priori, but he thought that economics was useful for doing history or evaluating policies).

I'm not sure that I agree that mathematics is a "thing" that's "out there;" mathematics is just a set of terms which relate to each other in specific ways. We created the meanings of the terms of mathematics, and when we use those terms we entail certain things which simply follow from what we started with. So "1+2=3" isn't true because of anything that exists, per se; it's true because the meaning of "1," "+," "2," "=," and "3" are such you can't "add" a "one" and a "two" and "equal" anything besides a "three." To deny that, you'd need to reject the definition of at least one of the terms that's involved.

And we do have things like that in other sciences. For example, in biology, you can't say something like, "Immediately after the lion killed the gazelle, and without being resuscitated, the gazelle went on with its life as if nothing ever happened." To say that the gazelle got "killed" "without resuscitation" rules out, by the definition of those terms, the possibility that the gazelle went on with its life afterwords. In economics, similarly, we have "She was completely satiated with what she had, but she still wanted more." If she wanted more, then she clearly wasn't completely satiated; that's true without going out to look.

But clearly, economics and biology go beyond simply playing with semantics when they try to apply their descriptions of things to the real world. And in doing this, it becomes clear how crude those descriptions are. Mises says that all action is purposeful, but surely this doesn't imply that I weigh my ends and means when my hand instinctively darts to my mouth to stifle a sneeze. In the natural and social sciences, we necessarily make use of auxiliary assumptions to connect theory to prediction or observation, and it's in doing so that we open up a new realm of possible mistakes that are not available to the theoretical mathematician.

4. James Rothfeld6:52 PM

Mathematics is a mental construct based on the impression that different objects in space-time are identical. There identity is defined in an arbitrary manner not taking into account their unique 'location' in space time.

Mathematics is a self-contained argument, and consequently allows for something that does not exist in the natural sciences: proof.

In the natural sciences, there is no proof, but in mathematics there is.

At the same time, mathematics does 'work', since it is created in reference to the real world. However, mental constructs such as '1' or '9.8765948734256' do not correspond to anything real.

Mathematics is not 'out there', but it's 'in here'. Matter and energy exists independently of thought, mathematics does not.

Like consciousness, it's built into the structure of the bio-chemical processes in our brain. Consciousness requires - consciousness, as does mathematics.

Strictly speaking, computers don't do mathematics - they simply execute orderly mechanical processes set into motion by humans.

It is not by accident that mathematics is the one branch of human thinking compatible with all religions and ideologies - (I have yet to come across a belief system that rejects the truth of Mathematics) - the self-referential nature of all Mathematics allows it to be absolutely perfect in terms of its consistency. No other branch of human thinking is as parsimonious - and hence God-like.

5. James Rothfeld7:52 PM

one other aspect differentiates mathematics from all other sciences: it is conceivable that given enough time an individual in isolation can recreate the entire body of mathematical knowledge without ever needing to look at nature. It is a branch of knowledge that can be expanded indefinitely without any reference to the physical world.

It has happened in the past that a mathematical genius was able to independently re-create doctoral level mathematics from the knowledge contained in a grade school mathematics primer. This is possible because the whole of mathematics is contained within the human mind.

This is not possible in the natural sciences, since the natural science depend on observations of phenomena that are outside the human mind.

6. Contrasting mathematics and biology in terms of human invention seems odd to me. Oh, well...

Dumbest wisdom ever: "Mathematics is the Queen of the Sciences." Mathematics is not a science at all.

7. > Mathematics is a self-contained argument.

I think this sentence says it all. Also, I'd add that Maths is a tool - albeit a cognitive one. It's such an adaptive tool that it can be used to sculpt reliable knowledge from any other sciences, including biology.

However, the way that tool is used (the methods) and what determines its outcome will still depend on the person and his/her creativity - so we sometimes end up with a grand sculpture, but most of the time we end up with a boring ashtray ;)

8. Anonymous7:07 PM

Well, gee Wabulon, I guess it doesn't matter that the meaning of the word 'science' has changed quite drastically since it was coined in Latin (not always for good reason) or that being the 'queen of the sciences' doesn't imply being a science (whoops). The phrase is supposed to mean that mathematics rules the sciences because they all use it but it doesn't use them. So it goes.

I'm a bit confused by the initial suggestion here. Bob said:

"So I want to say biology is the study of life. But I also want to say mathematics is the study of mathematics. In other words, it seems as if mathematics is an independent thing that's out there in the objective reality of the universe, and we just discover its truths. In contrast, it seems as if biology is a human invention, a fairly crude approach to grappling with the complexities of life that are "out there."

The 'in other words' makes no sense. If biology is the study of life, then the science is not its own object. If mathematics is the study of mathematics, then the science is its own object. In other words, the initial characterization of biology and mathematics suggests precisely the opposite of what you say it does.

I don't think this is an issue that anybody has a right to pronounce upon as though the answers were obvious. I'm extremely suspicious of any strong sort of realism in mathematics, but there does seem something very mistaken in the idea that we create mathematics in the way that we create, say, automobiles or poems or voluntary societies. I'm also very suspicious of the application of theoretical mathematics to the natural world and, in my less patient moments, am perfectly willing to dismiss most of theoretical physics as sheer construction. But again, I welcome some serious discussion of this stuff...

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