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This might be a purely philosophical question, but still...

Up to some points in geometry, Euclid's axioms were accepted, even though the 5th one was causing headache.

Then someone comes up and say "hey, what if we consider it's wrong ?", and bam!, here come spherical and hyperbolical geometries. Such theories ave a meaning and can even be used in physics to describe our universe.

In another fiels, mathematicians have known for quite some time that we can't make the square root of a negative number.

Then someone comes up and say "hey, what if we consider $\sqrt{-1}$ has a solution ?" and bam!, here come complex numbers, very useful in describing a lot of things, in particular waves in physics.

What is so special about these "rules" (5th axiom of Euclid and "no squared real number squared is negative") that we can suddenly consider them as false and gets something useful out of it ?

It seems to me, if I say "hey, let's consider $\pi=4$" or "hey, let's consider $x=x+1$ has a solution in some new set of numbers", it might bring some new theories, but they will be a complete waste of time.

Is there a reason why it is so ?

xdutoit
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  • Surely are the new geometries useful, but it was not a good idea to try to describe our universe this way. Surely are imgainary numbers useful but physical measurands are always real, it was again not a good idea to combine them for physical purposes. – Peter Jan 17 '21 at 10:29
  • Depending on how $\pi$ is defined (ratio of circumference to diameter, ratio of disk area to radius, sum of angles of a triangle, etc.), $\pi = 4$ is possible --- use an appropriate norm on ${\mathbb R}^2,$ consider triangles only on a certain curved surface, etc. And $x = x + 1$ is true for any infinite cardinal number and $+$ being cardinal addition (and $x = 1 + x$ is true for any nonzero limit ordinal and $+$ being ordinal addition). – Dave L. Renfro Jan 17 '21 at 11:09
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    @DaveL.Renfro: Depending on how $1$ is defined, $1 = 0$ is possible. Use an appropriate definition of the symbol "$1$"... – user21820 Jan 17 '21 at 12:17
  • @user21820: The examples I gave correspond to fairly well known mathematical "theories" where the results stated can arise. For $\pi$ see $\pi$ is the minimum value for pi by Adler/Tanton (2000) and On the values of pi for norms on ${\mathbb R}^2$ by Duncan/Luecking/McGregor (2004). As for cardinal and ordinal addition, see pretty much any set theory text. – Dave L. Renfro Jan 17 '21 at 15:36
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    @DaveL.Renfro: I'm not questioning the mathematical content to which you are referring. But I am objecting to you bringing up such stuff when it is clear that the asker does not even have basic mathematical proficiency. What I stated is also correct, from a logician's viewpoint. So what? Your rationale for saying that "π=4" can be meaningful in some contexts is also applicable to my saying that "1=0" can be meaningful in some contexts. Even better if it is "2=0". – user21820 Jan 17 '21 at 15:41
  • @user21820: I realize the OP probably doesn't have the background, but I didn't think the existing answers led to anything others with more background coming to this question in the future would find of much interest, plus I was hoping by bringing up these examples that maybe someone with more time than I have now would expand on it. One of the issues, as I see it, is that when generalizing we're (always?) focusing on generalizing one (or a few) specific properties at the expense of others. Some properties when generalized lead to interesting avenues to pursue, others appear to be dead ends. – Dave L. Renfro Jan 17 '21 at 15:51
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    @DaveL.Renfro: Well, I don't see a problem if you want to post a more advanced answer, but I definitely think you should first give a precise answer at the asker's level (to cater for the vast majority of readers), and then go on to more advanced stuff but making it clear that in those cases the "π" is not the same as the usual "π". If you don't make things clear, it is no better than saying that we can bend any and every thing we like, which is certainly not the case. – user21820 Jan 17 '21 at 16:31
  • @user21820: hehehe... now I feel ashamed (I'm actually a math teacher, and I did understood the different comments, don't worry). But I agree my question is naïve and was phrased in a naïve way. I'm still wondering though if, taking some meta-mathematical viewpoint, some theorems have an intrinsic quality of being "softer" than others and allow for ignoring them while still having a meaningful representation of our reality. But I guess it's a both philosophical and open question. – xdutoit Jan 19 '21 at 09:01
  • What's the meaning of "ignoring them"? Rigorous mathematics is always done within some foundational system (whether you realize it or not), which is typically some FOL system, and in this system every symbol is either defined or not at any point during your proof. If $0,1$ are defined and you have proven that $0 ≠ 1$, then there is absolutely nothing you can or ought to try to do to 'ignore it'. It is exactly the same for $π$. This has nothing to do with philosophy or reality. I am afraid that you have to learn proper FOL reasoning if you want to get this right. – user21820 Jan 20 '21 at 03:23
  • Feel free to ask me further in chat. – user21820 Jan 20 '21 at 03:23

2 Answers2

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An extension makes sense (and can in the best case turn out to be very useful), if it respects the structures we currently have and creates no contradictions. Consistency is necessary to create anything meaningful.

The complex numbers are , for example , useful because they form an algebraically closed field.

Another very important example is the system ZFC , which is a much stronger theory than the PA that are contained in ZFC.

Because you mentioned $\pi=4$ , actually in the USA, this was defined by law (!) but I do not exactly remember where and when. But such ideas , as you correctly pointed out, have no merit whatsoever.

Peter
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    $\pi = 4$ defined by law in the USA. That is a false story. A bill (with some mixed-up geometry in it, but not $\pi=4$) was introduced into the Indiana state legislature, but never passed. See https://en.wikipedia.org/wiki/Indiana_Pi_Bill – GEdgar Jan 17 '21 at 10:57
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The reason these rules were bend was because the original set of rules did not suffice for every purpose. Euclid's postulates were designed and deducted specifically for plane geometry. It is then reasonable that if one wants to do geometry on a curved space, the set of rules have to be adapted to suit this new situation.

As Peter mentions, it is mostly useful to extend an existing structure in such a way that there are no contradictions in the extended structure. However, one can also modify the existing structure in order to obtain a new structure which has maybe a bit other implications and philosophical interpretations.

It is hard to know beforehand bending which rules gives you meaningful insights, and which not, but experimenting and trying new things without knowing the outcome has been characteristically of science since day one.

EBP
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