input output, result x := e si; S a if B then Sl else S a while B do S data structure data type value of a data type a : A integer real Boolean (cis ..., cn) array [I] of T record sl:Tl; so :TB end record case s : (cl, c,) of set of T cl:(sl:Tl); Ca:(sa:Tz) end Mathematics function argument value x = e composition of functions definition by cases definition by recursion element, object set, type element of a set, object of a type a E A Z R {0,1) {ci, -, 4 TI, I+T TlX T o Ti+ T a curious situation is, I think, that the mathematical notions have gradually received an interpretation, the interpretation which we refer to as classical, which makes them unusable for programming. Fortunately, I do not need Constructive Mathematics and Computer Programming, 1985
input output, result x := e si; S a if B then Sl else S a while B do S data structure data type value of a data type a : A integer real Boolean (cis ..., cn) array [I] of T record sl:Tl; so :TB end record case s : (cl, c,) of set of T cl:(sl:Tl); Ca:(sa:Tz) end Mathematics function argument value x = e composition of functions definition by cases definition by recursion element, object set, type element of a set, object of a type a E A Z R {0,1) {ci, -, 4 TI, I+T TlX T o Ti+ T a curious situation is, I think, that the mathematical notions have gradually received an interpretation, the interpretation which we refer to as classical, which makes them unusable for programming. Fortunately, I do not need Constructive Mathematics and Computer Programming, 1985 “[programming] used to be looked down upon as the rather messy job of instructing this or that physically existing machine”
input output, result x := e si; S a if B then Sl else S a while B do S data structure data type value of a data type a : A integer real Boolean (cis ..., cn) array [I] of T record sl:Tl; so :TB end record case s : (cl, c,) of set of T cl:(sl:Tl); Ca:(sa:Tz) end Mathematics function argument value x = e composition of functions definition by cases definition by recursion element, object set, type element of a set, object of a type a E A Z R {0,1) {ci, -, 4 TI, I+T TlX T o Ti+ T a curious situation is, I think, that the mathematical notions have gradually received an interpretation, the interpretation which we refer to as classical, which makes them unusable for programming. Fortunately, I do not need Constructive Mathematics and Computer Programming, 1985 “[programming] used to be looked down upon as the rather messy job of instructing this or that physically existing machine” “It has made programming an activity akin in rigour and beauty to that of proving mathematical theorems”
programming ✦ Inter-disciplinary cross-pollination and unification Typed Functional Languages Are Nice “Nevertheless, numerous programmers continue to choose untyped scripting languages”
itself rather than through other facts” “We normally cannot give a full account why the fact should be what it is, but must accept it without explanation” The Blackwell Dictionary of Western Philosophy
a problem? The answer is, any way that works. This attitude [is] the attitude of mind of a truly creative pure mathematician. It does not appear in his final proofs […] These are the things that he publishes, but they in no way reflect the way that he works in order to obtain a guess as to what it is he is going to prove” New Textbooks for the “New” Mathematics
There are no set patterns of procedure. We try this and that. We guess. We try to generalize the result in order to make the proof easier. We try special cases to see if any insight can be gained in this way. J.B. Roberts
Number 1, July 1993 "THEORETICAL MATHEMATICS": TOWARD A CULTURAL SYNTHESIS OF MATHEMATICS AND THEORETICAL PHYSICS ARTHUR JAFFE AND FRANK QUINN Abstract. Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathe- matical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and destructive consequences. Serious caution is required, and the issue should be considered before, rather than after, obvious damage occurs. With the hazards carefully in mind, we propose a framework that should allow a healthy and positive role for speculation. Modern mathematics is nearly characterized by the use of rigorous proofs. This practice, the result of literally thousands of years of refinement, has brought to mathematics a clarity and reliability unmatched by any other science. But it also makes mathematics slow and difficult; it is arguably the most disciplined of human intellectual activities. Groups and individuals within the mathematics community have from time to time tried being less compulsive about details of arguments. The results have been mixed, and they have occasionally been disastrous. Yet today in certain areas there is again a trend toward basing mathematics on intuitive reasoning without proof. To some extent this is the old pattern of history being repeated by those unfamiliar with it. But it also may be the beginning of fundamental changes in the way mathematics is organized. In either case, it is vital at this time to reexamine the role of proofs in mathematical understanding and to develop a constructive context for these trends. We begin with a discussion of physics, partially because some of the current AMS Bulletin, 1993
Number 1, July 1993 "THEORETICAL MATHEMATICS": TOWARD A CULTURAL SYNTHESIS OF MATHEMATICS AND THEORETICAL PHYSICS ARTHUR JAFFE AND FRANK QUINN Abstract. Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathe- matical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and destructive consequences. Serious caution is required, and the issue should be considered before, rather than after, obvious damage occurs. With the hazards carefully in mind, we propose a framework that should allow a healthy and positive role for speculation. Modern mathematics is nearly characterized by the use of rigorous proofs. This practice, the result of literally thousands of years of refinement, has brought to mathematics a clarity and reliability unmatched by any other science. But it also makes mathematics slow and difficult; it is arguably the most disciplined of human intellectual activities. Groups and individuals within the mathematics community have from time to time tried being less compulsive about details of arguments. The results have been mixed, and they have occasionally been disastrous. Yet today in certain areas there is again a trend toward basing mathematics on intuitive reasoning without proof. To some extent this is the old pattern of history being repeated by those unfamiliar with it. But it also may be the beginning of fundamental changes in the way mathematics is organized. In either case, it is vital at this time to reexamine the role of proofs in mathematical understanding and to develop a constructive context for these trends. We begin with a discussion of physics, partially because some of the current AMS Bulletin, 1993 SPECULATION Labelling of Speculative Work
Number 1, July 1993 "THEORETICAL MATHEMATICS": TOWARD A CULTURAL SYNTHESIS OF MATHEMATICS AND THEORETICAL PHYSICS ARTHUR JAFFE AND FRANK QUINN Abstract. Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathe- matical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and destructive consequences. Serious caution is required, and the issue should be considered before, rather than after, obvious damage occurs. With the hazards carefully in mind, we propose a framework that should allow a healthy and positive role for speculation. Modern mathematics is nearly characterized by the use of rigorous proofs. This practice, the result of literally thousands of years of refinement, has brought to mathematics a clarity and reliability unmatched by any other science. But it also makes mathematics slow and difficult; it is arguably the most disciplined of human intellectual activities. Groups and individuals within the mathematics community have from time to time tried being less compulsive about details of arguments. The results have been mixed, and they have occasionally been disastrous. Yet today in certain areas there is again a trend toward basing mathematics on intuitive reasoning without proof. To some extent this is the old pattern of history being repeated by those unfamiliar with it. But it also may be the beginning of fundamental changes in the way mathematics is organized. In either case, it is vital at this time to reexamine the role of proofs in mathematical understanding and to develop a constructive context for these trends. We begin with a discussion of physics, partially because some of the current AMS Bulletin, 1993 SPECULATION Labelling of Speculative Work Standard Nomenclature “conjecture” vs “theorem”
Number 1, July 1993 "THEORETICAL MATHEMATICS": TOWARD A CULTURAL SYNTHESIS OF MATHEMATICS AND THEORETICAL PHYSICS ARTHUR JAFFE AND FRANK QUINN Abstract. Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathe- matical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and destructive consequences. Serious caution is required, and the issue should be considered before, rather than after, obvious damage occurs. With the hazards carefully in mind, we propose a framework that should allow a healthy and positive role for speculation. Modern mathematics is nearly characterized by the use of rigorous proofs. This practice, the result of literally thousands of years of refinement, has brought to mathematics a clarity and reliability unmatched by any other science. But it also makes mathematics slow and difficult; it is arguably the most disciplined of human intellectual activities. Groups and individuals within the mathematics community have from time to time tried being less compulsive about details of arguments. The results have been mixed, and they have occasionally been disastrous. Yet today in certain areas there is again a trend toward basing mathematics on intuitive reasoning without proof. To some extent this is the old pattern of history being repeated by those unfamiliar with it. But it also may be the beginning of fundamental changes in the way mathematics is organized. In either case, it is vital at this time to reexamine the role of proofs in mathematical understanding and to develop a constructive context for these trends. We begin with a discussion of physics, partially because some of the current AMS Bulletin, 1993 SPECULATION Labelling of Speculative Work Standard Nomenclature “conjecture” vs “theorem” Responsible Publication Journal of XYZ [email protected]
Number 2, April 1994 RESPONSES TO "THEORETICAL MATHEMATICS: TOWARD A CULTURAL SYNTHESIS OF MATHEMATICS AND THEORETICAL PHYSICS", BY A. JAFFE AND F. QUINN MICHAEL ATIYAH ET AL. Michael Atiyah The Master's Lodge Trinity College Cambridge CB2 1TQ England, U.K. I find myself agreeing with much of the detail of the Jaffe-Quinn argument, especially the importance of distinguishing between results based on rigorous proofs and those which have a heuristic basis. Overall, however, I rebel against their general tone and attitude which appears too authoritarian. My fundamental objection is that Jaffe and Quinn present a sanitized view of mathematics which condemns the subject to an arthritic old age. They see an inexorable increase in standards of rigour and are embarrassed by earlier periods of sloppy reasoning. But if mathematics is to rejuvenate itself and break exciting new ground it will have to allow for the exploration of new ideas and techniques which, in their creative phase, are likely to be as dubious as in some of the great eras of the past. Perhaps we now have high standards of proof to aim at but, in the early stages of new developments, we must be prepared to act in more buccaneering style. The history of mathematics is full of instances of happy inspiration triumph- ing over a lack of rigour. Euler's use of wildly divergent series or Ramanujan's insights are among the more obvious, and mathematics would have been poorer AMS Bulletin, 1994 1977; Oberwolfach Collection “Jaffe and Quinn present a sanitized view of mathematics which condemns the subject to an arthritic old age” “They […] are embarrassed by earlier periods of sloppy reasoning. But if mathematics is to rejuvenate itself and break exciting new ground it will have to allow for the exploration of new ideas and techniques” Michael Atiyah
Number 2, April 1994 RESPONSES TO "THEORETICAL MATHEMATICS: TOWARD A CULTURAL SYNTHESIS OF MATHEMATICS AND THEORETICAL PHYSICS", BY A. JAFFE AND F. QUINN MICHAEL ATIYAH ET AL. Michael Atiyah The Master's Lodge Trinity College Cambridge CB2 1TQ England, U.K. I find myself agreeing with much of the detail of the Jaffe-Quinn argument, especially the importance of distinguishing between results based on rigorous proofs and those which have a heuristic basis. Overall, however, I rebel against their general tone and attitude which appears too authoritarian. My fundamental objection is that Jaffe and Quinn present a sanitized view of mathematics which condemns the subject to an arthritic old age. They see an inexorable increase in standards of rigour and are embarrassed by earlier periods of sloppy reasoning. But if mathematics is to rejuvenate itself and break exciting new ground it will have to allow for the exploration of new ideas and techniques which, in their creative phase, are likely to be as dubious as in some of the great eras of the past. Perhaps we now have high standards of proof to aim at but, in the early stages of new developments, we must be prepared to act in more buccaneering style. The history of mathematics is full of instances of happy inspiration triumph- ing over a lack of rigour. Euler's use of wildly divergent series or Ramanujan's insights are among the more obvious, and mathematics would have been poorer AMS Bulletin, 1994 2007; Wikimedia Commons “I find the JQ prescription appalling because it would bring havoc into living branches of science” “[A] flow of young people […] kept being introduced to me for advice. They were acknowledged as brilliant and highly promising; but they could not stomach the Bourbaki credo, hence saw no future for themselves in mathematics” Benoît Mandelbrot
Number 2, April 1994 RESPONSES TO "THEORETICAL MATHEMATICS: TOWARD A CULTURAL SYNTHESIS OF MATHEMATICS AND THEORETICAL PHYSICS", BY A. JAFFE AND F. QUINN MICHAEL ATIYAH ET AL. Michael Atiyah The Master's Lodge Trinity College Cambridge CB2 1TQ England, U.K. I find myself agreeing with much of the detail of the Jaffe-Quinn argument, especially the importance of distinguishing between results based on rigorous proofs and those which have a heuristic basis. Overall, however, I rebel against their general tone and attitude which appears too authoritarian. My fundamental objection is that Jaffe and Quinn present a sanitized view of mathematics which condemns the subject to an arthritic old age. They see an inexorable increase in standards of rigour and are embarrassed by earlier periods of sloppy reasoning. But if mathematics is to rejuvenate itself and break exciting new ground it will have to allow for the exploration of new ideas and techniques which, in their creative phase, are likely to be as dubious as in some of the great eras of the past. Perhaps we now have high standards of proof to aim at but, in the early stages of new developments, we must be prepared to act in more buccaneering style. The history of mathematics is full of instances of happy inspiration triumph- ing over a lack of rigour. Euler's use of wildly divergent series or Ramanujan's insights are among the more obvious, and mathematics would have been poorer AMS Bulletin, 1994 ~1972; Oberwolfach Collection “The sequence for the understanding of mathematics may be: intuition, trial, error, speculation, conjecture, proof. The mixture and the sequence of these events differ widely in different domains, but there is general agreement that the end product is rigorous proof—which we know and can recognize, without the formal advice of the logicians” Saunders MacLane
the importance of rigorous proof. But a vocal group of research mathematicians insists on the vital roles of intuition, speculation and experimentation.
the importance of rigorous proof. But a vocal group of research mathematicians insists on the vital roles of intuition, speculation and experimentation.
proof checking ✦ Interactive proof development ✦ Libraries of mathematical content “Our vision is that formalising a mathematical proof may become as easy as writing mathematics in a mathematical text editor such as LATEX and that a mathematical proof will only be accepted for publication when it has been formally checked.”
proof checking ✦ Interactive proof development ✦ Libraries of mathematical content “Our vision is that formalising a mathematical proof may become as easy as writing mathematics in a mathematical text editor such as LATEX and that a mathematical proof will only be accepted for publication when it has been formally checked.”
tools for codifying mathematical proofs in order to communicate mathematical insight. They enter the picture after the creative phase of mathematical discovery.
optimise for the communication/codification phase. 2. Programmers want and need to use PLs for experimentally developing new ideas and discovering programs. Poke
There are no set patterns of procedure. We try this and that. We guess. We try to generalize the result in order to make the proof easier. We try special cases to see if any insight can be gained in this way. J.B. Roberts
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