How much freedom does math have?

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How much freedom does math have?
A couple of examples to explain.
1) With Pythagorean's theorem, dozens of proofs exist (including one by a US President). So you have the choice of which proof you favor.
2) In calculus, certain problems can be solved in more than one way (algebra, etc.)
There are other situations where certain proofs aren't fully accepted by all mathematicians. So what do you think?
PhilX
1) With Pythagorean's theorem, dozens of proofs exist (including one by a US President). So you have the choice of which proof you favor.
2) In calculus, certain problems can be solved in more than one way (algebra, etc.)
There are other situations where certain proofs aren't fully accepted by all mathematicians. So what do you think?
PhilX
Re: How much freedom does math have?
Philosophy Explorer wrote: ↑Tue Nov 14, 2017 11:43 pmA couple of examples to explain.
1) With Pythagorean's theorem, dozens of proofs exist (including one by a US President). So you have the choice of which proof you favor.
This theorem may have more known proofs than any other (the law of quadratic reciprocity being another contender for that distinction); the book The Pythagorean Proposition contains 370 proofs.[11] https://en.wikipedia.org/wiki/Pythagorean_theorem
2) In calculus, certain problems can be solved in more than one way (algebra, etc.)
There are other situations where certain proofs aren't fully accepted by all mathematicians. So what do you think?
PhilX
I once heard a phrase: "Math isn't about finding limits, but rather about finding possibilities." I hold that opinion as a personal axiom.
1) If we look at the nature of number as stemming strictly from 1
2) and all number stemming from 1, in itself being composed of 1
3) and all number being composed of 1 manifesting ad infinitum
4) Ad infinition is 1 revolving into itself to produce all possible numbers.
5) All possible number exist through all mathematical functions, as all mathematical functions are strictly extensions of positive (addition) and negative (subtraction) values.
6) As all mathematical functions are extensions of positive and negative values (founded in basic arithmatic), these mathematical functions provide foundations for further mathematical functions (multiplication, division) ad infinitum in correspondence to the ad infinitum nature of number (considering form and function are interjoined).
7)In theory there are infinite mathematics stemming from a core base synonymous to "1", and these infinite numbers/functions are a result of 1 revolving through itself.
Re: How much freedom does math have?
I think that an example which can provide material for thought for this topic would be the series:
Series: (i=1, i=infinity) ∑ i; i.e. 1+2+3+4... = ??
Is the result of this series: infinity or 1/12!!
The infinity part we can easily grasp. But the 1/12 is counterintuitive. One can watch the following Youtube videos to know what it is about. I have provided three but only one will do. I have provided three because I like all of these explanations, and I could not bring myself to chose only one of them! But anyway, the first one is an explanation from a physicist and the other two are from maths professors. Each professor provides some valuable insights.
1. https://www.youtube.com/watch?v=wI6XTVZXww
2.https://www.youtube.com/watch?v=Ed9mgo8FGk
3. https://www.youtube.com/watch?v=0Oazb7IWzbA
It appears that choice/freedom extends to mathematics as well and is not confined to just everyday life.
Re: How much freedom does math have?
Averroes wrote: ↑Wed Nov 15, 2017 5:14 pmI think that an example which can provide material for thought for this topic would be the series:
Series: (i=1, i=infinity) ∑ i; i.e. 1+2+3+4... = ??
How can "i" be any different than a "point" considering all imaginary structures at the micro and macro level are reduced to a point, with the intermediate being composed of "points"?
The observation I am trying to make, is how can what we understand of number be seperated from the spatial nature of reality we observe?
If all number is strictly composed of "1"....well what is "1"?
Is the result of this series: infinity or 1/12!!
The infinity part we can easily grasp. But the 1/12 is counterintuitive. One can watch the following Youtube videos to know what it is about. I have provided three but only one will do. I have provided three because I like all of these explanations, and I could not bring myself to chose only one of them! But anyway, the first one is an explanation from a physicist and the other two are from maths professors. Each professor provides some valuable insights.
1. https://www.youtube.com/watch?v=wI6XTVZXww
2.https://www.youtube.com/watch?v=Ed9mgo8FGk
3. https://www.youtube.com/watch?v=0Oazb7IWzbA
It appears that choice/freedom extends to mathematics as well and is not confined to just everyday life.
Re: How much freedom does math have?
"i" here is just an index, it can be replaced by another letter of the alphabet if you do not like "i", lets say "n" or "k". Please do not be offended, but I am talking mathematical stuff with PhilX, and I do not understand anything that you write. Please, forgive me if I do not reply to your other questions, it is just because I have absolutely no idea of what you talk about. May be you operate at a level which is too high for me!Eodnhoj wrote:How can "i" be any different than a "point" considering all imaginary structures at the micro and macro level are reduced to a point, with the intermediate being composed of "points"?
Re: How much freedom does math have?
Nothing to forgive, no offense was made on your part, none was intended on mine.Averroes wrote: ↑Wed Nov 15, 2017 7:13 pm"i" here is just an index, it can be replaced by another letter of the alphabet if you do not like "i", lets say "n" or "k". Please do not be offended, but I am talking mathematical stuff with PhilX, and I do not understand anything that you write. Please, forgive me if I do not reply to your other questions, it is just because I have absolutely no idea of what you talk about. May be you operate at a level which is too high for me!Eodnhoj wrote:How can "i" be any different than a "point" considering all imaginary structures at the micro and macro level are reduced to a point, with the intermediate being composed of "points"?
Re: How much freedom does math have?
To understand this one should study complex analysis and in particular the subject of analytic continuation to understand the context of this unfortunate meme floating around the internet and confusing people.
Did the videos you linked bring out this context, or were they silly internet woo?
Re: How much freedom does math have?
Well, if you had watched the videos and made your own research, you would not need to be asking that question! To understand this one must make effort, and study! The videos I have linked to are short interviews/lectures of physics and mathematics professors; I would not say that they were confusing people when they were showing that these mathematical equations are being used to make sense of the real world in physics!wtf wrote: ↑Wed Nov 15, 2017 9:26 pm
To understand this one should study complex analysis and in particular the subject of analytic continuation to understand the context of this unfortunate meme floating around the internet and confusing people.
Did the videos you linked bring out this context, or were they silly internet woo?
Now, of course the context and the analytic continuation is important, but for the purposes of this thread, it is still an example which shows that in mathematics we have the choice, at least from my perspective. Moreover, the analytic continuation spot on illustrates my point. Isn't it freedom in mathematics the subject of this thread? The thing is that you mentioned the context principle by bringing up the analytic continuation concept, but then why did you not apply the same context principle to the example with respect to the subject of this thread? This does not show consistency on your part.
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