Let's discuss Mathematics

[Lohrenswald]

I understand now.

I made myself a bit confused, in the original example that made me wonder, one of the vectors was a unit vector, so then it makes sense that the scalar product of a vector times the unit vector is the component of that vector along the given axis

 

[onejayhawk]

Thought experiment. There are certain absurdities which mathematicians take as granted. One is that the segment [0,1) has no endpoint at 1. The more correct statement is that [0,1) has no defined endpoint. It's there, but we don't have the language to talk about it.

So, can you prove that 0.999...is not an element of [0,1) without using 0.999... = 1?

J

 

[monikernemo]

Thought experiment. There are certain absurdities which mathematicians take as granted. One is that the segment [0,1) has no endpoint at 1. The more correct statement is that [0,1) has no defined endpoint. It's there, but we don't have the language to talk about it.

So, can you prove that 0.999...is not an element of [0,1) without using 0.999... = 1?

J

Firstly, can you define your notion of having no end point? Or having no defined endpoint? It seems vague what you are trying to point at mathematically. It is bold to say that the language isn't there. It is there. ZFC is the language/backbone of most modern mathematics.

Secondly, the modern construction of real numbers relies on ZFC, either dedekind cuts or equivalence classes of Cauchy sequence. If you don't know what are those, look them up.

In fact modern mathematics define the field of real numbers by having some satisfactory properties, that first it contains the rationals as an ordered subfield and secondly, every set that is bounded above has a least upper bound. Equivalently, every Cauchy sequence is convergent. So in particular, if one refers the element 0.9999... repeating, it can be written as the limit as n goes to infinity sum of 9*10^-i for i = 1 to n, which is contained in the reals since it is a Cauchy sequence, and it is in fact 1.

The two construction mentioned above can be shown to be isomorphic (in other words, essentially the same) and any other field that satisfy the above two properties are necessarily isomorphic as fields.

Tldr: these are not firstly, absurdities. It's just that someone outside of math lack understanding and calls it absurdities. You know, akin to how anti vaxxers say vaccination is a sham. Secondly this assumptions are certainly not taken for granted, in fact it is the crystallisation of the work of many generations, to show that one can construct the field of real numbers from rationals and to show that they have certain desirable properties. And of course to prove 0.999... not equals to 1 doubters that 0.999... = 1, in our definition of real numbers.

 

[Samson]

Thought experiment. There are certain absurdities which mathematicians take as granted. One is that the segment [0,1) has no endpoint at 1. The more correct statement is that [0,1) has no defined endpoint. It's there, but we don't have the language to talk about it.

So, can you prove that 0.999...is not an element of [0,1) without using 0.999... = 1?

J

Can you define 0.999...? If 0.333... = 1/3, and 0.666... = 2/3 should not 0.999... = 3/3?

 

[Ferocitus]

Tldr: these are not firstly, absurdities. It's just that someone outside of math lack understanding and calls it absurdities.

Or they're ultrafinitists or actualists who reject the vagueness of many of the claims made by followers of Cauchy, Dedekind, and Hilbert.

 

[Kyriakos]

A question would be if at some future development of math it would be more practical to have 0.999.... not equal 1 than equal it.
After all, what matters is internal consistence. If 0.999...=1 doesn't create a chasm in some future breakthrough which would be worth obliterating anything that backs 0.999...=1, then fine.
Without being aware of the system which proves 0.999... and 1 to be one and the same, I can still very easily realize that what matters are the tools one is given (and also ones they may be denied) by adhering to that system.

I am better equipped to approach such issues from a philosophical standpoint, given that is my university background. While math are discovered (because they are part of human thinking, regardless of this being obvious/conscious or not when it is more advanced math) I am quite sure that the actually crystallized math we have are only a small subset of the possible ones, and this includes larger or even cataclysmic changes (which probably aren't practical).

 

[Kyriakos]

Anyway, I have a less philosophical thing to add, that is an actual question about math

Has to do with formal logic.

I want to ask which way - in your view - is more helpful (more practical, in that it is more compatible with the entire system) to think of P ---> Q (if P then Q) as being a true statement (also) when P is false.

I am aware of two main ways to go about this. One I like considerably (cause I am prone to link something less defined to something more defined, if the link isn't suspect) is to identify P ---> Q as a subset issue. Ie say that Q is the larger set, including P-tied parts, but also having parts which don't need P to exist: If someone plays loud music next to you (P), you won't be able to concentrate (Q). While you won't be able to concentrate if one is playing loud music, that is only a subset of the causes for Q, so the fact P implies Q alone doesn't matter in the system and thus you can afford to identify the statement as true (it creates no problem with anything that would work).
What creates some sense of discomfort is that the entire system (formal logic) rests on true and false, and it has to be realized that this is above everything else. False-imples-x being a true statement always, is a result of the true-false backbone, not of the form of implication. Many people try to view this the other way round.

So, to be brief, my question is if you can help me prove Riemann using formal logic. If I can prove it then you will not die in two seconds (see what I did there)

 

[monikernemo]

I don't think it's unsettling. We are just trying to capture relationships here that are meaningful, which one of them is to say that Q is a necessary condition for P (or P is sufficient for Q). And one thing to note is that often in practice, P and Q are not arbitrary statements; P and Q are statements linked together by some other object. For example:
If a polynomial f is of degree n then f can have at most n roots. Then this "if P then Q" is alot more meaningful since there are polynomials f that are of degree n and polynomials that are not of degree n.

Of course one can churn out true but "gibberish" statements for example: If Kyriakos replies to my post then 5= 2+3. This statement is true regardless whether Kyriakos replies my post because indeed 5= 2 + 3. It is hard to capture causality, I think implication is good enough. And we only study a subset of statements that we find interesting or meaningful.

 

[Kyriakos]

I don't think it's unsettling. We are just trying to capture relationships here that are meaningful, which one of them is to say that Q is a necessary condition for P (or P is sufficient for Q). And one thing to note is that often in practice, P and Q are not arbitrary statements; P and Q are statements linked together by some other object. For example:
If a polynomial f is of degree n then f can have at most n roots. Then this "if P then Q" is alot more meaningful since there are polynomials f that are of degree n and polynomials that are not of degree n.

Of course one can churn out true but "gibberish" statements for example: If Kyriakos replies to my post then 5= 2+3. This statement is true regardless whether Kyriakos replies my post because indeed 5= 2 + 3. It is hard to capture causality, I think implication is good enough. And we only study a subset of statements that we find interesting or meaningful.

Sure, the point I made was that it isn't just a random biproduct of the system that you have random statements (as you called them), but that this is an automatic product of a system that rests on a true-false structure. That you won't be able to make use of random conditionals to prove something is true, yet from a hypothetical point of view of the actual system... it is of no importance. At least this is how I view it, and it helps one to not forget that formal logic isn't something tied to natural language [or, if you will, natural language isn't (as) well-defined]

Furthermore, one has to suppose that not just in current levels of formal language, but likely in all future possible advanced levels, the "random" conditionals and "random" statements in any system will always be vastly more than the ones of use in a proof. Still, they cannot be taken out of the system, for the system only happens to produce/allow some non-random conditionals and statements along with the rest.

 

[monikernemo]

Furthermore, one has to suppose that not just in current levels of formal language, but likely in all future possible advanced levels, the "random" conditionals and "random" statements in any system will always be vastly more than the ones of use in a proof. Still, they cannot be taken out of the system, for the system only happens to produce/allow some non-random conditionals and statements along with the rest.

Yes that is the case and one has to live with it. Of course there are many "bad" and "uncontrollable", "wild" objects. I don't think that is surprising.

 

[Kyriakos]

Yes that is the case and one has to live with it. Of course there are many "bad" and "uncontrollable", "wild" objects. I don't think that is surprising.

I am not saying it is surprising. It actually may become useful in some later development. I mean those random objects are every bit as much part of the system, so we shouldn't be so dismissive.
Anyway, I am actually posting from the distant future. Know how I managed to post in the past? Has to do with making use of those objects you thought of as useless and random

 

[monikernemo]

I am not saying it is surprising. It actually may become useful in some later development. I mean those random objects are every bit as much part of the system, so we shouldn't be so dismissive.

I thought that you were initially suggesting that having many "uninteresting" object is the flaw of the "true false" logic. Then again being "interesting" is subjective.

 

[Kyriakos]

I thought that you were initially suggesting that having many "uninteresting" object is the flaw of the "true false" logic. Then again being "interesting" is subjective.

No, I never said they are a flaw (ironically, you can even form this in the P--->Q fashion). How can something that produces stuff be itself responsible for a flaw when it is a mental construction? Mental constructions don't have unintended byproducts like physical constructions do (again from the hypothetical point of view of the construction itself ) and this because in a mental construction the observer (you) is very linked to the construct (your mental world).

Anyway, the first to create a formal logic system was Aristotle, and his included more than true/false (also had "indistinct"). The dynamics of that system were different due to using a larger grammar (basically that of the greek language). But it should be noted that the first to present a formal system where stuff like that in the current one are true, was Philon of Megara, who famously argued that a false conditional will always bring about a true statement.

 

[Kyriakos]

A very general question.
What kind (ie relatively more tied to which current) of "future math" do you think Erdos had in mind, when he spoke of those possibly being needed to prove the Collatz Conjecture?

 

[Samson]

Linear: y = ax + b
Quadratic: y = ax^2 + bx + c
Cubic: y = ax^3 + bx^2 + cx + d

Why is quadratic called quadratic, rather than planar? Should not quadratic have x^4 in it?

 

[AdrienIer]

Quadratic's root is the same as square. So 4 sides, but literally x^2

 

[Kyriakos]

Here periphrasis is used, and they are called "first degree", "second degree" etc

 

[Kyriakos]

I think it is also worth noting that virtually no highschool student is actually aware of why the formulae for third-degree equations work. Even second degree are learned mainly as formulae leading to two roots for real numbers. It would help if students actually relied less on memorizing some formulae, cause no one is going to examine something new by just parroting those.
Then again, what do you expect when almost all of the math "problems" to be solved in exams merely ask you to turn the equation into a difference or addition of squares? You don't need insight on how lego are created, if all you want to do is model the same hut 1000 times.

 

[AdrienIer]

I think it is also worth noting that virtually no highschool student is actually aware of why the formulae for third-degree equations work. Even second degree are learned mainly as formulae leading to two roots for real numbers. It would help if students actually relied less on memorizing some formulae, cause no one is going to examine something new by just parroting those.
Then again, what do you expect when almost all of the math "problems" to be solved in exams merely ask you to turn the equation into a difference or addition of squares? You don't need insight on how lego are created, if all you want to do is model the same hut 1000 times.

I'm not sure how my colleagues do it but I guarantee that I prove all the second degree formulas to my students every year (those who are on the year where they get taught how to solve them of course), showing them how you find the roots (if they exist) and why in other cases they don't.

 

[Kyriakos]

I'm not sure how my colleagues do it but I guarantee that I prove all the second degree formulas to my students every year (those who are on the year where they get taught how to solve them of course), showing them how you find the roots (if they exist) and why in other cases they don't.

What about the third degree?
Also, it is a gimmick to have a problem which just needs you to memorize the third degree formulaes or you are stuck, when you don't even know why they work. Rewarding isolated memory isn't a good idea, non?