\nAn examination of mathematical methods and the search for mathematical meaning.<\/strong><\/p>\n
This article curates a reading list (most sources available freely online1<\/a><\/sup>) organized into a set of encounters that lie outside the standard mathematics curriculum. They are intended to enrich the reader’s understanding of mathematics and its place in scientific inquiry, increase her\/his connection to the historical and philosophical questions behind the mathematics of the past and present, and gain greater satisfaction from further mathematical study. The reader should come away with a better understanding of the culture of mathematics: what mathematics is, mathematical method and meaning, and the relation of mathematics to the empirical world and to science. <\/p>\n We look at seven topics. These may be covered in any order, to suit your particular interests. There is no core body of technical material to master in this course; the important thing is a feel for how, why, and in what context the core ideas of mathematics evolved, getting to the essence of their motivation, and understanding the fruits of these efforts. The course such as the below should appeal to all those who have an itch to scratch<\/a> beneath the surface of mathematics, who find themselves asking “but why?”. It could be useful in all three tiers of education: secondary, post-secondary (undergraduate), and graduate, appropriately restructured.<\/p>\n <\/p>\n <\/a><\/p>\n <\/a><\/p>\n To be a mathematician meant to participate in a great common effort, to continue work begun thousands of years ago, to shed new light on old discoveries, and to seek new ways to prepare the developments of the future. — Richard Brauer (mathematician), “Emil Artin”<\/a>, Bulletin of the American Mathematical Society, Vol. 73, No.1, January 1967<\/p><\/blockquote>\n <\/a><\/p>\n One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than we originally put into them. — Heinrich Hertz (physicist)<\/p><\/blockquote>\n <\/a><\/p>\n “It’s one thing to be certain, but you can be certain and you can still be wrong.” <\/a><\/p>\n The interplay between generality and individuality, deduction and construction, logic and imagination—this is the profound essence of live mathematics. … In a far reaching development all of them will be involved … start from the “concrete” ground, then discard ballast by abstraction and rise to the lofty layers of thin air where navigation and observation are easy; after this flight comes the crucial test of landing and reaching specific goals in the newly surveyed low plains of individual “reality”. In brief, the flight into abstract generality must start from and return again to the concrete and specific. — Richard Courant (mathematician)<\/p><\/blockquote>\n <\/a><\/p>\n I have yet to see any problem, however complicated, which when you looked at it in the right way, did not become still more complicated. — Paul Anderson, New Scientist<\/p><\/blockquote>\n <\/a><\/p>\n It is true that Fourier has the opinion that the principal object of mathematics is the public utility and the explanation of natural phenomena; but a scientist like him ought to know that the unique object of science is the honor of the human spirit and on this basis a question of the theory of numbers is worth as much as a question about the planetary system. — C.J. Jacobi (mathematician)<\/p><\/blockquote>\n <\/a><\/p>\n The important thing in science is not so much to obtain new facts as to discover new ways of thinking about them. — Sir William Lawrence Bragg, Beyond Reductionism<\/p><\/blockquote>\n Share Your experiences in the Comments section below if you have enjoyed the materials or have suggestions, if you’re a student and have taken a course such as this, or if you have taught a similar course. >> Continue reading: <\/em>Meta-Questions and their Impact on Successful Mathematics, Science & Technology Education<\/a><\/p>\n An examination of mathematical methods and the search for mathematical meaning.<\/p>\n This article curates a reading list (most sources available freely online1) organized into a set of encounters that lie outside the standard mathematics curriculum. They are intended to enrich the reader’s understanding of mathematics and its place in scientific inquiry, increase her\/his [Read More…]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","footnotes":""},"categories":[7,4],"tags":[49,121,52,15,53,126,16,51,54,11,50],"coauthors":[112],"class_list":["post-1095","post","type-post","status-publish","format-standard","hentry","category-education","category-mathematics-phenomenology","tag-course","tag-education","tag-foundations-of-mathematics","tag-history-of-science","tag-logic","tag-mathematics","tag-phenomenology","tag-philosophy-of-science","tag-proof","tag-science","tag-syllabus","odd"],"views":14842,"_links":{"self":[{"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/posts\/1095","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/comments?post=1095"}],"version-history":[{"count":101,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/posts\/1095\/revisions"}],"predecessor-version":[{"id":8967,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/posts\/1095\/revisions\/8967"}],"wp:attachment":[{"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/media?parent=1095"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/categories?post=1095"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/tags?post=1095"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/coauthors?post=1095"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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\nSyllabus and Reading List<\/h3>\n
Topic 1. What is Mathematics? Its Nature and Characteristics.<\/h4>\n
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Topic 2. Reality, Truth, and the Nature of Mathematical Knowledge<\/h4>\n
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Topic 3. What is Proof? Proof and the Problem of Certainty<\/h4>\n
\n– John Kerry, 2004 Presidential Election against George W. Bush<\/a><\/p><\/blockquote>\n\n
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Topic 4. Some Readings in the History of Mathematics and the Evolution of Its Ideas<\/h4>\n
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\nAstronomy as a catalyst driving the increasing sophistication of arithmetic, algebra, geometry and calculational methods in Ancient Mathematics (Babylonia, Assyria, Egypt, Vedic India, Greece, and Alexandria), Muslim mathematics (Arabia, Persia, Central Asia), and in Renaissance Europe. [Read Ch.1–5 online (Ancient Mathematics)]<\/a> (Google Books)<\/p>\nTopic 5. The Search for Foundations in Mathematics<\/h4>\n
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Topic 6. Mathematics and Science<\/h4>\n
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\n[Preview Article — first page only]<\/a> (from JSTOR)\n<\/ol>\nTopic 7. Thoughts on Mathematical Practice and Mathematical Style<\/h4>\n
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\n[Download PDF]<\/a> (Originating server is Author’s collected works page<\/a>) [Read online]<\/a><\/p>\n8. Other Readings<\/h4>\n
\nThe following 3 articles of Brillouin address the problem of determinism and error. Is nature continuous of discrete? Are the real numbers useful? Which should dominate: empirics and description vs. axiomatics and deduction?<\/p>\n
\nShare your thoughts<\/h4>\n
\n<\/a>
\nIf collaborating on future work in this direction sounds interesting, drop a note in the comments section.<\/a> <\/p>\n
\nRelated Articles:<\/h4>\n
\nFootnotes:<\/h3>\n
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