when you think about it, formal systems are weird.

[alexr_rwx]

Many [programming languages], such as Pascal and LISP, look quite different from one another in style and structure. Can some algorithm be programmed in one of them and not the others? Of course not -- we can compile LISP into Pascal and Pascal into LISP, which means that the two languages describe exactly the same class of algorithms. So do all other reasonable programming languages. The widespread equivalence of computational models holds for precisely the same reason. Any two computational models that satisfy certain reasonable requirements can simulate one another and hence are equivalent in power...

-- Sipser, Introduction to the Theory of Computation

When we talk about mathematical ideas, we often say that we're "discovering" them, as if they were always there just waiting for us, and that when I talk about Turing Machines or differential calculus or set theory, I mean the same thing that yourusername means... but why? Rationalists of different stripes have long held maths to be a fundamental thing-in-reality -- but if you were going to be an empiricist about it, how do you account for this whole math thing? Hume (patron saint of hard empiricists) calls apodictic-type knowledge "consequences of names" but even that presupposes that there's such a thing as "consequence" when it comes to thinking about things. Is there such a thing as "what you must think, if you think"?

Are all of these things just implications of some hardwired logic bits that humans have in their brains, or is it a property of the world? Or is it that we learn to do the same sort of reasoning as other humans from having the particular wetware that we do and interacting with a reasonably-similar environment? Is it conceivable that another intelligent species (in a different kind of environment, I suppose) would come up with a different sort of reasoning where they would find the math that we think up illogical?

Particularly, the λ-calculus and the TM describe exactly the same sort of computation -- where did we get that computation from in the first place, and why were those cats Church and Turing just waiting to describe it with their different computational models?

Discuss. eponis and realitycalls particularly, this is me looking at you.

Comments

[neuroticmonk.livejournal.com]

Hume (patron saint of hard empiricists)

I'm not sure I'd call Hume the patron saint of empiricists. My impression of him was always that he essentially said, "you really can't know anything." I always associated Locke more with empiricism. Wikipedia agrees with you though, so

Is there such a thing as "what you must think, if you think"?

Obviously, though perhaps not quite what you've said. Consider two points connected by a plain. I ask you to find the shortest distance between those two points. Clearly, you will tell me that I need the straight line between those points. Why? Not because there is a definite "what you must think, if you think," but rather a definite, "what you must think, if you think in this way." Thus, you find that people with similar interests very often agree on basic fundamental things. It no doubt results from shared assumptions and goals. There are many routes from one place to another, but only so many optimal (or interesting for that matter) ones.

Generalizing: You're a child, wanting food. To get food you need to communicate that you're hungry to your mother. This necessitates you learning what will communicate this to her. Thus, from birth (if not before) you are programmed to understand the thinking of the people around you. The better your understanding people, the better you get along with them, even if you don't like them. Therefore, if you grow up with humans, you will tend to think in a certain way. It's evolution!

Happy Birthday Alex

[(Anonymous)]

Alex has a birthday today. If you are in Atlanta, be sure to bake him something! Have fun.

[eponis.livejournal.com]

Math seems to be a process of extracting really basic rules by trial-and-error in nature, then deriving more rules from those rules, then getting results from the rules, then testing those results against nature. I mean, if you can take observations about peeled oranges and the comparison of the size of the peels for different sizes of orange, then derive rules that verifiably predict the peel size of apples, then you seem to have something empirically true. The alternative would be a entirely complete yet different logical system that came up with precisely the same findings, which seems highly unlikely.

What's more interesting is the assumption that this logical system for predicting physical behavior and attributes is necessarily applicable to philosophical conclusions. :-)

Sorry about the incredibly late response; I've been really busy lately, and I had this window open to reply to forever.