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Algorithm - Wikipedia, the free encyclopedia

At the moment this concept seems important to me. It seems the the symbols of logic and math are manipulated according to flexible but precisely defined algorithms. And that the symbols often represent algorithms.

It seems to me that positional notation is an algorithm for translating any base into Unary. Unary numeral system - Wikipedia, the free encyclopedia

We are used to the decimal system and perhaps for lower numbers this "translation" is automatic. But do we not experience number in a concrete sense as unary? Of course as soon as we count, as even children can, we move beyond that.

Also I look at math symbols as algorthms. For instance, multiplication is a short-hand algorithm for repeated addition, and exponentiation (especially in simpler cases) is just repeated multiplication. (To oversimplify?) Can math be boiled down to one operation, one essential number concept? Obviously, the system has been richly developed, but what is it at its core?

I'm also interested in algorithm as it relates to computer programming. Has anyone else messed with programming? Behind the contingent syntax lurks the logical algorithm. At the moment, precise algorithm fascinates me. And so I attempt to fire up a conversation on the matter... Smile

@Reconstructo,

A system of precise instructions....

If we load it with "if thens" and "repeat untils," we can do quite a bit. I see now that the for next loop in basic is just the introduction of the capital Sigma used for summation in math. And it's logically the same. A definite summation formula is just a dense/efficient way to write a Value/magnitude/quantity.

Machines made of thought.

@Reconstructo,

I have done programming in Basic and primarily now in COBOL; any language seems to involve itself in logical structures, and finds useful sequences of instructions replicating a decision flow-chart. It is an interesting task to take everyday decisions or processes and translate these into code.

@jgweed,

jgweed;153940 wrote:

I have done programming in Basic and primarily now in COBOL; any language seems to involve itself in logical structures, and finds useful sequences of instructions replicating a decision flow-chart. It is an interesting task to take everyday decisions or processes and translate these into code.

I agree! I learned Basic first, a little COBOL long ago. Pascal. I always liked the stripped down syntax (and name) of C, but never really dove into it. Of course you know that syntax is not of the essence....but perhaps there are exceptions. I was just looking into lamba calculus which some newer languages are based on. They don't name functions. I won't try to explain it, but it seems like something the purist would dream up. Nested loops are beautiful. The math is simple, but with an array variable, the power is amazing. Perhaps the efficiency of such is why i find it beautiful. And experiencing the syntax as accident rather than essence is a move toward the notion of machines made of pure thought.

I never messed with machine code(or assembly), but I bet it has its charms. I used to make 2-dimensional video games with text graphics. I'm living in the past. Smile

@Reconstructo,

This is great. "Truth is fairest naked."
Knuth's up-arrow notation - Wikipedia, the free encyclopedia

Quote:

In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976. It is closely related to the Ackermann function and especially to the hyperoperation sequence. The idea is based on the fact that multiplication can be viewed as iterated addition and exponentiation as iterated multiplication. Continuing in this manner leads to iterated exponentiation (tetration) and to the remainder of the hyperoperation sequence, which is commonly denoted using Knuth arrow notation.

@Reconstructo,

What makes an algorithm important to me is its precision. It's this sort of algorithm that concerns me. Turing machines strike me as algorithmic. The beauty of math, chess, exact logic, and so on, is exactly this ideal preciseness. There is no blur, no noise. Our digital age has moved in this direction. Digital data is perfect, fresh, eternal, as long as its physical vessels remain intact. The 20 millionth copy of a picture is exactly like the first. It's this crystalline quality that makes math beautiful, algorthms beautiful, logic beautfiful, and many aspects of philosophy beautiful.

@Reconstructo,

Reconstructo;152515 wrote:

Algorithm - Wikipedia, the free encyclopedia

At the moment this concept seems important to me. It seems the the symbols of logic and math are manipulated according to flexible but precisely defined algorithms. And that the symbols often represent algorithms.

It seems to me that positional notation is an algorithm for translating any base into Unary. Unary numeral system - Wikipedia, the free encyclopedia

We are used to the decimal system and perhaps for lower numbers this "translation" is automatic. But do we not experience number in a concrete sense as unary? Of course as soon as we count, as even children can, we move beyond that.

Also I look at math symbols as algorthms. For instance, multiplication is a short-hand algorithm for repeated addition, and exponentiation (especially in simpler cases) is just repeated multiplication. (To oversimplify?) Can math be boiled down to one operation, one essential number concept? Obviously, the system has been richly developed, but what is it at its core?

I'm also interested in algorithm as it relates to computer programming. Has anyone else messed with programming? Behind the contingent syntax lurks the logical algorithm. At the moment, precise algorithm fascinates me. And so I attempt to fire up a conversation on the matter...

Reconstructo:
"Can math be boiled down to one operation, one essential number concept? Obviously, the system has been richly developed, but what is it at its core?"

Wittgenstein claims to extend the Pierce Arrow (nor) to include quantification theory. Indeed, he claims that all propositions, including mathematical propositions, are a result of repeated application of this extension.

All propositions are truth functions of elementary propositions.
See: Logical Atomism ..both Wittgenstein and Russell proposed a system based on atomic facts.

@Reconstructo,

Reconstructo;154234 wrote:

This is great. "Truth is fairest naked."
Knuth's up-arrow notation - Wikipedia, the free encyclopedia

This'll make your head spin:
GRAHAM'S NUMBER AND RAPIDLY GROWING FUNCTIONS - sci.math | Google Groups

@Owen phil,

Owen;172418 wrote:

"Can math be boiled down to one operation, one essential number concept? Obviously, the system has been richly developed, but what is it at its core?"

Just an off the top of my head shot in the dark - zero or not zero? :hmm:

@Owen phil,

Owen;172418 wrote:

Reconstructo:
"Can math be boiled down to one operation, one essential number concept? Obviously, the system has been richly developed, but what is it at its core?"

Wittgenstein claims to extend the Pierce Arrow (nor) to include quantification theory. Indeed, he claims that all propositions, including mathematical propositions, are a result of repeated application of this extension.

All propositions are truth functions of elementary propositions.
See: Logical Atomism ..both Wittgenstein and Russell proposed a system based on atomic facts.

Ah yes, I know the Peirce arrow. Great stuff. I have been reading about various philosophies of mathematics. I like them all. Logicism, formalism, and intuitism. But I have to go with the intuitionists, because numbers are like diamonds in my brain. I think Plato was right, but so was Aristotle. Most forms are abstracted from experience. But the source of number is always already there. Of course I think this is also the source of logic. I think logic is the unification of unities, according to a truth table. We zoom in and out of higher or lower level unities. You probably know of Zermelo's /Russell's paradox. That's why I can't go with logicism for math. But all "absolute form" is beautiful in my book.

---------- Post added 06-03-2010 at 02:32 PM ----------

Twirlip;172424 wrote:

This'll make your head spin:
GRAHAM'S NUMBER AND RAPIDLY GROWING FUNCTIONS - sci.math | Google Groups

It was great to see all those unthinkable numbers. And those are infinitesimals relative to what we could write. How strange it is. How about 5 trillion factorial to the power of (five trillion factorial to the power of (googleplex factorial to the power of (googleplex factorial to the power of (skewe's number factorial to the power of skewe's number factorial.)))?

---------- Post added 06-03-2010 at 02:34 PM ----------

thack45;172458 wrote:

Just an off the top of my head shot in the dark - zero or not zero? :hmm:

I think 0 is just about as naked as number gets. It's the presence of an absence. It's the mathematical version of the empty (or absolute) concept. And negation is also primal, I think. Smile

@Reconstructo,

Reconstructo;172542 wrote:

How about 5 trillion factorial to the power of (five trillion factorial to the power of (googleplex factorial to the power of (googleplex factorial to the power of (skewe's number factorial to the power of skewe's number factorial.)))?

Peanuts! :deflated:

@Reconstructo,

We have an exact language for the unthinkable. Or the thinkable-in-slow-motion.

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