A talk at the 6th Mathematical Transgressions meeting on the role of Arithmetic | N J Wildberger
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- čas přidán 3. 06. 2024
- This is a short heads-up for a talk that I gave recently at the 6th Mathematical Transgressions meeting (June 3-4 2024) which was online and organized by Barbara Baranska from the Department of Mathematics, University of the National Education Commission, Krakow, Poland. The title of my talk was "Towards a logical rational arithmetic as a foundation for maths educations and research".
My talk is followed by an interesting talk by Lucja Farnik of the University of the National Education Commission, Poland entitled " Selected manifestations of Lakatos' quasi-empiricism in contemporary algebraic geometry". I hope you enjoy it as well.
The chair of this session was Prof. P. Blaszczyk whose work is also quite interesting in these foundational directions.
Here is the link to these talks:
czcams.com/users/liveyAmhpyAmsj4
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I enjoyed the talk and your answers to the questions. I particularly liked the different angle you gave on what you have been saying recently in your lecture on how a belief in infinite processes allow one to “prove” (not prove) famous open pure mathematics problems - when you described the idea of having two algorithms ostensibly for computing the same thing, the matter of how you prove they are the same algorithm, if you start with an algorithm and add in a twist which would prove an unsolved problem if true, then the difficulty of proving identity of the algorithms includes the difficulty of determine the truth of the unsolved difficult problem. Very interesting point and a nice way to demonstrate it.
Thanks for sharing your talk.
I have followed you (NJW) since I came across your work of Rational Triginormy many years ago. It was put out in the video, the question, just what is a number?
A very fundamental question and one that everyone will ask when first exposed to numbers and their uses. But when I think about that question, I do not have a concise answer, or know how to answer it concisely. I know of many analogies that I can use and remember but a clear, concise, and logically correct definition of a number (of anytype) I do not have or can recall ever hearing one.
The closest I have come to logically sorting this out is with Zero and One only.
Zero is the null / origin of reference for relationship with a number. One is the measure / value (in some given frame of reference) that sets the unit of all other numbers in their relationship to Zero. Other than this I can not come up with anything that is logically sound.
When I think about the basic operations of arithmetic using the framework stated prior, I can understand division by Zero and One much more logically.
When divide by one, you are dividing by what sets its value, and as such there is a singular and unique relationship between One and that number (the number itself). When you divide by Zero you are dividing by the reference for ALL the numbers, and as such there is no singular relationship or result, so it is undefined.
I am no mathematician by any measure, but I do see the lack of solid logical underpinnings that are pointed out.
Great talk! Polish is the best accent for a mathematical intro, the Egyptian bread example was quite cool, and as usual you inspire by demanding mathematics be logical.
In my thinking about the fundamentals, I've been quite curious to learn about cutting sequences, as they seem to be at the intersection of geometric and numeric representation. I was wondering if you were aware of any work being done studying basic arithmetic in this context of cutting sequences (aka Sturmian words)?
The integer sequences are all essentially periodic unitary numbers, and the qualities of rationals, irrationals and transcendentals are all apparent in the length and complexity of their subsequences. m
With respect to understanding arithmetic and computability,what stands out is the constructibility of certain sequences (even irrational ones). With respects to the foundations of math, who knows, but with respect to computation it's very useful to be able to generate high precision values for certain constants.
Famously,the continued fraction [1;1,1,1,...] is constructible via a simple substitution sequence
Continued Fraction [1;1,1,1,...]
1011010110110101101011011010...
I should think it does, because it's more famous inverse sequence* can be constructed with the substitution rule 0 -> 1, 1 -> 10
Continued Fraction 1/[1;1,1,1,...]
0100101001001010010100100101...
(The ones and zeroes flipped.)
Meanwhile, the continued fraction for 2 has no known construction via substitution schemes.
Continued Fraction [1;2,2,2,…]
1010110101101010110101101010...
If it were possible to construct these sequences in the same or similar fashion, I think it would have considerable impact to our understanding.
Here are some more sequences in case anyone is curious. ChatGPT is happy to generate more for you, but be careful to verify its work. In particular for anything irrational or transcendental, as floating point precision will fail (unless you teach it how to do it right.)
Hope you enjoyed the comment!
0
000000000000000000000000000000
1
010101010101010101010101010101
2
001001001001001001001001001001
3
000100010001000100010001000100
4
000010000100001000010000100001
5
000001000001000001000001000001
1/3
111011101110111011101110111011
2/3
110110110110110110110110110110
1/5
111101111011110111101111011110
5/6
111101110111011101110111011101
5/7
111011101101110111011101101110
8/7
111110101111101011111010111110
Π
1110111011101110111011101110
π/4
0101010010101010010101001010
e
1101110111011011101110111011
1/e
0010001000100100010001000100
e/2
1010110101011010101101010110
root 2
1010110101101010110101101010
1/root 2
0101001010010101001010010101
Neega shut up