The QED Manifesto after Two Decades - Version 2.0

Ittay Weiss
2016 Journal of Software  
In 1994 the QED Manifesto described an ideal whereby mathematics is communicated via a computerized system in a fully formalized fashion complete with automatic proof checking and other derived tools facilitating profound improvements to the way mathematics is taught, the way new results are disseminated, and ultimately to how mathematics is practiced. Two decades later it is safe to say the dream is not yet a reality. Analyzing some of the difficulties met thus far in the realization of the
more » ... ginal Manifesto we propose here Version 2.0, presenting a view of a different ideal and a description of a system realizing part of that ideal. Journal of Software how cognitively understandable it is. For longer proofs the term exp (ℓ) becomes dominant and as soon as it passes a threshold determined by h, a fully formalized proof, i.e., having Φ = 1, receives a high value of Γ, i.e., it is cognitively doomed. Clearly, the value of h varies from person to person and is dependent on personal experience and expertise. Nonetheless, judging by the uniformity of style in the communication of modern mathematics, h appears to be a cognitive constant across the population of mathematicians (it is perhaps a topic for neuropsychology to conduct experiments to estimate h). In the polarity ensuing from this uncertainty principle it can be said that the original Manifesto seeks to minimize Φ. Version 2.0, the subject matter of this article, is concerned with minimizing Γ. We note that the formal aspects of mathematics communication are dealt with extensively as a proper mathematical area of study (see [4] for an excellent overview). The cognitive aspects of mathematics communication are also addressed, albeit typically not by research mathematicians. This unfortunate disparity in practice represents a miscommunication which is perhaps yet another incarnation of the cognitive uncertainly principle. It is crucial to state that QED Version 2.0 is neither meant to replace Version 1.0 nor to indicate that it is in any sense better. By analogy to programming languages, the difference between QED 1.0 and QED 2.0 is akin to the differences between low level programming paradigms and higher level ones, respectively. If formalizability and cognitive readability can be unified by some currently unknown clever mechanism (as [5] seems to suggest may be the case), or if the process of turning a 'higher level' proof into a formal 'lower level' one can be automated, then QED 1.0 and QED 2.0 will merge into a single product. Otherwise, each paradigm will have its own merits and its own applications, and products will coevolve utilizing various mixtures of the two versions. The presentation of the QED 2.0 Manifesto given below follows in spirit the structure of the original Manifesto. In particular, we describe the ambition, contemplate and respond to objections to the proposed ideal, and discuss some technicalities. A significant difference from the original Manifesto is in the final section describing the Mathropolis system at its current stage of development, a system which is QED 2.0 in a strong sense. I developed Mathropolis and the ideas expressed below in tandem, one influencing the other, and I hope its inclusion here elucidates the abstract ideas presented. Much as was the case with the original Manifesto, the ideas presented herein are certainly already in circulation, if not in print then in the minds of practitioners. I thus happily relinquish all claims of originality. To insert images in Word, position the cursor at the insertion point and either use Insert | Picture | From File or copy the image to the Windows clipboard and then Edit | Paste Special | Picture (with "Float over text" unchecked). "QED is the very tentative title of a project to build a computer system that effectively represents all important mathematical knowledge and techniques. The QED system will conform to the highest standards of mathematical rigor, including the use of strict formality in the internal representation of knowledge and the use of mechanical methods to check proofs of the correctness of all entries in the system." QED 2.0 is the very tentative title of a project to build a computer system that effectively represents all important mathematical knowledge and techniques. The QED 2.0 system will conform to the highest standard of clarity of exposition and accepted rigor, including paradigms for reusability, encapsulation, uniformization, and language independence in the communication of mathematics. The QED 2.0 project will be a major scientific undertaking requiring the cooperation and efforts of hundreds of mathematicians, considerable ingenuity by many computer scientists, and a careful analysis of 804
doi:10.17706/jsw.11.8.803-815 fatcat:s4tvqzfbirf4nlhhjdb6on7pcm