DOCSLIB.ORG

What We Don't Know About 4-Dimensions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
What We Don't Know About 4-Dimensions What we know and what we don’t know about 4-dimensions My plan for this lecture is to tell you a little about what we do know and something of what we don’t know about 4-dimensional spaces. 1. Manifolds, dimensions and coordinates I must digress first to tell you what I mean by a 4-dimensional space. To this end, let me start with the notion of a manifold. This is a space that at small scales looks flat and featureless. Imagine the surface of a ball. If you look at it at very small scales, you won’t see that it is curved. To an ant, it seems to all intents and purposes like the surface of a table. Figure 1 below depicts a surface in space, an example of a manifold. A region that looks flat has been magnified. Figure 1. When I say featureless, I mean featureless in the sense that the straight line or a flat plane or any Euclidean space is featureless. There is nothing to distinguish one point from another or one direction from another. Figure 2 depicts the line and the plane, the two smallest Euclidean spaces. Figure 2. A technical definition has a manifold being a space whose points have neighborhoods that can be identified with a Euclidean space. The number of coordinates of the relevant Euclidean space is the dimension of the manifold. Said differently, the dimension a given manifold is the number of local 1 coordinates that you need to specify your position in a neighborhood of any one of its points. For example, if you were to live on a circle, then you would need to give but one coordinate to specify your position. You would need only to tell your friends the angle of your abode from some reference point. See Figure 3; the house is 138º counter clockwise from the reference point. Figure 3. The circle is an example of a 1-dimensional manifold. It is denoted by S1; the S standing for sphere (not sircle) and the 1 being the dimension. The surface of the earth is an example of a 2-dimensional manifold. You need to give locally 2 parameters--latitude and longitude--to completely specify the position of a point. For example, the position of this Math Forum building is 18º 19´ 42´´ north lattitude and 109º 24´ 37´´ east longitude. If I wanted to give someone directions as to where to find this lecture room, I would also have to give them a third coordinate, this being the height over the surface of the earth. According to Google earth, this lecture hall is 153 meters about sea level. So, we need not two, but three coordinates to describe our world because we can move in three directions, not two. Now suppose that someone wanted to actually see my lecture. If this is the case, then they would have to also know a fourth coordinate, this being the time of the lecture, December 19, 2013 at 6:30 GMT which is 14:30 local time. This illustrates the fact that our observable universe is 4-dimensional, one needs 3 spatial coordinates and a fourth temporal coordinate to completely specify the position of any given point. In general, a 4-dimensional manifold is characterized by the property that each of its points has a neighborhood that looks like 4-dimensional Euclidean space. This means that each point of this neighborhood is completely specified by four numbers and that any given set of four numbers specifies a point. These 4 numbers are the coordinates of the points. Note however that the coordinates that are used for a neighborhood of a chosen point can differ from those that are used near some other point. If there were just one 2 coordinate system that worked everywhere, then the manifold would be either the whole of 4-dimensional Euclidean space, or a part of 4-dimensional Euclidean space. This fact that one coordinate set may not work over the whole manifold signals the existence of structure at large scales. This is true in any dimension. The case of the circle is perhaps the simplest example whereby the Euclidean coordinate given by the counter-clockwise angle is only locally single valued. Go around the circle once and the angle increases by 360º. This is a manifestation of the fact that the circle looks like the 1- dimensional Euclidean space only at small scales. It has a non-Euclidean structure at large scales. By way of a second example, the coordinates of latitude and longitude for the surface of the earth don’t work as local coordinates near the north and south poles. You will need to use a different set of local coordinates near these two points. This is a manifestation of the fact that the surface of the earth has non-Euclidean structure at large scales. 2. The large scale shapes of the earth and the universe The earth is a solid sphere and its surface is just like the surface of a ball. Modulo relatively small bumps (i.e hills and mountains), the surface of the earth looks to our eyes like a flat surface, the 2-dimensional Euclidean space. Indeed, many people in antiquity thought the earth was flat. Even so, the ancient Greeks already knew that the earth was round. There is an annotated version of a lecture course given by a Greek savant by the name of Gemino. The annotated version is by James Evans and J. Lennart Berggren and it is called Gemino’s “Introduction to the Phenomena”A Translation and Study of a Hellenistic Survey of Astronomy (Princeton University Press 2006). Gemino was the equivalent of a college professor and his lectures were meant to teach his students about the ‘phenomena’, the phenomena being astronomy and geometry. In Hellenistic times, astronomy and geometry were taught together as aspects the same thing. Gemino in his lectures gives three pieces of evidence for the earth being round. The first is the following fact: On the summer solstice, if you put a vertical stake in the ground at Alexandria in Egypt at noon local time and do the same thing due north in central Anatolia (modern Turkey) and then somewhat farther north, you will find that the shadows pointing north have different lengths. The distances between the stakes and the shadow lengths can be used to compute the curvature of the earth. See Figure 4. 3 Figure 4. The second piece of evidence can be seen when there is an eclipse of the moon. The shadow of the earth on the moon is curved. The Greeks knew that an eclipse is due to the earth getting between the sun and the moon. The third piece of evidence concerns elephants. If you go west from Alexandria towards the Atlantic ocean, you will first get to lands where there are elephants. The famous army of Hannibal (from Carthage in what is now Lybia) crossed the Alps into Italy coming from Spain with African elephants. Meanwhile, if you go east from Alexandria, across Anatolia towards India, you will again get to lands with elephants. For example, Alexander from Macedon fought against armies with Indian elephants during his campains in what is now Pakistan. If there are elephants to the west and elephants to the east and none in between, then the east and west must be connected around the back. Gemino makes a rather amazing prediction in his lectures from the fact that the earth is round. There was much commerce between the far flung parts of Europe and Asia during Hellenistic times, and as a consequence the Greeks knew that the climate got ever colder as you went to the north. For example, they knew that there are lands in the far north where there is perpetual ice and snow. They also knew that the climate got ever hotter to the south. In particular, the climate in the Sahara Desert was such that it was a barrier to travel and thus to knowledge. Gemino makes the bold statement that if you were to cross the Sahara, you would again get to lands with temperate climate, and then eventually to lands with perpetual snow and cold as in the far north. He then conjectures that the hypothetical southern temperate lands could be populated with people, people whom we (the Greeks) know nothing about. The surface of the earth is an example of a manifold that is curved in the large so as to have the shape of the boundary of a ball. This is to say that it is a 2-dimensional sphere. As I said, our universe is a 4-dimensional manifold, and this begs the question as 4 to what is its shape in the large? This is a question for astronomers to sort out. Even so, it is a mathematical question as to the list of all possible 4-dimensional spaces. 3. Lists of manifolds When talking about the listing manifolds, it is important to distinguish between the intrinsic structure and the shape that appears when a manifold is viewed as a subset of some larger, ambient space. By way of an example, the circle can be depicted in the plane in its standard round form and also with various bumps as in the left and right hand drawing of Figure 5. From an intrinsic point of view, both drawings are depictions of the same circle since in both cases, any given point is specified by an angular coordinate. Figure 5 By the same token, the standard unknotted circle in 3-space and a knotted circle in the 3- dimensional Euclidean space are both standard circles.
Load more

Recommended publications
  • The Materiality & Ontology of Digital Subjectivity
    THE MATERIALITY & ONTOLOGY OF DIGITAL SUBJECTIVITY: GRIGORI “GRISHA” PERELMAN AS A CASE STUDY IN DIGITAL SUBJECTIVITY A Thesis Submitted to the Committee on Graduate Studies in Partial Fulfillment of the Requirements for the Degree of Master of Arts in the Faculty of Arts and Science TRENT UNIVERSITY Peterborough, Ontario, Canada Copyright Gary Larsen 2015 Theory, Culture, and Politics M.A. Graduate Program September 2015 Abstract THE MATERIALITY & ONTOLOGY OF DIGITAL SUBJECTIVITY: GRIGORI “GRISHA” PERELMAN AS A CASE STUDY IN DIGITAL SUBJECTIVITY Gary Larsen New conditions of materiality are emerging from fundamental changes in our ontological order. Digital subjectivity represents an emergent mode of subjectivity that is the effect of a more profound ontological drift that has taken place, and this bears significant repercussions for the practice and understanding of the political. This thesis pivots around mathematician Grigori ‘Grisha’ Perelman, most famous for his refusal to accept numerous prestigious prizes resulting from his proof of the Poincaré conjecture. The thesis shows the Perelman affair to be a fascinating instance of the rise of digital subjectivity as it strives to actualize a new hegemonic order. By tracing first the production of aesthetic works that represent Grigori Perelman in legacy media, the thesis demonstrates that there is a cultural imperative to represent Perelman as an abject figure. Additionally, his peculiar abjection is seen to arise from a challenge to the order of materiality defended by those with a vested interest in maintaining the stability of a hegemony identified with the normative regulatory power of the heteronormative matrix sustaining social relations in late capitalism.
    [Show full text]
  • Group Knowledge and Mathematical Collaboration: a Philosophical Examination of the Classification of Finite Simple Groups
    Group Knowledge and Mathematical Collaboration: A Philosophical Examination of the Classification of Finite Simple Groups Joshua Habgood-Coote and Fenner Stanley Tanswell Forthcoming in Episteme, please refer to final version. Abstract In this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but have received very little attention. In this paper, we will consider the philosophical tensions that Steingart uncovers, and use them to argue that the best account of the epistemic status of the Classification Theorem will be essentially and ineliminably social. This forms part of the broader argument that in order to understand mathematical proofs, we must appreciate their social aspects. 1 Introduction Popular conceptions of mathematics are gripped by the myth of the ‘lone genius’. This myth is inspired by famous figures like Andrew Wiles, Grigori Perelman, and Srinivasa Ramanujan whose individual efforts are taken to be both representative and exemplary. In this paper, we push back against this individualistic view of how mathematics is and should be practiced, examining the significance of social and group level features of mathematical practices. In particular, we will discuss the epistemology of mathematics, and argue that in order to understand mathematics and its epistemology, we need to pay attention to collaboration, group knowledge, and other social factors.
    [Show full text]
  • Fundamental Theorems in Mathematics
    SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635].
    [Show full text]
  • Alexander Grothendieck Heng Li
    Alexander Grothendieck Heng Li Alexander Grothendieck's life Alexander Grothendieck was a German born French mathematician. Grothendieck was born on March 28, 1928 in Berlin. His parents were Johanna Grothendieck and Alexander Schapiro. When he was five years old, Hitler became the Chancellor of the German Reich, and called to boycott all Jewish businesses, and ordered civil servants who were not of the Aryan race to retire. Grothendieck's father was a Russian Jew and at that time, he used the name Tanaroff to hide his Jewish Identity. Even with a Russian name it is still too dangerous for Jewish people to stay in Berlin. In May 1933, his father, Alexander Schapiro, left to Paris because the rise of Nazism. In December of the same year, his mother left Grothendieck with a foster family Heydorns in Hamburg, and joined Schapiro in Paris. While Alexander Grothendieck was with his foster family, Grothendieck's parents went to Spain and partici- pated in the Spanish Civil War. After the Spanish Civil War, Johanna(Hanka) Grothendieck and Alexander Schapiro returned to France. In Hamburg, Alexander Grothendieck attended to elementary schools and stud- ied at the Gymnasium (a secondary school). The Heydorns family are a part of the resistance against Hitler; they consider that it was too dangerous for young Grothendieck to stay in Germany, so in 1939 the Heydorns sent him to France to join his parents. However, because of the outbreak of World War II all German in France were required by law to be sent to special internment camps. Grothendieck and his parents were arrested and sent to the camp, but fortunately young Alexander Grothendieck was allowed to continue his education at a village school that was a few miles away from the camp.
    [Show full text]
  • Nominations for President
    Nominations for President combinatorics which was then starting to become fashion- Nomination of able; and some number theory, which has never gone out George Andrews of fashion although the parts of it George dealt with were not fashionable then. George spent a year on leave at MIT. Richard Askey At Rota’s suggestion, he edited the “Collected Papers” of P. A. MacMahon. He also wrote a book, “The Theory of Who is George Andrews? According to Freeman Dyson, Partitions”, for the Encyclopedia of Mathematics and Its George Andrews is the chief gardener in Ramanujan’s Gar- Applications series which Rota edited. den. This is true, but is only part of who George Andrews George came to the University of Wisconsin-Madison for is. He is a number theorist with an honorary doctorate in a year, 1976-77. In the spring he went to Europe for two physics. He is a long-time user of computers in his own meetings in France. Since he was not teaching and airfare research, who has written about the harm technology can abroad was less expensive if you spent 21 to 45 days, he do in mathematics education. He is primarily a problem also went to Cambridge to look for old work in the Wren solver, yet one paper with Rodney Baxter and Peter For- Library of Trinity College. This changed his life. rester has had a major impact in mathematical physics What George found was a bit over 100 pages of math- with currently 490 citations in the Web of Science. ematical claims in the distinctive handwriting of Srinivasa George Andrews received his Ph.D.
    [Show full text]
  • Applications at the International Congress by Marty Golubitsky
    From SIAM News, Volume 39, Number 10, December 2006 Applications at the International Congress By Marty Golubitsky Grigori Perelman’s decision to decline the Fields Medal, coupled with the speculations surrounding this decision, propelled the 2006 Fields Medals to international prominence. Stories about the medals and the award ceremony at the International Congress of Mathematicians in Madrid this summer appeared in many influential news outlets (The New York Times, BBC, ABC, . .) and even in popular magazines (The New Yorker). In Madrid, the topologist John Morgan gave an excellent account of the history of the Poincaré conjecture and the ideas of Richard Hamilton and Perelman that led to the proof that the three-dimensional conjecture is correct. As Morgan pointed out, proofs of the Poincaré con- jecture and its direct generalizations have led to four Fields Medals: to Stephen Smale (1966), William Thurston (1982), Michael Freedman (1986), and now Grigori Perelman. The 2006 ICM was held in the Palacio Municipal de Congressos, a modern convention center on the outskirts of Madrid, which easily accommodated the 3600 or so participants. The interior of the convention center has a number of intriguing views—my favorite, shown below, is from the top of the three-floor-long descending escalator. Alfio Quarteroni’s plenary lecture on cardiovascular mathematics was among the many ses- The opening ceremony included a welcome sions of interest to applied mathematicians. from Juan Carlos, King of Spain, as well as the official announcement of the prize recipients—not only the four Fields Medals but also the Nevanlinna Prize and the (newly established) Gauss Prize.
    [Show full text]
  • Prize Is Awarded Every Three Years at the Joint Mathematics Meetings
    AMERICAN MATHEMATICAL SOCIETY LEVI L. CONANT PRIZE This prize was established in 2000 in honor of Levi L. Conant to recognize the best expository paper published in either the Notices of the AMS or the Bulletin of the AMS in the preceding fi ve years. Levi L. Conant (1857–1916) was a math- ematician who taught at Dakota School of Mines for three years and at Worcester Polytechnic Institute for twenty-fi ve years. His will included a bequest to the AMS effective upon his wife’s death, which occurred sixty years after his own demise. Citation Persi Diaconis The Levi L. Conant Prize for 2012 is awarded to Persi Diaconis for his article, “The Markov chain Monte Carlo revolution” (Bulletin Amer. Math. Soc. 46 (2009), no. 2, 179–205). This wonderful article is a lively and engaging overview of modern methods in probability and statistics, and their applications. It opens with a fascinating real- life example: a prison psychologist turns up at Stanford University with encoded messages written by prisoners, and Marc Coram uses the Metropolis algorithm to decrypt them. From there, the article gets even more compelling! After a highly accessible description of Markov chains from fi rst principles, Diaconis colorfully illustrates many of the applications and venues of these ideas. Along the way, he points to some very interesting mathematics and some fascinating open questions, especially about the running time in concrete situ- ations of the Metropolis algorithm, which is a specifi c Monte Carlo method for constructing Markov chains. The article also highlights the use of spectral methods to deduce estimates for the length of the chain needed to achieve mixing.
    [Show full text]
  • AROUND PERELMAN's PROOF of the POINCARÉ CONJECTURE S. Finashin After 3 Years of Thorough Inspection, Perelman's Proof of Th
    AROUND PERELMAN'S PROOF OF THE POINCARE¶ CONJECTURE S. Finashin Abstract. Certain life principles of Perelman may look unusual, as it often happens with outstanding people. But his rejection of the Fields medal seems natural in the context of various activities followed his breakthrough in mathematics. Cet animal est tr`esm¶echant, quand on l'attaque, il se d¶efend. Folklore After 3 years of thorough inspection, Perelman's proof of the Poincar¶eConjecture was ¯nally recognized by the consensus of experts. Three groups of researchers not only veri¯ed its details, but also published their own expositions, clarifying the proofs in several hundreds of pages, on the level \available to graduate students". The solution of the Poincar¶eConjecture was the central event discussed in the International Congress of Mathematicians in Madrid, in August 2006 (ICM-2006), which gave to it a flavor of a historical congress. Additional attention to the personality of Perelman was attracted by his rejec- tion of the Fields medal. Some other of his decisions are also non-conventional and tradition-breaking, for example Perelman's refusal to submit his works to journals. Many people ¯nd these decisions strange, but I ¯nd them logical and would rather criticize certain bad traditions in the modern organization of science. One such tradition is a kind of tolerance to unfair credit for mathematical results. Another big problem is the journals, which are often transforming into certain clubs closed for aliens, or into pro¯table businesses, enjoying the free work of many mathemati- cians. Sometimes the role of the \organizers of science" in mathematics also raises questions.
    [Show full text]
  • The New York Times Book of Mathematics Reviewed by Ian Stewart
    Book Review The New York Times Book of Mathematics Reviewed by Ian Stewart an order that helps to The New York Times Book of Mathematics tell a story—which is, Edited by Gina Kolata after all, what journal- Sterling, 2013 ism is about. US$24.95, 496 pages This choice makes ISBN-13: 978-1402793226 the book far more ac- cessible to the general Mathematicians often complain that their subject reader, though I’m is neglected in the mass media, although when it tempted to photocopy does get reported there always seem to be a few the contents list and of us who find it impossible to resist the urge to shuffle the articles complain bitterly about inaccuracies and “hype”, back into the order which usually seems to mean the promotion of the of publication. Read- area concerned instead of their own. As this sump- ing them in that order tuous volume demonstrates, neither complaint would give a dramatic can sensibly be directed at the New York Times perception of how our subject has changed over [NYT ]. For more than a century this high-quality the last 120 years; however, this comes over any- newspaper has done sterling work in the service way. Paul Hoffman’s foreword and Gina Kolata’s in- of our profession, the public, and the cause of sci- troduction provide concise and helpful overviews ence journalism. of such questions, and the first group of topics This collection contains more than a hundred addresses general issues of what mathematics is NYT articles on mathematics and its applications and what it’s for.
    [Show full text]
  • Of the European Mathematical Society
    NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Feature History Interview ERCOM Google and Linear Algebra The ICM through History J. Morgan, A. Connes, R. Jeltsch Rényi Institute Budapest p. 10 p. 16 p. 22 p. 36 March 2007 Issue 63 ISSN 1027-488X S E European M M Mathematical E S Society TopologyJournal of It is with great pleasure that we announce the launch of the Journal of Topology, a new journal that will publish its first issue in January 2008. The Board of the Journal will be: Michael Atiyah • Martin Bridson Ralph Cohen • Simon Donaldson Nigel Hitchin • Frances Kirwan Marc Lackenby • Jean Lannes Wolfgang Lück • John Roe • Graeme Segal Ulrike Tillmann (Managing Editor) Aims and Scope The Journal of Topology will publish papers of high quality and significance in topology, geometry and adjacent areas of mathematics. Interesting, important and often unexpected links connect topology and geometry with many other parts of mathematics, and the editors welcome submissions on exciting new advances concerning such links, as well as those in the core subject areas of the journal. The Journal of Topology will appear in quarterly issues with articles posted individually online. Guidelines for Paper Submission Submissions are currently being welcomed. For information on submitting a paper, please visit www.lms.ac.uk/publications/jtop.html If you have further enquiries about the journal, please contact Susan Hezlet, Publisher LMS, at [email protected], or the Managing Editor at [email protected]. The price of the Journal for Volume 1 (four issues) including electronic access will be £300 or $570 in 2008, although we plan further discounts for libraries who take our other journals.
    [Show full text]
  • Perelman, Poincare, and the Ricci Flow
    PERELMAN, POINCARE,´ AND THE RICCI FLOW SCOTT DUKE KOMINERS ABSTRACT. In this expository article, we introduce the topological ideas and context central to the Poincare´ Conjecture. Our account is intended for a general audience, providing intuitive defini- tions and spatial intuition whenever possible. We define surfaces and their natural generalizations, manifolds. We then discuss the classification of surfaces as it relates to the Poincare´ and Thurston Geometrization conjectures. Finally, we survey Perelman’s results on Ricci flows with surgery. In July 2003, Russian mathematician Grigori Perelman announced the third and final paper of a series which solved the Poincare´ Conjecture, a fundamental mathematical problem in three- dimensional topology. Perelman was awarded the Fields Medal—mathematics’s equivalent of the Nobel Prize—for his discoveries [ICM], which were also declared the “breakthrough of the year” by Science [Ma]. The Poincare´ Conjecture is an easily-stated, elegant question. Nonetheless, the problem stood open for nearly a century, leading mathematicians to develop more and more advanced techniques of attack. Perelman’s papers [P1, P2, P3] build on the recently developed theory of the Ricci flow, extending methods pioneered by Hamilton [H1, H2, H3]. Perelman’s arguments are highly innovative and technical, so much so that the mathematical community spent years evaluating and fleshing out the work. In this paper, we will introduce and set in context the topological ideas central to the Poincare´ Conjecture. We focus on surfaces and their generalization, manifolds, providing the necessary intuition at each step. We will then discuss the rich history of the classification of manifolds, a topic which has driven the evolution of topology as a field of mathematics.
    [Show full text]
  • Downloaded From: Publisher: Springer Verlag DOI
    Pais, Alexandre (2018) Book Review: The limits of pluralism. Bharath Sri- raman (Ed.) (2017) Humanizing mathematics and its philosophy: Essays celebrating the 90th birthday of Reuben Hersh. Educational Studies in Math- ematics, 99 (2). pp. 235-240. ISSN 0013-1954 Downloaded from: https://e-space.mmu.ac.uk/621815/ Publisher: Springer Verlag DOI: https://doi.org/10.1007/978-3-319-61231-7 Please cite the published version https://e-space.mmu.ac.uk Book Review: The limits of pluralism. Bharath Sriraman (Ed.) (2017) Humanizing mathematics and its philosophy: Essays celebrating the 90th birthday of Reuben Hersh Basel: Birkhäuser. 363 pages. doi:10.1007/978-3-319-61231-7 Hardcover: ISBN: 978-3-319-61230-0. eBook: ISBN: 978-3-319- 61231-7 Alexandre Pais I remember being a masters student at the University of Lisbon in the early 2000s when, in one of the classes, the teacher, João Pedro da Ponte, introduced, not without reluctance, Philip J. Davis and Reuben Hersh’s The Mathematical Experience as the textbook for the course. His reluctance was due to the idea that the book might be a bit too Bnon-academic^ and Bpopulistic^ for a postgraduate course. For me, having just come from a highly formalistic mathematical training, where mathematics was presented as a finished science, and all the work consisted of proving theorems with no attempt to problematise the implications, the history, or the meaning of such results, this book brought a breath of fresh air into my mathematics education, and sparked a different way of engaging with this science. I realised that one can talk about mathematics outside the straitjacket of deductive proof and formalism.
    [Show full text]