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DAWKINS, SAGAN REVIEWS • SCIENCE & SECULARISM • MACROEVOLUTION THE MAG A ZINE FOR SCI ENCE AND REASON Vol ume 31, No. 2 • March/April 2007 • INTRODUCTORY PRICE U.S. $4.95 • Canada $5.95 Special Issue SCIENCE,SCIENCE, Daniel C. ANDAND DENNETT GOD,GOD, (NON)(NON) Carl BELIEFBELIEF SAGAN Frederick CREWS Michael RUSE Lee M. SILVER Alan Dean FOSTER Peter OLOFSSON Allan MAZUR Mark NEWBROOK 03> David C. HAAS Pub lished by the Commit tee for Skeptical Inquiry 0556698 80575 THE COM MIT TEE FOR SKEPTICAL INQUIRY AT THE CENTER FOR IN QUIRY–TRANSNATIONAL (AD JA CENT TO THE UNIVER SI TY AT BUF FA LO) AN IN TER NA TION AL OR GAN I ZA TION Paul Kurtz, Chair man; pro fes sor emer i tus of phi los o phy, Uni ver si ty at Buf fa lo Bar ry Karr, Ex ec u tive Di rect or Joe Nick ell, Sen ior Re search Fel low Massi mo Pol id oro, Re search Fel low Rich ard Wis e man, Re search Fel low Lee Nisbet, Spe cial Pro jects Di rect or FEL LOWS James E. Al cock,* psy chol o gist, York Univ., Tor- Con sult ants, New York, NY Mar burg, Ger ma ny on to Sus an Haack, Coop er Sen ior Schol ar in Arts Lor en Pan kratz, psy chol o gist, Or e gon Health Jer ry An drus, ma gi cian and in vent or, Al ba ny, Or e gon and Sci en ces, Professor of Phi los o phy and Sci en ces Univ. Mar cia An gell, M.D., former edi tor-in-chief, New Professor of Law, Univer si ty of Mi ami Robert L. Park, professor of physics, Univ. of Eng land Jour nal of Med i cine C. E. M. Hansel, psy chol o gist, Univ. of Wales Maryland Steph en Bar rett, M.D., psy chi a trist, au thor, David J. Helfand, professor of astronomy, John Pau los, math e ma ti cian, Tem ple Univ. con sum er ad vo cate, Al len town, Pa. Columbia Univ. Stev en Pink er, cog ni tive sci en tist, Harvard Willem Betz, professor of medicine, Univ. of Doug las Hof stad ter, pro fes sor of hu man un der- Mas si mo Pol id oro, sci ence writer, au thor, Brussels stand ing and cog ni tive sci ence, In di ana Univ. ex ec u tive di rect or CI CAP, It a ly Bar ry Bey er stein,* bi o psy chol o gist, Si mon Fra ser Ger ald Hol ton, Mal linc krodt Pro fes sor of Phys ics Mil ton Ro sen berg, psy chol o gist, Univ. of Chic a go Univ., Van cou ver, B.C., Ca na da and pro fes sor of his to ry of sci ence, Har vard Wal la ce Sam pson, M.D., clin i cal pro fes sor of Ir ving Bie der man, psy chol o gist, Univ. of South ern Univ. med i cine, Stan ford Univ., ed i tor, Sci en tif ic Cal i for nia Ray Hy man,* psy chol o gist, Univ. of Or e gon Re view of Al ter na tive Med i cine Sus an Black more, Vis it ing Lec tur er, Univ. of the Le on Jar off, sci en ces ed i tor emer i tus, Time Am ar deo Sar ma, manager NEC Europe Ltd., West of Eng land, Bris tol Ser gei Ka pit za, former ed i tor, Rus sian edi tion, ex ec u tive di rect or, GWUP, Ger ma ny. Hen ri Broch, phys i cist, Univ. of Nice, France Sci en tif ic Amer i can Ev ry Schatz man, former pres i dent, French Phys ics Jan Har old Brun vand, folk lor ist, pro fes sor Law rence M. Krauss, au thor and pro fes sor of phys ics As so ci a tion emer i tus of Eng lish, Univ. of Utah and as tron o my, Case West ern Re serve Uni ver si ty Eu ge nie Scott, phys i cal an thro pol o gist, ex ec u tive Mar io Bunge, phi los o pher, McGill Uni ver si ty Ed win C. Krupp, as tron o mer, di rect or, Grif fith di rect or, Na tion al Cen ter for Sci ence Ed u ca tion John R. Cole, an thro pol o gist, ed i tor, Na tion al Ob ser va to ry Rob ert Sheaf fer, sci ence writer Cen ter for Sci ence Ed u ca tion Paul Kurtz,* chair man, Cen ter for In quiry El ie A. Shne our, bi o chem ist, au thor, president and Fred er ick Crews, lit er ary and cul tur al crit ic, Law rence Kusche, sci ence writer research director, Bi os ys tems Re search In sti tute, pro fes sor emer i tus of Eng lish, Univ. of Le on Le der man, emer i tus di rect or, Fer mi lab; La Jol la, Ca lif. Cal i for nia, Berke ley No bel lau re ate in phys ics Dick Smith, film pro duc er, pub lish er, Ter rey Hills, Rich ard Dawk ins, zo ol o gist, Ox ford Univ. Scott Lil i en feld, psy chol o gist, Emory Univ. N.S.W., Aus tral ia Ge of frey Dean, tech ni cal ed i tor, Perth, Aus tral ia Lin Zix in, former ed i tor, Sci ence and Tech nol o gy Rob ert Stein er, ma gi cian, au thor, El Cer ri to, Ca lif. Dan i el C. Den nett, Uni ver si ty Pro fes sor and Aus tin Dai ly (Chi na) Vic tor J. Sten ger, emer i tus pro fes sor of phys ics B. Fletch er Pro fes sor of Phi los o phy, Di rect or of Je re Lipps, Mu se um of Pa le on tol o gy, Univ. of and as tron o my, Univ. of Ha waii; ad junct the Cen ter for Cog ni tive Stud ies at Tufts Uni v. Cal i for nia, Berke ley pro fes sor of phi los o phy, Univ. of Col o ra do Ann Druyan, writer and producer, and CEO, Eliz a beth Loft us, pro fes sor of psy chol o gy, Univ. of Jill Cor nell Tar ter, as tron o mer, SE TI In sti tute, Cosmos Studios, Ithaca, New York Cal i for nia, Ir vine Moun tain View, Ca lif. Cor nel is de Ja ger, pro fes sor of as tro phys ics, Univ. Paul Mac Cready, sci en tist/en gi neer, Car ol Tav ris, psy chol o gist and au thor, Los Ange les, of Utrecht, the Neth er lands Aer o Vi ron ment, Inc., Mon rov ia, Ca lif. Ca lif. Ken neth Fed er, pro fes sor of an thro pol o gy, John Mad dox, ed i tor emer i tus of Na ture Da vid Thom as, phys i cist and math e ma ti cian, Cen tral Con nec ti cut State Univ. Da vid Marks, psy chol o gist, City Uni ver si ty, Lon don. Per al ta, New Mex i co An to ny Flew, phi los o pher, Read ing Univ., U.K. Mar io Men dez-Acos ta, jour nal ist and Steph en Toul min, pro fes sor of phi los o phy, Univ. An drew Fra knoi, as tron o mer, Foot hill Col lege, sci ence writer, Mex i co City, Mex i co of South ern Cal i for nia Los Al tos Hills, Ca lif. Marv in Min sky, pro fes sor of me dia arts and Neil de Gras se Ty son, as tro phys i cist and di rect or, Kend rick Fra zi er, sci ence writer, ed i tor, SKEP TI CAL sci en ces, M.I.T. Hay den Plan e tar i um, New York City IN QUIR ER Da vid Mor ri son, space sci en tist, NA SA Ames Ma ri lyn vos Sa vant, Pa rade mag a zine Yv es Gal i fret, vice-pres i dent, Af fil i at ed Re search Cen ter con trib ut ing ed i tor Or gan i za tions: France Rich ard A. Mul ler, pro fes sor of phys ics, Univ. of Stev en Wein berg, pro fes sor of phys ics and Mar tin Gard ner, au thor, crit ic Ca lif., Berke ley as tron o my, Univ. of Tex as at Aus tin; Mur ray Gell-Mann, pro fes sor of phys ics, San ta Fe Joe Nick ell, sen ior re search fel low, CSI No bel lau re ate In sti tute; No bel lau re ate Lee Nis bet,* phi los o pher, Med aille Col lege E.O. Wil son, Uni ver si ty Pro fes sor Emer i tus, Thom as Gi lov ich, psy chol o gist, Cor nell Univ. Bill Nye, science ed u ca tor and tele vi sion host, Nye Labs Har vard Uni ver si ty Hen ry Gor don, ma gi cian, col um nist, Tor on to James E. Oberg, science writer Rich ard Wis e man, psy chol o gist, Uni ver si ty of Saul Green, Ph.D., bi o chem ist, pres i dent of ZOL Irm gard Oe pen, pro fes sor of med i cine (re tired), Hert ford shire • • • Vis it the CSI Web site at www.csi cop.org • • • The SKEP TI CAL IN QUIR ER (ISSN 0194-6730) is published bi month ly by the Commit tee for of the November/December 2006 is sue.
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    HYPERNUMBERS AND OTHER EXOTIC STUFF Photo by Mateusz Dach (1) MORE ON THE "ARITHMETICAL" SIDE Tropical Arithmetics Introduction to Tropical Geometry - Diane Maclagan and Bernd Sturmfels http://www.cs.technion.ac.il/~janos/COURSES/238900-13/Tropical/MaclaganSturmfels.pdf https://en.wikipedia.org/wiki/Min-plus_matrix_multiplication https://en.m.wikipedia.org/wiki/Tropical_geometry#Algebra_background https://en.wikipedia.org/wiki/Amoeba_%28mathematics%29 https://www.youtube.com/watch?v=1_ZfvQ3o1Ac (friendly introduction) https://en.wikipedia.org/wiki/Log_semiring https://en.wikipedia.org/wiki/LogSumExp Tight spans, Isbell completions and semi-tropical modules - Simon Willerton https://arxiv.org/pdf/1302.4370.pdf (one half of the tropical semiring) Hyperfields for Tropical Geometry I. Hyperfields and dequantization - Oleg Viro https://arxiv.org/pdf/1006.3034.pdf (see section "6. Tropical addition of complex numbers") Supertropical quadratic forms II: Tropical trigonometry and applications - Zur Izhakian, Manfred Knebusch and Louis Rowen - https://www.researchgate.net/publication/ 326630264_Supertropical_Quadratic_forms_II_Tropical_Trigonometry_and_Applications Tropical geometry to analyse demand - Elizabeth Baldwin and Paul Klemperer http://elizabeth-baldwin.me.uk/papers/baldwin_klemperer_2014_tropical.pdf International Trade Theory and Exotic Algebras - Yoshinori Shiozawa https://link.springer.com/article/10.1007/s40844-015-0012-3 Arborescent numbers: higher arithmetic operations and division trees - Henryk Trappmann http://eretrandre.org/rb/files/Trappmann2007_81.pdf
  • On the Principle Qf Excluded Middle U* I-Lro 6*

    On the Principle Qf Excluded Middle U* I-Lro 6*

    ON THE PRINCIPLE 011' EXCLUDED MIDDLE 415 trans~ prove that classical mathemati('J3 is ~..IV, g, with rules ofinfemnce) of qnanti­ latahle into intuitionistic mathematics. ficatwn theory and extend his iHustra_ For this purpose, with each formula €> of tim.l to cover also the rule P and the mathemat,ics there is associated a trans­ remaining axioms, and if we let .t>:O be lat~on <0* in a perfectly general manner JEO plus Axiom 6) t,hen we have also: ,~, (IIi, § 2). If (a), and (Ea) are the Theorem III. If U f--I\£' \15, then only symbols we use for forming new On the principle qf excluded middle u* I-lRO 6*. formulas from given formulas, the defini. In that case, all the derivatioIh9 in V 6~ ANDREI NIKOLAEVICH KOLMOGOROV tion amounts to: for atomic e* is its § 4, can be dispensed with because the; donble negation is, or n@5; (B)* is n( B*); would follow from Theorem III. (1925) (A -'>- B)* is n(A* -'>- B*); ((a)A(a))* is A very suggestive remark (beginning n(a)(A(a»*; «(Ea)A(a)* is n(Ea)(A(a»*. of IV, § 5, and last two paragraphs of The following T'C'suHs are proved ex~ IV, § 6) is that every axiom A of matbe­ To " I"rge extent, this paper antici­ foundation since it asserts something ""tly (III, § 3) : matics is of type .!(., that is, A. is (intni­ p"ted not only Heytiog's form"liz"tion of about the consequences of something im~ Lemma 1. 1-i8 nA" ~ ir.