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How has mathematics gotten so abstract?

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Re: How has mathematics gotten so abstract?

#211

Post by godelski »

btilly wrote: post #140
godelski wrote: post #128

Earlier quoted context omitted.

> I guarantee that a naive presentation doesn't actually include the axioms But you said "modern math courses". Are you now talking about a casual conversation? I mean the OP's story is that his wife just liked listening to him talk about his passions. > Uncountable need not mean more. Sure. But that doesn't mean that there aren't differing categories. However you slice it, we can operate on these things in different…

Your guesses at what I seem to think are completely off base and insulting. When I say "modern math courses", I mean like the standard courses that most future mathematicians take on their way to various degrees. For all that we mumble ZFC, it is darned easy to get a PhD in mathematics without actually learning the axioms of ZFC. And without learning anything about the historical debates in the foundations of mathema…

Honestly it's difficult to understand exactly what you're arguing. Because I understand laymen not understanding your argument about infinities not being real (and even many HN users don't understand code is math bit a CS degree doesn't take you far in math. Some calc and maybe lin alg) but are we concerned about laymen? I too am frustrated by nonexperts having strong opinions and having difficulties updating them, but that's not a culture problem. We're on HN and we know the CS stereotypes, right?

If instead you're talking about experts then I learned about what you're talking about in my Linear 2 course in a physics undergrad and have seen the topic appear many times since even outside my own reading of set theory. The axiom of choice seems to have even entered more main stream nerd knowledge. It's very hard to learn why AoC is a problem without learning about how infinities can be abused. But honestly I don't know any person that's even an amateur mathematician that thinks infinities are physical

Re: How has mathematics gotten so abstract?

#212

Post by btilly »

godelski wrote: post #211
btilly wrote: post #140

Earlier quoted context omitted.

Your guesses at what I seem to think are completely off base and insulting. When I say "modern math courses", I mean like the standard courses that most future mathematicians take on their way to various degrees. For all that we mumble ZFC, it is darned easy to get a PhD in mathematics without actually learning the axioms of ZFC. And without learning anything about the historical debates in the foundations of mathema…

Honestly it's difficult to understand exactly what you're arguing. Because I understand laymen not understanding your argument about infinities not being real (and even many HN users don't understand code is math bit a CS degree doesn't take you far in math. Some calc and maybe lin alg) but are we concerned about laymen? I too am frustrated by nonexperts having strong opinions and having difficulties updating them, b…

The fact that you think I'm talking about the axiom of choice, demonstrates that you didn't understand what I'm talking about. I would also be willing to bet a reasonable sum of money that this topic did not come up in your Linear 2 course in physics undergrad.

The arguments between the different schools of philosophy in math are something that most professional mathematicians are unaware of. Those who know about them, generally learned them while learning about either the history of math, or the philosophy of math. I personally only became aware of them while reading https://www.amazon.com/Mathematical-Experience-Phillip-J-Dav.... I didn't learn more about the topic until I was in grad school, and that was from personal conversations. It was never covered in any course that I took on, either in undergraduate or graduate schools.

Now I'm curious. Was there anything that I said that should have been said more clearly? Or was it hard to understand because you were trying to fit what I said into what you know about an entirely unrelated debate about the axiom of choice?

Re: How has mathematics gotten so abstract?

#213

Post by fooker »

My hypothesis for this is the disconnect between mathematics and fields like physics and theoretical computer science.

We likely need new mathematics for making progress in physics or ..say.. have a better understanding of the PvsNP kind of problems, but very few high caliber mathematicians are motivated to do this.

Which makes sense, as it’s way easier and prestigious to define and solve your own abstract problems, publish one paper per grad student per year and coast through research life.

Re: How has mathematics gotten so abstract?

#214

Post by godelski »

btilly wrote: post #212
godelski wrote: post #211

Earlier quoted context omitted.

Honestly it's difficult to understand exactly what you're arguing. Because I understand laymen not understanding your argument about infinities not being real (and even many HN users don't understand code is math bit a CS degree doesn't take you far in math. Some calc and maybe lin alg) but are we concerned about laymen? I too am frustrated by nonexperts having strong opinions and having difficulties updating them, b…

The fact that you think I'm talking about the axiom of choice, demonstrates that you didn't understand what I'm talking about. I would also be willing to bet a reasonable sum of money that this topic did not come up in your Linear 2 course in physics undergrad. The arguments between the different schools of philosophy in math are something that most professional mathematicians are unaware of. Those who know about the…

  > The fact that you think I'm talking about the axiom of choice, demonstrates that you didn't understand what I'm talking about.
Dude... just a minute ago you were complaining about ZFC... Sure, I brought up AoC but your time to protest was then.

The reason I brought up AoC is because it is a common way to learn about the abuse of infinity and where axioms need be discussed. Both things you brought up. I think you are reading further into this than I intended.

  > Now I'm curious. Was there anything that I said that should have been said more clearly?
Is this a joke?

When someone says

  >> Honestly it's difficult to understand exactly what you're arguing.
That's your chance to explain. It is someone explicitly saying... I'm trying to understand but you are not communicating efficiently.

This is even more frustrating as you keep pointing out that this is not common knowledge. So why are you also communicating like it is?! If it is something so few know about then be fucking clear. Don't make anyone guess. Don't link a book, use your own words and link a book if you want to suggest further reading, but not "this is the entire concept I'm talking about". Otherwise we just have to guess and you getting pissed off that we guess wrong is just down right your own fault.

So stop shooting yourself in the foot and blaming others. If people aren't understanding you, try assuming they can't read your mind and don't have the exact same knowledge you do. Talk about fundamental principles...

Re: How has mathematics gotten so abstract?

#215

Post by btilly »

godelski wrote: post #214
btilly wrote: post #212

Earlier quoted context omitted.

The fact that you think I'm talking about the axiom of choice, demonstrates that you didn't understand what I'm talking about. I would also be willing to bet a reasonable sum of money that this topic did not come up in your Linear 2 course in physics undergrad. The arguments between the different schools of philosophy in math are something that most professional mathematicians are unaware of. Those who know about the…

> The fact that you think I'm talking about the axiom of choice, demonstrates that you didn't understand what I'm talking about. Dude... just a minute ago you were complaining about ZFC... Sure, I brought up AoC but your time to protest was then. The reason I brought up AoC is because it is a common way to learn about the abuse of infinity and where axioms need be discussed. Both things you brought up. I think you ar…

I've had a lot of chances to explain. I've posted a lot of explanations. For example see https://news.ycombinator.com/item?id=45435534 for an explanation that I posted 11 hours ago. See https://plato.stanford.edu/entries/mathematics-constructive/ for a link that I gave. See https://news.ycombinator.com/item?id=45434701 for someone with a different point of view, attempting to explain the same key point.

That point being that what we mean by "exists" is fundamentally a philosophical question. And our conclusions about what mathematical things exist will depend on how we answer that question. And very specifically, there are well-studied mathematical philosophies in which uncountable sets do not have larger cardinalities than countable ones.

If none of those explanations wind up being clear for you, then I'm going to need feedback from you to have a chance to explain this to you. Because you haven't told me enough for me to make any reasonable guess what the sticking point is between you and understanding. And without that, I have no chance of guessing what would clarify this for you.

Re: How has mathematics gotten so abstract?

#216

Post by drdeca »

btilly wrote: post #191
drdeca wrote: post #188

Earlier quoted context omitted.

> Things only exist when you can construct them. This is exactly what I’m saying is presumptive! If constructivism is to earn the merit of being less presumptive by virtue of not assuming the existence of various things, it should also not assume the non-existence of those things. Which, I think many visions of constructivism do earn this merit, but not your description of it.

So having a different philosophy from you makes me presumptive? What makes you presume that you have any business telling someone with different beliefs from you, what is OK to believe? You may believe in the existence of whatever you like. Whether that be numbers that cannot be specified, or invisible pink unicorns. I'll be over in the corner saying that your belief does not compel me to agree with you on the questi…

No, that’s not what I said. Thinking you can determine a-priori that something that is logically self-consistent, cannot exist, if there is no reason that such a thing being physically instantiated would imply a logical contradiction, is the thing I think is presumptive.

Merely believing that such a thing (a halting oracle) doesn’t exist, isn’t something I meant to call presumptive, only believing that you can know a-priori (with certainty) that such things cannot exist.

I don’t claim that you are obligated to agree with me that they do exist. Someone who believes they don’t, but doesn’t believe they can know this as certain a-priori knowledge, would be no more presumptive than I am, and someone who is agnostic on the question of whether they exist would be less presumptive than I am.

Also, I disagree with your notion of “meaningfulness”. At a minimum, all statements in the arithmetic hierarchy are meaningful. The continuum hypothesis might in a certain sense not be meaningful.

Re: How has mathematics gotten so abstract?

#217

Post by btilly »

drdeca wrote: post #216
btilly wrote: post #191

Earlier quoted context omitted.

So having a different philosophy from you makes me presumptive? What makes you presume that you have any business telling someone with different beliefs from you, what is OK to believe? You may believe in the existence of whatever you like. Whether that be numbers that cannot be specified, or invisible pink unicorns. I'll be over in the corner saying that your belief does not compel me to agree with you on the questi…

No, that’s not what I said. Thinking you can determine a-priori that something that is logically self-consistent, cannot exist, if there is no reason that such a thing being physically instantiated would imply a logical contradiction, is the thing I think is presumptive. Merely believing that such a thing (a halting oracle) doesn’t exist, isn’t something I meant to call presumptive, only believing that you can know a…

> Merely believing that such a thing (a halting oracle) doesn’t exist, isn’t something I meant to call presumptive, only believing that you can know a-priori (with certainty) that such things cannot exist.

If you think that I was making that case, then you have misunderstood something important.

Constructivism is a statement about what kinds of arguments will convince me that things exist.

Could things exist that I don't believe in? Absolutely! There could well be a bank account with my name on it that I don't know about. Its existence is possible, and my lack of belief in it is no skin off of its back. But I still don't believe that it exists.

Similarly, the Platonists could be correct. There could be an omniscient God whose perfect mind gives existence to a perfect system of mathematics, beyond human comprehension. I have no way to prove that there isn't such a God, and therefore that there isn't such a perfect mathematics.

However the potential for such things to exist is a point of theology. I do not believe in their existence. Just as I do not believe in the existence of Santa. In neither case can I prove that they don't exist. And if you choose to believe in them, that's your business. Not mine.

There is nothing presumptive in my laying out the rules of reason that I will accept as convincing to me. There is a lot of presumption if anyone else comes along and tells me that I should think differently about unprovable propositions.

Now it happens to be the case that from the rules of reason that I use, I provably can't be convinced of the existence of certain things. That's a mathematical theorem. But the fact that I can't be convinced, doesn't prove that you shouldn't be convinced. You are free to be convinced of all of the unprovable assertions that you wish. And it is also true that on something like this, I have no way to convince you that it doesn't exist.

On meaningfulness, meaning is in the eye of the beholder. For example there are people who are willing to pay a million dollars for a century old stamp which was misprinted with the airplane upside-down. (See https://en.wikipedia.org/wiki/Inverted_Jenny to verify that.) They clearly find great meaning in that stamp. But I don't.

So again, you're free to find meaning in whatever you want. But you're in the wrong to object that I don't find meaning in what you consider important.

Re: How has mathematics gotten so abstract?

#218

Post by dapperdrake »

godelski wrote: post #128
btilly wrote: post #103

Earlier quoted context omitted.

I guarantee that a naive presentation doesn't actually include the axioms, and doesn't address the philosophical questions dividing formalism from constructivism. Uncountable need not mean more. It can mean that there are things that you can't figure out whether to count, because they are undecidable.

> I guarantee that a naive presentation doesn't actually include the axioms But you said "modern math courses". Are you now talking about a casual conversation? I mean the OP's story is that his wife just liked listening to him talk about his passions. > Uncountable need not mean more. Sure. But that doesn't mean that there aren't differing categories. However you slice it, we can operate on these things in different…

Vitali's counter-example for the existence of a non-trivial measure requires the Axiom of Choice. But not assuming the Axiom of Choice then prohibits choosing and actually using that measure. There is more to it than just Banach-Tarski.

Re: How has mathematics gotten so abstract?

#219

Post by reverendjames »

dcchuck wrote: post #3
thadt wrote: post #2

This reminds of of that one time when I was on a date with a girl from the history department who somehow bemusedly sat through my entire mini-lecture on comparing infinite sets. Twenty years and three kids later, she'll still occasionally look me straight in the eye and declare "my infinity is bigger than your infinity."

This is the type of romcom I'd watch ;)

You want to watch him and his wife?

Re: How has mathematics gotten so abstract?

#220

Post by drdeca »

btilly wrote: post #217
drdeca wrote: post #216

Earlier quoted context omitted.

No, that’s not what I said. Thinking you can determine a-priori that something that is logically self-consistent, cannot exist, if there is no reason that such a thing being physically instantiated would imply a logical contradiction, is the thing I think is presumptive. Merely believing that such a thing (a halting oracle) doesn’t exist, isn’t something I meant to call presumptive, only believing that you can know a…

> Merely believing that such a thing (a halting oracle) doesn’t exist, isn’t something I meant to call presumptive, only believing that you can know a-priori (with certainty) that such things cannot exist. If you think that I was making that case, then you have misunderstood something important. Constructivism is a statement about what kinds of arguments will convince me that things exist. Could things exist that I d…

Ah, that I wouldn't call presumptuous. I just interpreted a previous statement you made as saying that one can prove that certain such things can't exist. So, I guess there was just a miscommunication (I guess I misunderstood.)
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