Mathematical Search Some thoughts from a Wolfram|Alpha perspective Michael Trott Wolfram|Alpha Slide 1 of 16 A simple/typical websearch in 2020? Slide 2 of 16 Who performs mathematical searches? Not only mathematicians, more and more engineers, biologists, …. Slide 3 of 16 The path/cycle of mathematical search:
An engineer, biologist, economist, physicist, … needs to know mathematical results about a certain problem
Step 1: find relevant literature citations (Google Scholar, Microsoft Academic Search, MathSciNet, Zentralblatt) fl Step 2: obtain copy of the book or paper (printed or electronic) fl Step 3: read, understand, apply/implement the actual result (identify typos, misprints, read references fl Step 1) The ideal mathematical search shortens each of these 3 steps.
Assuming doing the work; no use of newsgroups, http://stackexchange.com, …. Ideal case: Mathematical search gives immediate theSlide answer,4 of 16 not just a pointer to a paper that might contain the answer; either by computation of by lookup Current status of computability:
• fully computable (commutative algebra, quantifier elimination, univariate integration, …)
• partially computable (nonlinear differential equations, proofs of theorem from above, …)
• barely or not computable (Lebesgue measures, C* algebras, …), but can be semantically annotated How are the 100 million pages of mathematics distributed?
J. Borwein: Recalibrate Watson to solve math problems Cost : $500 million Timeframe : Five years Last years: Watson, Wolfram|Alpha Slide 5 of 16 Slide 6 of 16 Slide 7 of 16 Ask your smartphone (right now only SIRI) for the gravitational potential of a cube Slide 8 of 16 And in the not too distant future play with orbits around a cube on your cellphone: find closed orbits of a particle in the gravitational potential of a cube
initial position
x0 1.3
y 0 0
z0 0
initial velocity
v x,0 0
v y ,0 -0.29
v z,0 0
charge
plot range
solution time 39.8 Slide 9 of 16 Returned results are complete semantic representations of the mathematical fact
And, within Mathematica, they are also fully computable.
WolframAlpha@"product representation snHu,mL", 88"ProductRepresentations:MathematicalFunctionIdentityData", 1<, "Input" HoldCompleteBJacobiSN@u, mD ã p u 1 - 2 CosB F EllipticNomeQ@mD2 k + EllipticNomeQ@mD4 k 1 ¶ EllipticK@mD p u 2 EllipticNomeQ@mD1ê4 ‰ SinB FF 1ê4 p u m k=1 1 - 2 CosB F EllipticNomeQ@mD-1+2 k + EllipticNomeQ@mD-2+4 k 2 EllipticK@mD EllipticK@mD Slide 10 of 16 How much math? ~ 100 million pages of mathematics ~ 2 …3 million theorems ~ 80,000 mathematicians ~ 15 doubling time mathematical knowledge Claim (hope): Over the time scale of 10 years, it should be possible to implement a large fraction of the current mathematical knowledge in a fully semantic way. (not fully computational and not all proofs) For comparison: Wolfram|Alpha knows now a few hundred thousand mathematical facts. Good fraction from literature, but also large amounts of substantial generalizations of known results. Based on six year work of a pretty small team of people. Conjectured side effect: Many results would be generalized from a modern perspective. Semantic forms (not fully computable) of mathematicalSlide knowledge11 of will16 allow for: 1) Semantic search The semantic representation does immediately allow for semantic searching. (Add a shallow NLP layer.) 2) Make mathematical knowledge alive 3) New data representation, data aggregation, and data comparisons 4) Deriving new mathematical knowledge programmatically Slide 12 of 16 2) Make mathematical knowledge alive first attempts Slide 13 of 16 Slide 14 of 16 Slide 15 of 16 3) Data representation, data aggregation, data comparisons example of probability distributions; contains 30,000+ formulas continuous in H-¶, ¶L H97L @a, ¶L H182L @a, bD H125L general distribution class: discrete in H-¶, ¶L H1L @ a, ¶L H74L @a, bD H27L general alphabetically A B C D E F G H I J K L M N O P Q R S T U V W X Y Z distribution NormalDistribution@m, sD default random all properties class PDFs NormalDistribution@m, sD 1 PDF ê; x œ flip-through all Hx-mL2 2 p s ‰ 2 s2 DistributionDomain Interval@8-¶, ¶ 1 -x+m CDF erfc ê; x œ 2 compact use selected 2 s Characteristics to show view properties Mean m 2 defining characteristics Variance s StandardDeviation s PDF z2 s2 H zL m- support CharacteristicFunction ‰ 2  m n n parameter conditions HH-ÂL s L Hn 2 s ê; n œ Ï n ¥ 0 main characteristics 2nê2 Moment CDF First raw moments mean in explicit form variance standard deviation characteristic function smaller plots larger plots moments plots and samples default sized plots generalized moments PDF 0.4 generalized means 0.3 generating functions 0.2 quantiles 0.1 differential equation 0.0 -3 -2 -1 0 1 2 3 entropies more characteristics typical names CDF Sketch 1.0 0.8 0.6 show single properties all occurring properties all substituted 0.4 0.2 0.0 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 m 1. s 1. plot also PDF derivatives plot also PDF-based Lorenz curve £ Hm-xL wHxL PDFODE w HxL ê; s2 wHxL PDF@NormalDistribution@m, sD, xD functions do not show definitions show special function definitions display traditionally typeset in Mathematica StandardForm Relations between distribution properties Slide 16 of 16 4) Deriving new mathematical knowledge programmatically today with little programming Examples: • J1HnL ‚H-1Ln n=1 n • HnH1L 4 ‚ „3 - 1 n n=1 2 n! 1 2 lnH∂x L lnHxL = Ip - 6 logHxL HlogHxL + ˝LM 6