The Man Who Invented Modern Probability • Hacker News Board

Re: The Man Who Invented Modern Probability

#71

Post by mtdewcmu » Sat, Aug 24, 2013, 7:30 AM UTC

nwhitehead wrote: post #67

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…

> 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 tha…

Interesting. I'm not all that up on philosophy, and I had to look up constructivism and the law of the excluded middle. Computers can only deal directly with discrete math, so that eliminates quite a bit of mathematics. Moreover, despite being regarded as Turing machines, you could in reality count all the states that a computer could be in, by counting the bits of memory, and you will find that it's a finite number. So computers are, in fact, only finite state machines. It's often overlooked that what makes a Turing machine a Turing machine is the infinitely-long tape. So computers occupy a fairly small sliver of the infinite universe of math.

Philosophically, my worldview is like that of science: the way to know something is by making observations and formulating and testing hypotheses. What we can observe is limited, and hypotheses are only models that are tested by successive approximations. I'm not sure I understand what you mean that that statement isn't required to have an ultimate truth. Presumably a statement like that has an answer, but the fact that something is knowable in principle doesn't mean that there's any way to get the answer. For instance, the Hubble telescope can see far away galaxies that we can never get an up close look at. The question of whether there's life somewhere else in the universe must have an answer, but we can't know what's in those galaxies; even with a better telescope, we'd be seeing what they looked like a billion years ago. Many things will never be known.

Re: The Man Who Invented Modern Probability

#72

Post by phaemon » Sat, Aug 24, 2013, 9:30 AM UTC

mtdewcmu wrote: post #68

phaemon wrote: post #66

Earlier quoted context omitted.

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 repres…

I was under the impression that irrational numbers are numbers that simply can't be represented as a ratio. This is independent of the number system used. I'm not familiar with the Taylor series: can you use it as a number system? Isn't it simply a representation of functions (in which case, you might as well just write "sqrt(2)" as invoke anything else)?

It's tough to Google for "Taylor series number system" as you just get pages about the guitars. ;-)

Re: The Man Who Invented Modern Probability

#73

Post by mtdewcmu » Wed, Aug 28, 2013, 7:31 PM UTC

phaemon wrote: post #72

mtdewcmu wrote: post #68

Earlier quoted context omitted.

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 repres…

I was under the impression that irrational numbers are numbers that simply can't be represented as a ratio. This is independent of the number system used. I'm not familiar with the Taylor series: can you use it as a number system? Isn't it simply a representation of functions (in which case, you might as well just write "sqrt(2)" as invoke anything else)? It's tough to Google for "Taylor series number system" as you…

I don't know if you're still reading this, but... Using Taylor series as a number system is just a hypothetical. Any systematic way of describing numbers could be a number system. Irrationals can't be represented as ratios, but beyond that, some have other finite representations, and others have none.