Why Do People Misunderstand Mathematical Proofs of Nonexistence?

[1MileCrash]

"There is no algebraic formula for the roots of a general quintic polynomial."

R: "That we KNOW of."

"No, we can confirm that one does not exist."

R: "But that's according to our current knowledge."

I just changed the topic since the quintic thing was an analogy, but this misunderstanding troubled me.

What would be a fitting response? I suppose a response is that all of our "current knowledge" is also developed through mathematical proof, and thus any result derived from our "current knowledge" cannot be nullified by "further knowledge" as this would require that our axioms have an underlying inconsistency. But I suspect this response would not satisfy someone who did not study mathematics.

Statements of nonexistence in math seem to be taken as "we haven't found it yet" by the general populace. Why is that?

 

[micromass]

Try asking him

"Are there any prime >2 that are even?"

If he says "No", then proceed telling him that that's due to our current knowledge and maybe later knowledge will reveal one. If he protests, then use this protest against him and tell him it's exactly the same with the quintic polynomial case.

Now, if he does agree that we might someday find a prime >2 that is even, then I would just stop arguing with him as it's useless.

(Or go for something simpler: are there any squares that are circles or are there any even numbers that are odd)

 

[ModusPwnd]

Statements of nonexistence in math seem to be taken as "we haven't found it yet" by the general populace. Why is that?

Math is unlike most or all other intellectual endeavors. Math is built upon axioms that are assumed for the sake of the math. Theorems derived under these axioms are (if derived correctly) air tight since the logic follows and the axiom is assumed or taken for granted. There is no empirical requirement to substantiate the axioms. In other endeavors the axioms (or postulates) are always subject to change or refinement since there is a requirement of empirical substantiation. If new observational evidence is found the postulates themselves change which is what "we haven't found it yet" assumes can or will happen. This will never happen in math since observational evidence doesn't change the axioms a theorem has been derived under. (Though it might inspire new axioms under which new theorems could be derived.) The general population doesn't, in general , make this distinction.

 

[Borek]

Statements of nonexistence in math seem to be taken as "we haven't found it yet" by the general populace. Why is that?

Making a mistake doesn't stop those making it from reproduce.

As opposed to eating cyanide or jumping from the roof.

 

[Ryan_m_b]

Statements of nonexistence in math seem to be taken as "we haven't found it yet" by the general populace. Why is that?

Because for every other field this is a perfectly normal thing to hear and the majority of people aren't educated in mathematics. Sure they may have gone through a bit at school but that just covers the basics, none of the theory and for many was a long time ago.

 

[AlephZero]

Making a mistake doesn't stop those making it from reproduce.

As opposed to eating cyanide or jumping from the roof.

Even eating cyanide or jumping from the roof doesn't stop you reproducing, if you do things in the correct order

 

[leroyjenkens]

Making a mistake doesn't stop those making it from reproduce.

As opposed to eating cyanide or jumping from the roof.

I used to jump off the roof all the time when I was a kid. Jackie Chan was a bad influence on me. But that didn't stop me from reproducing. Other things have stopped me from reproducing, such as being the kind of loser who jumps off roofs.

 

[DennisN]

https://www.youtube.com/watch?v=Iov3x_D7nxA

 

[disregardthat]

Just explain that it's logically impossible. If we were to find one, then we would have a logical contradiction on our hands.

 

[jmeps]

If it is understood that the world of mathematics is:
1. Purely a product of the mind.
2. Coupled with the observational power of "man".
3. And used as a tool for predicting and explaining.
Then no response is required. Other than the person does not understand 1,2,or 3.

 

[lendav_rott]

Statement
"There is no formula to generate primes indefinitely."

Counterargument
"Oh you are just saying that because it's not been discovered yet"

Defense
"It's been proven many times over that it can't be done"

Ignorance
"Ohhh, you are so shortsighted, I'll soon have my own spaceship that goes faster than the speed of light"

Irritability
"GRRRRRRRRRRRRR XXXXXX SSSPPFFFFF!@!@!@!@ %%$^$&&*$*I"


If there is a statement of existence - then prove it DOES exist. Do not defend the argument with something like "oh? Prove it doesn't exist, then"

If there is no proof of existence - then it does not exist.
If there is proof of non-existence, then it does not exist.

Simple math

 

[micromass]

Statement
"There is no formula to generate primes indefinitely."

Not sure if that's the best example since there are formulas to generate primes indefinitely. They're just not practical and simple. Who knows, maybe there is such a formula? I really doubt it has been proven that it can't be done.

 

[Borek]

What does it mean "formula"?

I mean, I guess P(i) - function that returns ith prime - doesn't count. But why it doesn't count, and what does?

 

[1MileCrash]

If there is no proof of existence - then it does not exist.

Going to have to disagree entirely on that one.

 

[micromass]

Going to have to disagree entirely on that one.

Agreed. I completely missed that sentence. It's very very very wrong.

 

[lendav_rott]

Let's avoid a philosophical debate. I claim that without proof, something does not exist. The basis of that claim is that if something really did exist, well then it shouldn't be a problem to prove it.

Let's say we Claim that there is a civilization in another galaxy, millions of light years away from us and that there is a pack of oreos in someone's hand at this very moment. Doesn't have to be oreos even, cookies of some sort - similar to the composition of ours. Now, we can't prove it is the case, but we Could claim such a thing - mainly because we Know we can't prove this to be correct or incorrect for some time now. Until it is proven, such a thing does not exist, which would, of course, make this "If no proof, doesn't exist" a false one, but so far it's working for me, unless you can bend my faith - I understand why you say it's wrong, well, should have your hands full then.

 

[Borek]

I claim that without proof, something does not exist.

No. Without a proof we simply don't know if it exists or not.

 

[lendav_rott]

In that case we have the extremists on our case with the existence of god and whatnot.

 

[micromass]

The basis of that claim is that if something really did exist, well then it shouldn't be a problem to prove it.

How could you possibly think that? I don't even know how to respond to that, it's just so illogical.

 

[micromass]

In that case we have the extremists on our case with the existence of god and whatnot.

Please leave religion out of this. The thread was about mathematics. It even says Math in the thread title.

 

[micromass]

How could you possibly think that? I don't even know how to respond to that, it's just so illogical.

OK, maybe I have a good response. So you think that if we can't prove something exists then it doesn't exist because proving it would be easy. Right?

So, how long should we try to prove the existence of something before we can decide it doesn't exist?

 

[1MileCrash]

Lendav_rott, it's not a matter that is up for debate. You said that if "there is no proof of existence, then it does not exist." This is simply a false statement, end of story. There is nothing to debate, you're wrong.

Probably the most immediate indicator that you are wrong is that it makes any existence conjecture in math automatically solved; no, the proposed object does not exist, because I just conjectured it to exist for the first time, and there is no proof that it exists. So I can confirm its nonexistence. QED.

Hell, you solved the Riemann hypothesis! No proof of the existence of any nontrivial zeros of the continuation of the Riemann zeta function without a real component of 1/2? Therefore they don't exist. QED. Collect the prize money.

In that case we have the extremists on our case with the existence of god and whatnot.

?

 

[1MileCrash]

So, how long should we try to prove the existence of something before we can decide it doesn't exist?

I shutter at the thought of how contrary it is to the spirit of mathematics to "decide" something. I suppose that's the point, though.

 

[Tobias Funke]

I don't think lendav_rott's views are as extreme as some of you are making them seem. There's a thread on stackexchange where quite a few people claim that a proof is not valid if only the writer understands it. The dislike of any sort of Platonism is so strong that some professional mathematicians will confidently say that there's some validity determining function--I guess on NxR with first entry number of mathematicians who understand the proof, and second entry their mathematical rank--that proofs are run through, and a proof can be invalid on Monday and valid on Tuesday with no changes. It may be just semantics, but I get the feeling it's deeper than that.

I don't agree, and I don't agree with lendav_rott, but I don't think what he's saying here is particularly crazy in comparison.

 

[1MileCrash]

I don't know what lendav rott's views are on anything, I just know he said a completely incorrect statement that no reasonable person could agree with, and addressed it. I also don't know what you mean by extreme, it's just wrong, and it isn't a view.

In mathematics, if you prove nonexistence, that thing does not exist. It does not mean that there is a strong indicator that it does not exist, it does not mean that a prevailing opinion of nonexistence is justified, it means that it doesn't exist.

If there is a lack of proof of existence, then argue that a prevailing opinion of nonexistence is justified, argue that this is an indicator of nonexistence. Do these things if you want, but do not say "if there is no proof of existence, then it does not exist." Mathematics is about saying what we mean and meaning what we say.

 

[Tobias Funke]

"If there is no proof of existence, then it does not exist" is exactly what I'd expect someone who claims that mathematics is invented, and not discovered, to say. I don't agree, but I don't see why it's so much more wrong (if it is wrong) than claiming that the unit circle didn't exist before it was "invented." It's a philosophical question at that point, and it was just one sentence in a response that agreed with the main topic of "if there is proof of non-existence, then it does not exist" anyway.

 

[1MileCrash]

I believe that mathematics is invented and not discovered. The natural numbers are an invention.

I would never say that a lack of proof of existence meant nonexistence, though. We invent the mathematics, but we do not invent the results that follow, they are the deductive result of axioms that we invented.

It does not follow, deductively, that an object does not exist because of the lack of proof confirming its existence. There is nothing to philosophize over. There is no branch of mathematical philosophy that holds the Riemann hypothesis as solved.

 

[lendav_rott]

If I am to assume lack of proof of existence leads to limbo (uncertainty whether something does or does not exist) then it very quickly becomes a philosophical topic. The only certainty is existence and proof of the latter.

It doesn't have to be material, even - how do we know emotions exist? Hate, anger, happiness, love? Billions of people have no reason to lie to me or anyone else and put on an act - the only logical conclusion is that these emotions do exist, we can all feel them.

What I've been saying is: "lack of proof means non-existence" in math exclusively - we are welcome to philosophize about what you or I or they hold true or theologize about the existence of god all we want, until there is solid proof, it stays out of the math field.

You don't go to work because you MAY be paid for your work - the only certainty you accept is you WILL get paid and you have proof of it from many other co-workers who also get paid - the money does exist regardless, however your salary will only exist once you get paid.

Should you not get paid, therefore you can't prove your salary exists - so you can't convince the bank to give you a house loan or all purpose loan either. If there is no proof, it does not exist.

Or maybe you like to deal a lot with maybes and I am the crazy one, works for me either way.

Might also be necessary - I don't think of math as punching numbers and theories together - to me, math is logic. So to answer the original question of the topic (not worded as a yes-or-no question, but essentially comes down to it) - No - I don't care for "none-that-we-know-of"s - Either 1 or 0. If you can't prove it, it's a 0, if you prove it's a 0, it's a 0 and naturally, if you prove it's a 1, it's a 1.

 

[1MileCrash]

If I am to assume lack of proof of existence leads to limbo (uncertainty whether something does or does not exist) then it very quickly becomes a philosophical topic. The only certainty is existence and proof of the latter.

There is certainty in nonexistence. There does not exist an even prime greater than 2.

It's not because there is a lack of proof for its existence, it's because there is proof of its nonexistence.

It doesn't have to be material, even - how do we know emotions exist? Hate, anger, happiness, love? Billions of people have no reason to lie to me or anyone else and put on an act - the only logical conclusion is that these emotions do exist, we can all feel them.

What I've been saying is: "lack of proof means non-existence" in math exclusively - we are welcome to philosophize about what you or I or they hold true or theologize about the existence of god all we want, until there is solid proof, it stays out of the math field.

You don't go to work because you MAY be paid for your work - the only certainty you accept is you WILL get paid and you have proof of it from many other co-workers who also get paid - the money does exist regardless, however your salary will only exist once you get paid.

Should you not get paid, therefore you can't prove your salary exists - so you can't convince the bank to give you a house loan or all purpose loan either. If there is no proof, it does not exist.

Or maybe you like to deal a lot with maybes and I am the crazy one, works for me either way.

Might also be necessary - I don't think of math as punching numbers and theories together - to me, math is logic. So to answer the original question of the topic (not worded as a yes-or-no question, but essentially comes down to it) - No - I don't care for "none-that-we-know-of"s - Either 1 or 0. If you can't prove it, it's a 0, if you prove it's a 0, it's a 0.

I have no idea what you're talking about. I have no idea why you are talking about emotions or god or salaries. I am beyond perplexed by your response and have no idea what to say in reply.

Do you hold the Riemann hypothesis as solved? You'd better say yes.

 

[lendav_rott]

These are examples from dear life outside the 4 walls. It makes no difference, I like to take examples from life - the logic behind it is all the same as if it were a math assignment.

 

[micromass]

If I am to assume lack of proof of existence leads to limbo (uncertainty whether something does or does not exist) then it very quickly becomes a philosophical topic. The only certainty is existence and proof of the latter.

It doesn't have to be material, even - how do we know emotions exist? Hate, anger, happiness, love? Billions of people have no reason to lie to me or anyone else and put on an act - the only logical conclusion is that these emotions do exist, we can all feel them.

What I've been saying is: "lack of proof means non-existence" in math exclusively - we are welcome to philosophize about what you or I or they hold true or theologize about the existence of god all we want, until there is solid proof, it stays out of the math field.

You don't go to work because you MAY be paid for your work - the only certainty you accept is you WILL get paid and you have proof of it from many other co-workers who also get paid - the money does exist regardless, however your salary will only exist once you get paid.

Should you not get paid, therefore you can't prove your salary exists - so you can't convince the bank to give you a house loan or all purpose loan either. If there is no proof, it does not exist.

Or maybe you like to deal a lot with maybes and I am the crazy one, works for me either way.

Might also be necessary - I don't think of math as punching numbers and theories together - to me, math is logic. So to answer the original question of the topic (not worded as a yes-or-no question, but essentially comes down to it) - No - I don't care for "none-that-we-know-of"s - Either 1 or 0. If you can't prove it, it's a 0, if you prove it's a 0, it's a 0 and naturally, if you prove it's a 1, it's a 1.

To be blunt: you're wrong. And your post has just proven that you don't know enough about mathematics to discuss this matter properly. I can't convince you of the way math really is since your opinion is just so illogical and since you are so badly informed about math. So I'm not going to try.

I wish to leave this topic open for other discussions. So please do not respond to this thread anymore. I will consider it a thread hijack.

 

[dkotschessaa]

These are examples from dear life outside the 4 walls. It makes no difference, I like to take examples from life - the logic behind it is all the same as if it were a math assignment.

In mathematics, terms are defined precisely enough for the logic to prove things with absolute certainty. We do not have this luxury in "life."

In real life examples you can use formal logic, but terms are not precisely defined and so argumentation is much sloppier. That's where informal logic is helpful (fallacies and other "thinking tools" to use Daniel Dennett's term).

Using real life examples to treat mathematical topics is like...is like... somebody help me out here. It's bad. Really bad.

Edit: Sorry to provoke lendav_rott any further as I do not wish to participate in thread hijacking.

My first sentence above I think is a pretty good contribution to the original post.

-Dave K