Axiom musings ... Hamming and SANE
Tim Daly <[email protected]> Wed, 27 Apr 2022 09:22:45 -0400
| Newsgroups | gmane.comp.mathematics.axiom.devel |
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| Message-ID | <CAJn5L=+pe5ebznrBZSLvycNkTKdy5_7xVLUirXdtRmgPQSesJA@mail.gmail.com> |
--00000000000032148705dda2b5e3 Content-Type: text/plain; charset="UTF-8" ... [snip] ... I guess that's where we differ. Axiom is a piece of research software. Axiom was never intended to be a product. IBM gave us two choices. Either find someone to market it (NAG) or throw it away. FOSS software did not exist at that time and we were not allowed to work on it in any capacity once it was sold. That stopped the research but not the desire. On the "little problem" front I have many things I would like to do with Axiom. For example, I have a hierarchical graph of about 50 different kinds of matrices arranged so Axiom's inheritence would fit perfectly. Within that set are things like unitary matrices which would have just the algorithms you need to do quantum computers. The DHMATRIX domain would be usefully extended to drive the several robots I have in my office. Fourier and Laplace transform algorithms would be very useful for robot dynamics. Graphics algorithms should be extended to show Bode plots. The graphics world could be extended to do 3D solid models to use with my 3D printer. Quantum-safe encryption uses lattices which would fit well with skew matrix algorithms. There are a lot of "little problems" I wish I had time to work on. Richard Hamming (of Hamming distance fame) has a youtube video [0]. He used to ask his Bell Labs colleagues "What are the fundamental problems in your area and why aren't you working on them?" So what are the "fundamental problems" of combining computers and math? Why aren't you working on them? I see two streams of computer mathematics. The computer algebra stream started in the 60s/70s and uses the "It works for me" approach. The proof stream started in the 90s and uses the "Lets prove my current theorem". The two streams have only James Davenport in common. "Computational Mathematics" would be a combination of both of these streams. Proofs would make Axiom less ad-hoc. Proven computer algebra algorithms would vastly expand proof assistants with trusted results. I view "Computational Mathematics", combining computer algebra and proofs as one of the "fundamental problems" of this era. The SANE version of Axiom is my attack on a fundamental problem. Axiom is the best research platform available to show both camps what is possible in future mathematics. Axiom showed the power that comes from using mathematics as a scaffold architecture guiding the organization. Grounding the group theory scaffold with proofs will make Axiom a much more solid structure for future mathematics. That said, my real motivation is that this research is fun. I am pretty sure nobody but me will ever care. I'm a "research rat" by nature. Tim [0] You and Your Research https://www.youtube.com/watch?v=a1zDuOPkMSw --00000000000032148705dda2b5e3 Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable <div dir=3D"ltr"><div><font color=3D"#888888">... [snip] ...<br></font></di= v><div><font color=3D"#888888"><br></font></div><div><font color=3D"#888888= ">I guess that's where we differ. Axiom is a piece of research software= .</font></div><div><font color=3D"#888888"><br></font></div><div><font colo= r=3D"#888888">Axiom was never intended to be a product. IBM gave us two cho= ices. Either</font></div><div><font color=3D"#888888">find someone to marke= t it (NAG) or throw it away. FOSS software did</font></div><div><font color= =3D"#888888">not exist at that time and we were not allowed to work on it i= n any <br></font></div><div><font color=3D"#888888">capacity once it was so= ld. That stopped the research but not the desire.</font></div><div><br></di= v><div><font color=3D"#888888">On the "little problem" front I ha= ve many things I would like to do with</font></div><div><font color=3D"#888= 888">Axiom. For example, I have a hierarchical graph of about 50 different<= /font></div><div><font color=3D"#888888">kinds of matrices arranged so Axio= m's inheritence would fit perfectly.</font></div><div><font color=3D"#8= 88888">Within that set are things like unitary matrices which would have ju= st</font></div><div><font color=3D"#888888">the algorithms you need to do q= uantum computers. The DHMATRIX</font></div><div><font color=3D"#888888">dom= ain would be usefully extended to drive the several robots I have</font></d= iv><div><font color=3D"#888888">in my office. Fourier and Laplace transform= algorithms would be very</font></div><div><font color=3D"#888888">useful f= or robot dynamics. Graphics algorithms should be extended</font></div><div>= <font color=3D"#888888">to show Bode plots. The graphics world could be ext= ended to do 3D solid</font></div><div><font color=3D"#888888">models to use= with my 3D printer. Quantum-safe encryption uses</font></div><div><font co= lor=3D"#888888">lattices which would fit well with skew matrix algorithms. = There are <br></font></div><div><font color=3D"#888888">a lot of "litt= le problems" I wish I had time to work on.<br></font></div><div><font = color=3D"#888888"><br></font></div><div><font color=3D"#888888">Richard Ham= ming (of Hamming distance fame) has a youtube video [0].<br></font></div><d= iv><font color=3D"#888888">He used to ask his Bell Labs colleagues "Wh= at are the fundamental</font></div><div><font color=3D"#888888">problems in= your area and why aren't you working on them?"</font></div><div><= font color=3D"#888888"><br></font></div><div><font color=3D"#888888">So wha= t are the "fundamental problems" of combining computers and math?= <br></font></div><div><font color=3D"#888888">Why aren't you working on= them?<br></font></div><div><font color=3D"#888888"><br></font></div><div><= font color=3D"#888888">I see two streams of computer mathematics. The compu= ter algebra</font></div><div><font color=3D"#888888">stream started in the = 60s/70s and uses the "It works for me" approach.</font></div><div= ><font color=3D"#888888">The proof stream started in the 90s and uses the &= quot;Lets prove my</font></div><div><font color=3D"#888888">current theorem= ". The two streams have only James Davenport</font></div><div><font co= lor=3D"#888888">in common.</font></div><div><font color=3D"#888888"><br></f= ont></div><div><font color=3D"#888888">"Computational Mathematics"= ; would be a combination of both of these</font></div><div><font color=3D"#= 888888">streams. Proofs would make Axiom less ad-hoc. Proven computer</font= ></div><div><font color=3D"#888888">algebra algorithms would vastly expand = proof assistants with trusted</font></div><div><font color=3D"#888888">resu= lts.</font></div><div><font color=3D"#888888"><br></font></div><div><font c= olor=3D"#888888">I view "Computational Mathematics", combining co= mputer algebra</font></div><div><font color=3D"#888888">and proofs as one o= f the "fundamental problems" of this era. The</font></div><div><f= ont color=3D"#888888">SANE version of Axiom is my attack on a fundamental p= roblem.<br></font></div><div><font color=3D"#888888"><br></font></div><div>= <font color=3D"#888888">Axiom is the best research platform available to sh= ow both camps</font></div><div><font color=3D"#888888">what is possible in = future mathematics. Axiom showed the power that</font></div><div><font colo= r=3D"#888888">comes from using mathematics as a scaffold architecture guidi= ng</font></div><div><font color=3D"#888888">the organization. Grounding the= group theory scaffold with proofs</font></div><div><font color=3D"#888888"= >will make Axiom a much more solid structure for future mathematics.</font>= </div><div><font color=3D"#888888"><br></font></div><div><font color=3D"#88= 8888">That said, my real motivation is that this research is fun. <br></fon= t></div><div><font color=3D"#888888">I am pretty sure nobody but me will ev= er care. <br></font></div><div><font color=3D"#888888">I'm a "rese= arch rat" by nature. <br></font></div><div><font color=3D"#888888"><br= ></font></div><div><font color=3D"#888888">Tim<br></font></div><div><font c= olor=3D"#888888"><br></font></div><div><font color=3D"#888888">[0] You and = Your Research</font></div><div><font color=3D"#888888"><a href=3D"https://w= ww.youtube.com/watch?v=3Da1zDuOPkMSw" target=3D"_blank">https://www.youtube= .com/watch?v=3Da1zDuOPkMSw</a></font></div><div><font color=3D"#888888"><br= ><br></font></div></div> --00000000000032148705dda2b5e3--