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The Man Who Invented Modern Probability

nautil.us

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Re: The Man Who Invented Modern Probability

#61

Post by mtdewcmu »

phaemon wrote: post #54
mtdewcmu wrote: post #48

Earlier quoted context omitted.

In some sense, the existence of irrational numbers with no pattern in their digits is an illusory artifact of the number system. We can talk about them collectively because decimals are not required to have an end, and we have to postulate them to fill in the gaps in the number line, but such numbers lack any description or means of being separated as individuals. But then, is any mathematical abstraction real? I gue…

This is wrong. The square root of two was identified as irrational by the ancient Greeks, when considering the length of the diagonal of the unit square. The proof has nothing to do with decimal fractions (they didn't use decimal fractions at all).

I'm referring to irrational numbers with no pattern. Square roots can be described and the digits can be enumerated by an algorithm. The diagonalization argument shows that there can't be a description for all irrational numbers.

Re: The Man Who Invented Modern Probability

#62

Post by mtdewcmu »

lutusp wrote: post #55
mtdewcmu wrote: post #48

Earlier quoted context omitted.

In some sense, the existence of irrational numbers with no pattern in their digits is an illusory artifact of the number system. We can talk about them collectively because decimals are not required to have an end, and we have to postulate them to fill in the gaps in the number line, but such numbers lack any description or means of being separated as individuals. But then, is any mathematical abstraction real? I gue…

> In some sense, the existence of irrational numbers with no pattern in their digits is an illusory artifact of the number system. But irrational numbers remain irrational in any integer number base. Therefore no, they aren't illusory at all, nor are they an artifact of the "number system". > ... such numbers lack any description or means of being separated as individuals. Also false. Many irrational numbers are easy…

"Many irrational numbers are easy to identify unambiguously."

Absolutely. Cantor proved that an infinite number of others can't be identified at all, because they outnumber all possible descriptions. Some numbers can only be described by an infinitely long list of digits, one that can't be produced by a Turing machine and contains an infinite amount of irreducible information. The Kolmogorov complexity is infinite.

Newton's equations describe a mathematical model of the universe that agrees well with measurements taken under familiar conditions. Einstein's equations describe a different mathematical model, one that has good agreement with experiment over a much wider range of conditions than Newton's. But general relativity has well-known problems. Its equations give nonsense solutions under some circumstances, e.g. singularities. Physical theories are models that predict the outcomes of experiments. They're reductionist out of necessity. It's unknown and probably unknowable as to whether a perfect model is possible, but there are likely things that can't be reduced.

Re: The Man Who Invented Modern Probability

#63

Post by lutusp »

mtdewcmu wrote: post #62
lutusp wrote: post #55

Earlier quoted context omitted.

> In some sense, the existence of irrational numbers with no pattern in their digits is an illusory artifact of the number system. But irrational numbers remain irrational in any integer number base. Therefore no, they aren't illusory at all, nor are they an artifact of the "number system". > ... such numbers lack any description or means of being separated as individuals. Also false. Many irrational numbers are easy…

"Many irrational numbers are easy to identify unambiguously." Absolutely. Cantor proved that an infinite number of others can't be identified at all, because they outnumber all possible descriptions. Some numbers can only be described by an infinitely long list of digits, one that can't be produced by a Turing machine and contains an infinite amount of irreducible information. The Kolmogorov complexity is infinite. N…

> Cantor proved that an infinite number of others can't be identified at all, because they outnumber all possible descriptions.

I'm tempted to say that that definition places those examples in a unique set, thus at least unambiguously identifying the set to which they belong.

Re: The Man Who Invented Modern Probability

#64

Post by hollerith »

consz wrote: post #24
nawitus wrote: post #20

Earlier quoted context omitted.

Your proof seems to only prove that a finite number of programs (described by you) which can produce a finite number of irrational numbers, while there are infinite number of irrational numbers. But we're surely not talking about a finite number of programs. I'm not even sure what the Kolmogorov complexity of a "complex irrational number" means. If you need the sequence of digits and you cannot use an algorithm to pr…

The set of irrational numbers which can be defined algorithmically are programable, eg. they are defined by a program of finite length.

>The set of irrational numbers which can be defined algorithmically are programable

You mean "computable".

Re: The Man Who Invented Modern Probability

#65

Post by mtdewcmu »

lutusp wrote: post #63
mtdewcmu wrote: post #62

Earlier quoted context omitted.

"Many irrational numbers are easy to identify unambiguously." Absolutely. Cantor proved that an infinite number of others can't be identified at all, because they outnumber all possible descriptions. Some numbers can only be described by an infinitely long list of digits, one that can't be produced by a Turing machine and contains an infinite amount of irreducible information. The Kolmogorov complexity is infinite. N…

> Cantor proved that an infinite number of others can't be identified at all, because they outnumber all possible descriptions. I'm tempted to say that that definition places those examples in a unique set, thus at least unambiguously identifying the set to which they belong.

You can identify the set of numbers with infinite Kolmogorov complexity. But you can't separate out an individual from the set.

Turing machines might not capture all numbers that can be described, but, interestingly, descriptions and Turing machines have the same cardinality.

Re: The Man Who Invented Modern Probability

#66

Post by phaemon »

mtdewcmu wrote: post #61
phaemon wrote: post #54

Earlier quoted context omitted.

This is wrong. The square root of two was identified as irrational by the ancient Greeks, when considering the length of the diagonal of the unit square. The proof has nothing to do with decimal fractions (they didn't use decimal fractions at all).

I'm referring to irrational numbers with no pattern. Square roots can be described and the digits can be enumerated by an algorithm. The diagonalization argument shows that there can't be a description for all irrational numbers.

Just to be clear: you think that there are many irrational numbers that exist independently of which number system you use, but that there are infinitely many that do depend on the number system you use? Is that right?

Re: The Man Who Invented Modern Probability

#67

Post by nwhitehead »

mtdewcmu wrote: post #48
nwhitehead wrote: post #34

Earlier quoted context omitted.

For each fixed length there are a finite number of programs of that length. If we use 8-bit bytes for the alphabet, there are 256^N programs of length N. The set of ALL programs is infinite (we don't limit the length of programs), but it is countable. The "countable" part means we can put the set into one-to-one correspondence with the natural numbers. The correspondence starts with 0 mapping to the empty program, th…

In some sense, the existence of irrational numbers with no pattern in their digits is an illusory artifact of the number system. We can talk about them collectively because decimals are not required to have an end, and we have to postulate them to fill in the gaps in the number line, but such numbers lack any description or means of being separated as individuals. But then, is any mathematical abstraction real? I gue…

> But then, is any mathematical abstraction real? I guess it's all beside the point.

I actually think the reality of mathematical abstractions is hugely important because of... computer programs! In a real way, programs are the embodiment of mathematics. I want my programs to work so I need the underlying math to work as well.

That's why I'm a constructivist. I reject the law of the excluded middle because proofs that use it don't translate into real programs; they translate into programs that ask an omnipotent oracle to decide which branch to take. Constructive proofs translate into working programs.

It also ties into philosophy. I am a skeptic, so when someone tells me either A or not A must be true even if we can never know which one, I ask for proof or justification of that fact. The justifications that I get are remarkably similar to logical "proofs" that god exists, and just as fallacious. This isn't to say that there couldn't be an ultimate truth about A or a god, just that it is not logically necessary.

Re: The Man Who Invented Modern Probability

#68

Post by mtdewcmu »

phaemon wrote: post #66
mtdewcmu wrote: post #61

Earlier quoted context omitted.

I'm referring to irrational numbers with no pattern. Square roots can be described and the digits can be enumerated by an algorithm. The diagonalization argument shows that there can't be a description for all irrational numbers.

Just to be clear: you think that there are many irrational numbers that exist independently of which number system you use, but that there are infinitely many that do depend on the number system you use? Is that right?

Irrational numbers, by definition, include decimal numbers that have infinitely many digits after the decimal point, and there are no rules about what those digits have to be. This is powerful enough to represent any irrational number regardless of the number base. However, if you're talking about number systems, not all number systems have equal ability to represent irrational numbers. Whatever the system, to represent all the irrationals, it would have to be capable of going on forever. I think the part of my point that you're asking about is my statement that they're a side effect of decimal numbers. Irrational numbers like sqrt(2) and e have concise representations as the limits of Taylor series. Only when converted into decimals do they appear to have infinite amounts of information. So if we used Taylor series as the number system instead of decimals, simple irrationals would look simple, and we might be less inclined to treat them the same as numbers that have no finite descriptions at all.

Re: The Man Who Invented Modern Probability

#69

Post by mtdewcmu »

lutusp wrote: post #63
mtdewcmu wrote: post #62

Earlier quoted context omitted.

"Many irrational numbers are easy to identify unambiguously." Absolutely. Cantor proved that an infinite number of others can't be identified at all, because they outnumber all possible descriptions. Some numbers can only be described by an infinitely long list of digits, one that can't be produced by a Turing machine and contains an infinite amount of irreducible information. The Kolmogorov complexity is infinite. N…

> Cantor proved that an infinite number of others can't be identified at all, because they outnumber all possible descriptions. I'm tempted to say that that definition places those examples in a unique set, thus at least unambiguously identifying the set to which they belong.

This appears to be exactly the concept I was talking about. Maybe I read about these numbers some time ago and forgot that they already had a name.

http://en.wikipedia.org/wiki/Definable_number

Re: The Man Who Invented Modern Probability

#70

Post by mtdewcmu »

phaemon wrote: post #66
mtdewcmu wrote: post #61

Earlier quoted context omitted.

I'm referring to irrational numbers with no pattern. Square roots can be described and the digits can be enumerated by an algorithm. The diagonalization argument shows that there can't be a description for all irrational numbers.

Just to be clear: you think that there are many irrational numbers that exist independently of which number system you use, but that there are infinitely many that do depend on the number system you use? Is that right?

Take a look at this.

http://en.wikipedia.org/wiki/Definable_number

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