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!

[oniugnip.livejournal.com]

I think on your first example, when it comes to planes, that's more of an empirically verifiable situation -- if you have something plane-like, a table perhaps, and you're looking for paths across it you could verify just by trying lots of different paths that the straightest one is the shortest. So that's something you can observe, and it wouldn't be too hard to argue that you get some leverage, as a living being, from generalizing from particulars so that you have a widely applicable rule for living (well, I want to get there fast, so I'll take a straight-line path)...

But you can't really observe, for example, algorithms or sets or numbers in the abstract, and so they seem to require some accounting-for from an empiricist perspective -- why does it seem like everybody comes up with similar concepts? Would you say that your understanding of fast mod arithmetic is a product of how you've been acculturated to deal with other people?

And if so, how does this cause-and-effect thing work out? Is that just a generalization that we've learned, or is Kant right in saying it's something hardwired into us that we apply to our experiences (and one might say we've been tuned by evolution to do that)?

[(Anonymous)]

But you can't really observe, for example, algorithms or sets or numbers in the abstract, and so they seem to require some accounting-for from an empiricist perspective -- why does it seem like everybody comes up with similar concepts?

Certainly, they do. I accounted for them if you think about it. The fundamental issue here is "where do people start" and "where do people want to go" and "what are they likely to try first that will work." Most people start from a similar place to the people around them. This is especially true among the very educated. Now, lots of people wind up trying to solve the same problem. Finally, people are programmed to think like other people. It's what we teach kids in school after all. Therefore, there is a high degree of probability that people will come up with the same solution. Also, there are *only* so many workable solutions to any given problem.

Would you say that your understanding of fast mod arithmetic is a product of how you've been acculturated to deal with other people?

More or less. I understand fast mod arithmetic via the proofs of Euclid and company. So, if I start trying to solve a problem using that kind of math, that's the basis from which I start. I don't have another place to begin from. It's the same place anyone else with my level of education an understanding would start from. School teaches you to solve problems from where you are. School and society aslo function to make sure that where you are is close to where everyone else is. There is a reason John Nash claimed "classes will dull you mind; destroy your potential for original thought."

- neuroticmonk

Happy Birthday Alex

[(Anonymous)]

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

Re: Happy Birthday Alex

[jane-doe-9.livejournal.com]

Happy Birthday Alex!

Re: Happy Birthday Alex

[samarin.livejournal.com]

I baked him something!

Re: Happy Birthday Alex

[(Anonymous)]

Happy Birthday!

- neuroticmonk

Re: Happy Birthday Alex

[(Anonymous)]

Haha! Posting two comments anonymously to this thread made LJ accuse me of being a spam bot! Enjoy!

[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.