Proof, in the form of step by step deduction, following the rules of logical reasoning, is the ultimate test of validity in mathematics. Some proofs, however, are so long or complex, or both, that they cannot be checked for errors by human experts. In response, a small but growing community of mathematicians, collaborating with computer scientists, have designed systems that allow proofs to be verified by machine. The success in certifying proofs of some prestigious theorems has led some mathematicians to propose a complete rethinking of the profession, requiring future proofs to be written in computer readable code. A few mathematicians have gone so far as to predict that artificial intelligence will replace humans in mathematical research, as in so many other activities.
One's position on the possible future mechanization of proof is a function of one's view of mathematics itself. Is it a means to an end that can be achieved as well, or better, by a competent machine as by a human being? If so, what is that end, and why are machines seen as more reliable than humans? Or is mathematics rather an end in itself, a human practice that is pursued for its intrinsic value? If so, what could that value be, and can it ever be shared with machines?
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This is a past event that happened on Saturday, December 1st, 2018 at 2:30pm.
Participants
Avner Ash
Professor of Mathematics, Boston College
Avner Ash is Professor of Mathematics at Boston College. His recent research is mostly in Number Theory, at an intersection of topology, group theory and Galois theory. His Ph.D. was awarded at Harvard under the direction of David Mumford. He has held faculty positions at Columbia and The Ohio State University. He has been a... read more! »
Garry Hagberg
James H. Ottaway Professor of Philosophy and Aesthetics, Bard College
Garry Hagberg is the James H. Ottaway Professor of Philosophy and Aesthetics at Bard College, and has also held a Chair in the School of Philosophy at the University of East Anglia. Author of numerous papers at the intersection of aesthetics and the philosophy of language, his books include Meaning and Interpretation: Wittgenstein, Henry James,... read more! »
Michael Harris
Professor of Mathematics, Columbia University
Michael Harris is Professor of Mathematics at Columbia University; before that he held positions at Brandeis University and Université Paris-Diderot. He obtained his Ph.D. in 1977 from Harvard University, under the direction of Barry Mazur. He has organized or co-organized more than 20 conferences, workshops, and special programs in his field of number theory. He... read more! »
Barry Mazur
Gerhard Gade University Professor, Harvard University
Barry Mazur is a mathematician at Harvard University who has often taught courses in History of Science and Philosophy. His books include: Imagining Numbers (particularly the squareroot of minus fifteen) (Farrar Straus and Giroux); Prime Numbers and the Riemann Hypothesis, written with William Stein (Cambridge University Press) and he has edited with Apostolos Doxiadis the... read more! »
Peter Pesic
Director of the Science Institute & Musician-in-Residence, St. John's College (Santa Fe, NM)
A writer, pianist, physicist, and educator, Peter Pesic is the director of the Science Institute and Musician-in-Residence at St. John’s College in Santa Fe, NM. His six books published by MIT Press concern the intersections of science, music, history, and ideas, including Abel’s Proof: A Search for the Meaning of Mathematical Unsolvability, Polyphonic Minds: Music... read more! »
Perhaps most nonartists think the beauty of math is in a math model such as an image of fractals, but i think most artists find them annoying. Why? You do not have an artist on the panel.
Mathematics emits a sens of Beauty, but mathematics is lacking of emit of sens of Sublime. Beauty is only affect of Pleasure. Sublime is mixt affect of pleasure mingle with Bitternes and Suffering,. Beauty suggest to us that the world, nevertheless of all disasters and ruins, will stay defended by Universal Harmony, which is horizon, alas ,only for physical, chemical and biological kingdom, which are saved because stil not yet granted by the horrable Gift of Knowledge, and freedom to exercize the power of Negation everywhere. But just that suprime gift of knowing is given simultaneously with Consciousness, which suddenly set upright man agaist the Opening Infinity and botomless cosmos , in which he recognizes as his own Psyche. Consciousness ,that at one go break the harmony and unveiled the veil of Proportions, showing abysm of Sorrow before unknown , which is comming to the man, who not only plays and enjoy of symmetry and Harmony, but also is urged on some divine-demonic passion for everlasting knowledge .Numbers are warriors of Beauty, while Concepts are warriors of Sublime. Numbers doesn’t interact with time and they remain confine into Pleasure as affect of Beauty . On the contrary, Sign Symbol-Concepts ever interact with time, and from this interaction they extract from time bitternes and suffering, Sens of Beauty is emanation of pure Signs as Numbers and Letters ,which Meaning coincides with its Signs ,and meaning of Mathematical signs is undistinguishable from its sign,. Because of lacking of their own meaning the Numbers mirror everything . The sens of Sublime is emits by Signs Symbol which are the Concepts.. The Difference between Numbers and Concepts as Sign- Symbols is that : Numbers are Signs which meanings coincides with its Signs, and the time of Numbers and the time of its meaning is the same: this is alowes time Present . Numbers cannot transcend the time Present , because they doesn’t interact with Time. Conversly, The Signs- Symbols-Concepts interact with time, such way, that Sign Concept remain in time Present, but Meaning of Cocept is transfered in time Future. Sign Symbol constantly distance and emits its meaning into time Future. Why ? Because, the General Rule, that executes Sign-Symbol-Concept ,cannot realizes itself complitely, and modus of its existence is Esse in Futuro. (Ch. S. Pierce) Mathematics is pure Ideas into mind of God. These ideas are Harmony ,proportions and symmetry. Until the pure syntax of ideas remains into the mind of god – there is only Esences and stil not Existence. Only by Performance of syntax of ideas universe can come into being, Who is those who performs the pure structures ? This is a Sensus, or Sensory Organs. The Performace of pure syntax of ideas not only transforme Esence into Being, it also endows the pure structures with previous lacking sens of Sublime. And what is Beauty and what is Sulime ? Beauty is affect of Symmetry, which serves as world articulator of Infinite into Finitude, of Negativity into Positivity, and of Non-repeatable into Repeatable ,Dissimilar into Simmilar. Individual into General and Indeterministic Freedom into deterministic necessity. Thus Symmery and Beauty console and atribute to the Infinite world finite image. and they succssed in that because of organizing the quantity, similar and general under the Principle of Game-Plaiyng , lending value to Pleasure and giving hope that the world is not sensless, . Conversly, the Sublime is indication that another much severe, Principle of Seriousness shifted us from sweet positivity to horrable negativity, from quit finitude to fever infinity , and from secure known to restlesness unknown, thus showiyng to us pefect different and danger world ,organizing the quality, dys-similar,non-repeatable and individual under the principle of Tragic sence ,that is keepeng the moral of seriousness by the prize of opennig Consciousness towards death.
It is quiet notable that the Unuverse is harmonic until Numbers cannot interact with time, and unnable to discover that Heaven is just a Hole , penetrated by morfless infinity . But only by interaction of mind with time the knowledge is born and possible.
Dysharmony of the universe is discovred by Concepts,. For mathematicians universe is woven by beauty, harmony and proportions. For philosophers Beauty is just a Spell to hide the ruthless image of the world. Yet, whitout Concepts the Numbers are only empty Game.Play. Only Concepts make Numbers Serious, but by the prize of meaningless of knowing.