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Most Math Problems Do Not Have a Unique Right Answer

devlinsangle.blogspot.com

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Re: Most Math Problems Do Not Have a Unique Right Answer

#61

Post by j2kun »

dbecker wrote: post #55
j2kun wrote: post #35

Earlier quoted context omitted.

Careful there. If you require one to be able to express a problem via some finite string (and via some universally fixed method for encoding problems as strings), then there are only countably many problems.

Though, if we apply your requirement, the set of problems without a unique answer must be countable as well. So, the cardinality of problems with a unique answer is still equal to the cardinality of problems without one.

You're just using the terminology imprecisely is all. (Indeed, two things being countable does not mean they have the same cardinality; finite sets are also countable)

Re: Most Math Problems Do Not Have a Unique Right Answer

#62

Post by hebbarp »

skierscott wrote: post #41
hebbarp wrote: post #31

Most Math Problems Do Not Have a Unique Right Answer The title is both right and wrong. You are actually comparing school or college math with math applied in real world. School or college math works with few variables, for instance, and we consider most others remaining constant. Rarely have I seen school or college level students working with, say, derivatives of more than three variables. School or college math is…

> Hence, a question like this may not have a unique right answer. If you allow philosophers in, you will definitely not have a unique answer :) I'm not sure if this is what you're describing, but many nonlinear[1] math problems have no closed form solution[2]. That means you can't use any regular function, all the operators and the infinitely real numbers to describe every solution: you can only use the infinitely re…

Yes. I read your blog with great interest. I have taken some notes and get back with comments, perhaps on your blog.

Re: Most Math Problems Do Not Have a Unique Right Answer

#63

Post by dbecker »

j2kun wrote: post #61
dbecker wrote: post #55

Earlier quoted context omitted.

Though, if we apply your requirement, the set of problems without a unique answer must be countable as well. So, the cardinality of problems with a unique answer is still equal to the cardinality of problems without one.

You're just using the terminology imprecisely is all. (Indeed, two things being countable does not mean they have the same cardinality; finite sets are also countable)

I think it's trivially obvious that both sets are infinite.

There are infinitely many addition problems involving two numbers, and that's a subset of the set of problems that the article claims to be smaller.

Re: Most Math Problems Do Not Have a Unique Right Answer

#64

Post by JadeNB »

j2kun wrote: post #35
dbecker wrote: post #23

Since uncountably many problems have unique right answers, I'd guess the cardinality of the set of problems with unique right answers is the same as the cardinality of the set of problems without. :)

Careful there. If you require one to be able to express a problem via some finite string (and via some universally fixed method for encoding problems as strings), then there are only countably many problems.

This, too, requires care. I think that we can all agree that, for each real number x, "is x normal?" is a problem (or at least a question that a mathematician might ask). That's uncountably many problems right there! (The fact that only countably many specific instances of it can be written down is, I think, a different matter.)

Re: Most Math Problems Do Not Have a Unique Right Answer

#65

Post by j2kun »

dbecker wrote: post #63
j2kun wrote: post #61

Earlier quoted context omitted.

You're just using the terminology imprecisely is all. (Indeed, two things being countable does not mean they have the same cardinality; finite sets are also countable)

I think it's trivially obvious that both sets are infinite. There are infinitely many addition problems involving two numbers, and that's a subset of the set of problems that the article claims to be smaller.

I'm not making claims about the truth or falsity of your statement, just critiquing the logic and terminology you're using (the lack of a definition for a "problem," the minor misuse of the term "countable"). Regardless, I really hope you saw the article as something besides a mathematical claim, because it clearly did not pretend to be one.

Re: Most Math Problems Do Not Have a Unique Right Answer

#66

Post by j2kun »

JadeNB wrote: post #64
j2kun wrote: post #35

Earlier quoted context omitted.

Careful there. If you require one to be able to express a problem via some finite string (and via some universally fixed method for encoding problems as strings), then there are only countably many problems.

This, too, requires care. I think that we can all agree that, for each real number x, "is x normal?" is a problem (or at least a question that a mathematician might ask). That's uncountably many problems right there! (The fact that only countably many specific instances of it can be written down is, I think, a different matter.)

The question is, "What do you define as a 'problem'?" You need to answer that before you can make claims like "we can all agree..." and give an example of something that I don't agree is a problem (if you're saying what I think you're saying).

Re: Most Math Problems Do Not Have a Unique Right Answer

#67

Post by dbecker »

j2kun wrote: post #65
dbecker wrote: post #63

Earlier quoted context omitted.

I think it's trivially obvious that both sets are infinite. There are infinitely many addition problems involving two numbers, and that's a subset of the set of problems that the article claims to be smaller.

I'm not making claims about the truth or falsity of your statement, just critiquing the logic and terminology you're using (the lack of a definition for a "problem," the minor misuse of the term "countable"). Regardless, I really hope you saw the article as something besides a mathematical claim, because it clearly did not pretend to be one.

Indeed. My original comment was meant as a joke, not as a serious mathematical claim.

Re: Most Math Problems Do Not Have a Unique Right Answer

#68

Post by Grue3 »

ssivark wrote: post #21
jordigh wrote: post #2

I want to add something here: great mathematicians compute too. They also know how to perform an algorithm. It is in performing, say, long division, that you start to notice things like when 10 is a primitive root modulo the divisor. Gauss spent his down time counting primes (in his head, he said). Riemann's notes were full of haphazard computations. Amidst his scratch work where the Riemann-Siegel formula appears, t…

I kinda agree with you, but there are counter-examples. For eg: Grothendieck. > One striking characteristic of Grothendieck’s mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called “Grothendieck prime”. In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendie…

The best part, 57 is trivially divisible by 3 (5+7=12)

Re: Most Math Problems Do Not Have a Unique Right Answer

#69

Post by Grue3 »

JadeNB wrote: post #64
j2kun wrote: post #35

Earlier quoted context omitted.

Careful there. If you require one to be able to express a problem via some finite string (and via some universally fixed method for encoding problems as strings), then there are only countably many problems.

This, too, requires care. I think that we can all agree that, for each real number x, "is x normal?" is a problem (or at least a question that a mathematician might ask). That's uncountably many problems right there! (The fact that only countably many specific instances of it can be written down is, I think, a different matter.)

By the same logic "What is the cardinality of the set X" is also a problem, thus the class of problems isn't even a set and doesn't have cardinality!

Re: Most Math Problems Do Not Have a Unique Right Answer

#70

Post by JadeNB »

j2kun wrote: post #66
JadeNB wrote: post #64

Earlier quoted context omitted.

This, too, requires care. I think that we can all agree that, for each real number x, "is x normal?" is a problem (or at least a question that a mathematician might ask). That's uncountably many problems right there! (The fact that only countably many specific instances of it can be written down is, I think, a different matter.)

The question is, "What do you define as a 'problem'?" You need to answer that before you can make claims like "we can all agree..." and give an example of something that I don't agree is a problem (if you're saying what I think you're saying).

My weak, but (I think) practical, implicit definition of 'problem' was:

> a question that a mathematician might ask

I agree that this is not a very useful definition, by virtue of its extreme and probably excessive inclusiveness, but I think that it's hard to do any better without using words like 'interesting' that themselves need careful definition. (It's fair to argue that my definition in turn requires clarification of the term 'mathematician', but I can weasel my way around that by replacing it with 'person', or else just declaring that anyone interested in trying to ascertain the normality of a number is mathematically minded enough to be called a mathematician.)

By this definition, I think that it is hard to argue with my claim to have produced an uncountable family of problems—simply because, at least classically, to do so you'd have to produce a specific number x about whose normality no mathematician could ever ask. I could then demolish that counterexample by asking you if that particular number x was normal. :-)

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