einschlie13lich aller seiner Teile ist urheberrechtlich geschUtzt. Jede Verwertung au13erhalb der... more einschlie13lich aller seiner Teile ist urheberrechtlich geschUtzt. Jede Verwertung au13erhalb der engen Grenzen des Urheberrechtsgesetzes ist ohne Zustimmung des Veri ages unzuliissig und strafbar. Das gilt besonders fUr Vervie1fii1tigungen, Ubersetzungen, Mikroverfilmungen und die Einspeicherung und Verarbeitung in elektronischen Systemen.
Library of Congress Cataloging-in-Publication Data Engeler, Erwin. [Metamathematik der E1ementarm... more Library of Congress Cataloging-in-Publication Data Engeler, Erwin. [Metamathematik der E1ementarmathematik. English] Foundations of mathematics: questions of analysis, geometry & algorithmics / Erwin Engeler ; translated by Charles B. Thomas. p. cm. Includes bibliographical references.
In den bisherigen Ausfuhrungen kam implizit die Grundhaltung zum Ausdruck, dass die Klasse der pa... more In den bisherigen Ausfuhrungen kam implizit die Grundhaltung zum Ausdruck, dass die Klasse der partiell-rekursiven Funktionen auf N nicht nur eine mathematisch reizvolle Klasse von Objekten darstellt, sondern uberhaupt den Begriff des programmierten Rechnens erst zu einem theoretischen Gegenstand macht. Was nun das Rechnen auf naturlichen Zahlen betrifft, so hat sich schon in den fruhen dreissiger Jahren aus verschiedenen Erfahrungen die Ansicht herausgebildet, dass die Klasse der partiell-rekursiven Funktionen nicht nur beispielhaft sondern exakt die Klasse der in irgendeinem vernunftigen Sinn “im Prinzip” ausrechenbaren Funktionen darstelle.
The usual modelling of the syllogisms of the Organon by a calculus of classes does not include re... more The usual modelling of the syllogisms of the Organon by a calculus of classes does not include relations. Aristotle may however have envisioned them in the first two books as the category of relatives, where he allowed them to compose with themselves. Composition is the main operation in combinatory logic, which therefore offers itself to logicians for a new kind of modelling. The resulting calculus includes also composition of predicates by the logical connectives.
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
... No need arose to write Term explicitly in the system, and it was judged (following the lead o... more ... No need arose to write Term explicitly in the system, and it was judged (following the lead of Floyd [2]) that for properties of the structure ~ it was convenient to choose the language of first-order predicate logic. ... Co., Chicago 1968. [2] RW Floyd: Assigning Meaning to Programs. ...
... by Erwin Engeler* We know that every program for the register machine of Shepherdson-Sturgis ... more ... by Erwin Engeler* We know that every program for the register machine of Shepherdson-Sturgis ... This attitude is taken in various theories of program schemes such as Yanov (1958), McCarthy (1963 a), Floyd (1967 a), and others. The contribution of the present author flowed ...
It is a commonplace, that the use of mathematical software Is in the process of influencing, tf n... more It is a commonplace, that the use of mathematical software Is in the process of influencing, tf not shaping, the working style of the mathematician, physicist and engineer. Considerable effort has been put into the creation of integrated systems by various groups. On the other hand, relatively small progress has been seen in making a telling impact on the larger scientific community with respect to widespread use of such systems. The computer workplace for the scientist has not quite happened yet. We feel, that this is a question foremost of education. It is imperative that we give the advanced student of the exact sciences and engineering a memorable experience of success with using mathematical software. Obviously the attainment of such an aim depends crucially on a very carefully thought-out collection of representative, well motivated projects originating in physics, pure and applied mathematics, electrical engineering, computer science etc.. And, of course, on the easy access to mathematical software, documentation and expert counselling. At ETH we have been working for some years to provide and enhance such an environment. We think that we have succeeded with this pilot project. The present report gives an overview of the concept of our mathematical laboratory and provides some details of the projects that are presently provided for our students, Since there are more than a hundred students that take the laboratory course during a given term, we also describe some of the software support for the administration of the lab. This algorithm is conve~ed to a program thatcMlsroutinesfromtheMP-package(Multi-Prectsion)that allows an arbitrary numberofdecimal places to be computed. The program asksforthe number ofdecimalstouse: c arbitrary number of decimals of PI (J. Waldvogel, ETHZ
Bürgi’s method for the computation of the sine function is b ased mathematically on the fact that... more Bürgi’s method for the computation of the sine function is b ased mathematically on the fact that this function satisfies a differential equation. T he solution is approximated by a sequence of functions computed simutaneously at discrete valu s. This observation motivates a generalisation by which we obtain a massively pa rallel algorithm for secondorder differential equations in general. We also show the ps ychology of invention may have originally lead to these approximations. Since the lim it of the sequence is close to, but does not coincide with the solution, Bürgi’s artificium consists of ”normalizing” the approximation by a constant factor, whereby he obtains the e xact values as discussed in [2]. This, however does not generalize. 1. APPROACHINGEULER (WITH A FILE ) Bürgi was a Swiss watchmaker, astronomer and mathematicia n. He was a perfect artisan who created ingenious, beautifully finished pieces of mathe matical and astronomical instruments1, and also very original math...
The Hilbert Program in Göttingen was winding down in the early 1930s. By then it was mostly in th... more The Hilbert Program in Göttingen was winding down in the early 1930s. By then it was mostly in the hands of Paul Bernays who was writing the first volume of Grundlagen der Mathematik. Hermann Weyl had succeeded David Hilbert. There were three outstanding doctoral students in logic: Haskell B.Curry, Saunders MacLane and Gerhard Gentzen. These three students are at the beginning of three threads in mathematical logic: Combinatory Logic (Curry), Proof Theory (Gentzen) and Algebra of Proofs (MacLane), the last one essentially forgotten, except perhaps for some technical results useful in computer algebra (cf. Newman’s Lemma). The present author, also a student of Bernays, when looking up this mathematical ancestry, was fascinated by the contrast between MacLane’s enthusiasm about the ideas in his thesis as expressed in his letters home, and the almost complete absence of any mathematical follow-up. Is it possible that mathematical development has passed something by, just because MacLan...
The mathematical model employed in this essay attempts to explain how complex scripts of behaviou... more The mathematical model employed in this essay attempts to explain how complex scripts of behaviour and conceptual content can reside in, combine and interact on large neural networks. The neural hypothesis attributes functions of the brain to sets of firing neurons dynamically: to sets of cascades of such firings, typically visualised by imaging technologies. Such sets are represented as the elements of what we call a neural algebra with their interaction as its basic operation. The neuro-algebraic thesis identifies "thoughts" with elements of a neural algebra and "thinking" with its basic operation. We argue for this thesis by a thought-experiment. It examines examples of human thought processes in proposed emulation by neural algebras. In particular we analyse problems such as controlling, classifying and learning. In neural algebras these may be posed as algebraic equations, whose solutions may lead to extensions of a neural algebra by new elements. The modelling of such extensions consists of formal analogues to familiar faculties such as reflection, distinction and comprehension which will be mades precise as operations on the algebra. An advantage of our approach is that this modelling leads directly to brain functions. From the cascades of such functions we obtain the neurons involved in them and their connective structure, and mathematically describe their behaviour.
Publisher Summary This chapter discusses combinatorial theorems for the construction of models. I... more Publisher Summary This chapter discusses combinatorial theorems for the construction of models. It provides a further study of the interconnections between model theory and combinatorics and attempt a systematic development. Since the compactness of first-order logic is one of the most successfully used properties in model theory, it is natural to investigate the role of compactness also in our more general framework. This leads to a program of finding simple combinatorial analogues of the main results of first-order model theory. Clearly, it is to be expected that some such results, because of their more general character, will become of independent interest.
Studies in Logic and the Foundations of Mathematics, 1987
The formation process of pure logic programs over a first-order language is iterated to give rise... more The formation process of pure logic programs over a first-order language is iterated to give rise to cumulative logic programs. Such programs turn out to be objects in a combinatory model and are therefore amenable to algebraic manipulation including equation solving. It is pointed out how this fact can be employed in a discipline of modeling for cooperative processes. Other results concern the representability of equational classes, universal classes and algorithmic classes as solution sets of a set of combinatory equations.
einschlie13lich aller seiner Teile ist urheberrechtlich geschUtzt. Jede Verwertung au13erhalb der... more einschlie13lich aller seiner Teile ist urheberrechtlich geschUtzt. Jede Verwertung au13erhalb der engen Grenzen des Urheberrechtsgesetzes ist ohne Zustimmung des Veri ages unzuliissig und strafbar. Das gilt besonders fUr Vervie1fii1tigungen, Ubersetzungen, Mikroverfilmungen und die Einspeicherung und Verarbeitung in elektronischen Systemen.
Library of Congress Cataloging-in-Publication Data Engeler, Erwin. [Metamathematik der E1ementarm... more Library of Congress Cataloging-in-Publication Data Engeler, Erwin. [Metamathematik der E1ementarmathematik. English] Foundations of mathematics: questions of analysis, geometry & algorithmics / Erwin Engeler ; translated by Charles B. Thomas. p. cm. Includes bibliographical references.
In den bisherigen Ausfuhrungen kam implizit die Grundhaltung zum Ausdruck, dass die Klasse der pa... more In den bisherigen Ausfuhrungen kam implizit die Grundhaltung zum Ausdruck, dass die Klasse der partiell-rekursiven Funktionen auf N nicht nur eine mathematisch reizvolle Klasse von Objekten darstellt, sondern uberhaupt den Begriff des programmierten Rechnens erst zu einem theoretischen Gegenstand macht. Was nun das Rechnen auf naturlichen Zahlen betrifft, so hat sich schon in den fruhen dreissiger Jahren aus verschiedenen Erfahrungen die Ansicht herausgebildet, dass die Klasse der partiell-rekursiven Funktionen nicht nur beispielhaft sondern exakt die Klasse der in irgendeinem vernunftigen Sinn “im Prinzip” ausrechenbaren Funktionen darstelle.
The usual modelling of the syllogisms of the Organon by a calculus of classes does not include re... more The usual modelling of the syllogisms of the Organon by a calculus of classes does not include relations. Aristotle may however have envisioned them in the first two books as the category of relatives, where he allowed them to compose with themselves. Composition is the main operation in combinatory logic, which therefore offers itself to logicians for a new kind of modelling. The resulting calculus includes also composition of predicates by the logical connectives.
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.
... No need arose to write Term explicitly in the system, and it was judged (following the lead o... more ... No need arose to write Term explicitly in the system, and it was judged (following the lead of Floyd [2]) that for properties of the structure ~ it was convenient to choose the language of first-order predicate logic. ... Co., Chicago 1968. [2] RW Floyd: Assigning Meaning to Programs. ...
... by Erwin Engeler* We know that every program for the register machine of Shepherdson-Sturgis ... more ... by Erwin Engeler* We know that every program for the register machine of Shepherdson-Sturgis ... This attitude is taken in various theories of program schemes such as Yanov (1958), McCarthy (1963 a), Floyd (1967 a), and others. The contribution of the present author flowed ...
It is a commonplace, that the use of mathematical software Is in the process of influencing, tf n... more It is a commonplace, that the use of mathematical software Is in the process of influencing, tf not shaping, the working style of the mathematician, physicist and engineer. Considerable effort has been put into the creation of integrated systems by various groups. On the other hand, relatively small progress has been seen in making a telling impact on the larger scientific community with respect to widespread use of such systems. The computer workplace for the scientist has not quite happened yet. We feel, that this is a question foremost of education. It is imperative that we give the advanced student of the exact sciences and engineering a memorable experience of success with using mathematical software. Obviously the attainment of such an aim depends crucially on a very carefully thought-out collection of representative, well motivated projects originating in physics, pure and applied mathematics, electrical engineering, computer science etc.. And, of course, on the easy access to mathematical software, documentation and expert counselling. At ETH we have been working for some years to provide and enhance such an environment. We think that we have succeeded with this pilot project. The present report gives an overview of the concept of our mathematical laboratory and provides some details of the projects that are presently provided for our students, Since there are more than a hundred students that take the laboratory course during a given term, we also describe some of the software support for the administration of the lab. This algorithm is conve~ed to a program thatcMlsroutinesfromtheMP-package(Multi-Prectsion)that allows an arbitrary numberofdecimal places to be computed. The program asksforthe number ofdecimalstouse: c arbitrary number of decimals of PI (J. Waldvogel, ETHZ
Bürgi’s method for the computation of the sine function is b ased mathematically on the fact that... more Bürgi’s method for the computation of the sine function is b ased mathematically on the fact that this function satisfies a differential equation. T he solution is approximated by a sequence of functions computed simutaneously at discrete valu s. This observation motivates a generalisation by which we obtain a massively pa rallel algorithm for secondorder differential equations in general. We also show the ps ychology of invention may have originally lead to these approximations. Since the lim it of the sequence is close to, but does not coincide with the solution, Bürgi’s artificium consists of ”normalizing” the approximation by a constant factor, whereby he obtains the e xact values as discussed in [2]. This, however does not generalize. 1. APPROACHINGEULER (WITH A FILE ) Bürgi was a Swiss watchmaker, astronomer and mathematicia n. He was a perfect artisan who created ingenious, beautifully finished pieces of mathe matical and astronomical instruments1, and also very original math...
The Hilbert Program in Göttingen was winding down in the early 1930s. By then it was mostly in th... more The Hilbert Program in Göttingen was winding down in the early 1930s. By then it was mostly in the hands of Paul Bernays who was writing the first volume of Grundlagen der Mathematik. Hermann Weyl had succeeded David Hilbert. There were three outstanding doctoral students in logic: Haskell B.Curry, Saunders MacLane and Gerhard Gentzen. These three students are at the beginning of three threads in mathematical logic: Combinatory Logic (Curry), Proof Theory (Gentzen) and Algebra of Proofs (MacLane), the last one essentially forgotten, except perhaps for some technical results useful in computer algebra (cf. Newman’s Lemma). The present author, also a student of Bernays, when looking up this mathematical ancestry, was fascinated by the contrast between MacLane’s enthusiasm about the ideas in his thesis as expressed in his letters home, and the almost complete absence of any mathematical follow-up. Is it possible that mathematical development has passed something by, just because MacLan...
The mathematical model employed in this essay attempts to explain how complex scripts of behaviou... more The mathematical model employed in this essay attempts to explain how complex scripts of behaviour and conceptual content can reside in, combine and interact on large neural networks. The neural hypothesis attributes functions of the brain to sets of firing neurons dynamically: to sets of cascades of such firings, typically visualised by imaging technologies. Such sets are represented as the elements of what we call a neural algebra with their interaction as its basic operation. The neuro-algebraic thesis identifies "thoughts" with elements of a neural algebra and "thinking" with its basic operation. We argue for this thesis by a thought-experiment. It examines examples of human thought processes in proposed emulation by neural algebras. In particular we analyse problems such as controlling, classifying and learning. In neural algebras these may be posed as algebraic equations, whose solutions may lead to extensions of a neural algebra by new elements. The modelling of such extensions consists of formal analogues to familiar faculties such as reflection, distinction and comprehension which will be mades precise as operations on the algebra. An advantage of our approach is that this modelling leads directly to brain functions. From the cascades of such functions we obtain the neurons involved in them and their connective structure, and mathematically describe their behaviour.
Publisher Summary This chapter discusses combinatorial theorems for the construction of models. I... more Publisher Summary This chapter discusses combinatorial theorems for the construction of models. It provides a further study of the interconnections between model theory and combinatorics and attempt a systematic development. Since the compactness of first-order logic is one of the most successfully used properties in model theory, it is natural to investigate the role of compactness also in our more general framework. This leads to a program of finding simple combinatorial analogues of the main results of first-order model theory. Clearly, it is to be expected that some such results, because of their more general character, will become of independent interest.
Studies in Logic and the Foundations of Mathematics, 1987
The formation process of pure logic programs over a first-order language is iterated to give rise... more The formation process of pure logic programs over a first-order language is iterated to give rise to cumulative logic programs. Such programs turn out to be objects in a combinatory model and are therefore amenable to algebraic manipulation including equation solving. It is pointed out how this fact can be employed in a discipline of modeling for cooperative processes. Other results concern the representability of equational classes, universal classes and algorithmic classes as solution sets of a set of combinatory equations.
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