The paper reviews the most consequential defects and rectification of traditional mathematics and its foundations. While this work is only the tip of the iceberg, so to speak, it gives us a totally different picture o...The paper reviews the most consequential defects and rectification of traditional mathematics and its foundations. While this work is only the tip of the iceberg, so to speak, it gives us a totally different picture of mathematics from what we have known for a long time. This journey started with two teasers posted in SciMath in 1997: 1) The equation 1 = 0.99… does not make sense. 2) The concept ?does not exist. The first statement sparked a debate that raged over a decade. Both statements generated a series of publications that continues to grow to this day. Among the new findings are: 3) There does not exist nondenumerable set. 4) There does not exist non-measurable set. 5) Cantor’s diagonal method is flawed. 6) The real numbers are discrete and countable. 7) Formal logic does not apply to mathematics. The unfinished debate between logicism, intuitionism-constructivism and formalism is resolved. The resolution is the constructivist foundations of mathematics with a summary of all the rectification undertaken in 2015, 2016 and in this paper. The extensions of the constructivist real number system include the complex vector plane and transcendental functions. Two important results in the 2015 are noted: The solution and resolution of Hilbert’s 23 problems that includes the resolution of Fermat’s last theorem and proof Goldbach’s conjecture.展开更多
We provide some more explicit formulae to facilitate the computation of Ohtsuki’s rational invariants λ<sub>n</sub> of integral homology 3-spheres extracted from Reshetikhin-Turaev SU（2） quantum invari...We provide some more explicit formulae to facilitate the computation of Ohtsuki’s rational invariants λ<sub>n</sub> of integral homology 3-spheres extracted from Reshetikhin-Turaev SU（2） quantum invari- ants. Several interesting consequences will follow from our computation of λ<sub>2</sub>. One of them says that λ<sub>2</sub> is always an integer divisible by 3. It seems interesting to compare this result with the fact shown by Murakami that λ<sub>1</sub> is 6 times the Casson invariant. Other consequences include some general criteria for distinguishing homology 3-spheres obtained from surgery on knots by using the Jones polynomial.展开更多
文摘The paper reviews the most consequential defects and rectification of traditional mathematics and its foundations. While this work is only the tip of the iceberg, so to speak, it gives us a totally different picture of mathematics from what we have known for a long time. This journey started with two teasers posted in SciMath in 1997: 1) The equation 1 = 0.99… does not make sense. 2) The concept ?does not exist. The first statement sparked a debate that raged over a decade. Both statements generated a series of publications that continues to grow to this day. Among the new findings are: 3) There does not exist nondenumerable set. 4) There does not exist non-measurable set. 5) Cantor’s diagonal method is flawed. 6) The real numbers are discrete and countable. 7) Formal logic does not apply to mathematics. The unfinished debate between logicism, intuitionism-constructivism and formalism is resolved. The resolution is the constructivist foundations of mathematics with a summary of all the rectification undertaken in 2015, 2016 and in this paper. The extensions of the constructivist real number system include the complex vector plane and transcendental functions. Two important results in the 2015 are noted: The solution and resolution of Hilbert’s 23 problems that includes the resolution of Fermat’s last theorem and proof Goldbach’s conjecture.
基金The first author is supported in part by NSFthe second author is supported by an NSF Postdoctoral Fellowship.
文摘We provide some more explicit formulae to facilitate the computation of Ohtsuki’s rational invariants λ<sub>n</sub> of integral homology 3-spheres extracted from Reshetikhin-Turaev SU（2） quantum invari- ants. Several interesting consequences will follow from our computation of λ<sub>2</sub>. One of them says that λ<sub>2</sub> is always an integer divisible by 3. It seems interesting to compare this result with the fact shown by Murakami that λ<sub>1</sub> is 6 times the Casson invariant. Other consequences include some general criteria for distinguishing homology 3-spheres obtained from surgery on knots by using the Jones polynomial.
