- Early life and education
- Jaime Nubiola: "The Continuity of Continuity: A Theme in Leibniz, Peirce and Quine"
- Gottfried Wilhelm Leibniz (1646-1716)

Managers are invited to rediscover the art of thinking.

### Early life and education

They should understand the role of mental models, realize the importance of cognitive bias, agree on clear definitions and efficient criteria etc. Creativity demands the ability to unshackle ourselves from conventional ways of thinking, to "think outside the box". But we need to go a step further. Once outside the box, we need to construct a new box or boxes that is, new intellectual frameworks or models to help us structure our thinking.

Only once we have done so can we generate truly game-changing ideas. Great course. I enjoyed thinking about management and philosophy. Backhouse and Jan L.

The next 3 MCP Conferences took place in the 90s, in the United Kingdom, in Germany, and in Sweden, respectively, and if I were an American, I would consider the fact that the first four of these conferences all took place in Europe a very alarming symptom indeed. On the other hand, we should be glad that the gospel of design by derivation, rather than by trial and error, is still preached. For me personally it is very gratiftying to see that the area of application has been extended by people whom I consider in moments of greater vanity as my pupils: I have in mind A.

Martin in connection with chip design, and W. Feijen and A. They all have done me proud. Back to personal history. During that decade, Computing Science, or "Informatics", as the topic became known on the continent, became a much more substantial discipline than my mathematical colleagues had envisaged or intended and, perhaps not totally without justification, part of the blame for this undesired development was heaped upon me.

Life in Eindhoven became a little difficult, but fortunately, Burroughs Corporation offered to rescue me. I stayed with Burroughs Corporation for 11 wonderful and very productive years, my primary charter being to do my own thing. I had one major disappointment: I had hoped to lead the life of a scientific "man of letters", but the frequency with which my letters were answered was, on the average, too low to maintain a lively interchange, and so I ended up travelling more than I had intended and was comfortable with.

But apart from that, they were, as said, wonderful years. It was too good to last, and it did not. After the company had changed its CEO, it rapidly lost interest in science and technology, and the groups with which I had built up my relations disbanded the one after the other. But it was not only Burroughs that changed during that decade, for I changed as well; my mathematical interests widened and eventually I was ready to return to the University Campus, and was fortunate that UT Austin offered me the opportunity to do so.

During my Burroughs years I changed very much as a mathematician. By the end of the 60s, R. Floyd and C. Hoare had shown us how to reason in principle about programs, but it took a few years before their findings penetrated through my skull. In the early 70s we studied Tony Hoare's correctness proof of the routine "FIND" and my friends still remember that at the end of that study I declared in disgust that "that meaningless ballet of symbols was not my cup of tea"; only gradually, my resistance to that kind of formula manipulation faded, until eventually, I began to love it.

Circumstances forced me to become a very different mathematician. I had invented predicate transformers as a formal tool for programming semantics and they served my purpose, but I lacked at the time the mathematical background needed to understand what I was doing, so much so that I published a paper with a major blunder. I was helped by C. Scholten, with whom I had worked since the 50s, and together we provided the mathematical background that had been lacking.

In doing so we developed a much more calculational style of doing mathematics than we had ever practised before. I stayed away from where I had blundered, but that hurdle the younger and mathematically better equipped R. Dijkstra took in his stride after he had extended our apparatus from predicates to relations.

## Jaime Nubiola: "The Continuity of Continuity: A Theme in Leibniz, Peirce and Quine"

Let me try to sketch my mathematical change. For centuries, mathematics has been what it says in the dictionary: the abstract science of space, number and quantity. In this conception, mathematics is very much about things, such as triangles, prime numbers, areas, derivatives and correspondences. Accordingly, such mathemetics is done in a language in which nouns prevail and which each time is tied rather closely to the subject matter at hand. In its orientation towards things , it invited pictures and visual representations. This has gone so far that the Greeks introduced, for instance, the hardly needed notion of the "triangular numbers" 1, 3, 6, 10, etc.

But at least since the 18th Century —personally I am aware of Augustus de Morgan and George Boole from Great Britain— there has emerged another view, representing a shift from the "quod" to the "quo modo", from the "what" to the "how": it views mathematics primarily as the art and science of effective reasoning. This was refreshing because the methodological flavour gave it a much wider applicability.

History has shown that the mathematical community needs a constant reminder of this more methodological conception of its trade, as many mathematicians invest so much of their time and energy in the study of a specific area that they tend to identify mathematics with its [always specific] subject matter. As far as I know, Gottfried Wilhelm Leibniz, who lived from to , has been the first to tackle effective reasoning as a technical problem.

As a youngster of 20 years of age he conceived, possibly inspired by the work of Descartes, a vision of reasoning as applying a calculus. Like modern computing scientists, he invented impressive names for what had still to be invented, and, for good reasons not overly modest, he called his system no more and no less than "Characteristica Universalis". And again like modern computing scientists, he grossly underestimated the time the project would take: he confidently prophesied that a few well-chosen men could do the job in five years, but the whole undertaking was at the time of such a radical novelty that even the genius of Leibniz did not suffice for its realization, and it was only after another two centuries that George Boole began to realize something similar to the subsystem that Leibniz had called the "calculus raticinator".

At the same time this would be a sort of universal language or script, but infinitely different from all those projected hitherto; for the symbols and even the words in it would direct reason; and errors, except those of fact, would be mere mistakes in calculation.

I think it absolutely astounding that he foresaw how "the symbols would direct the reasoning", for how strongly they would do so was one of the most delightful discoveries of my professional life. Around , the Dream of Leibniz came closer to realization as the necessary formalisms became available and the great German mathematician David Hilbert promoted the project. Hilbert made clear that the total calculation had to be achieved with the aid of manipulations from a well-defined repertoire.

## Gottfried Wilhelm Leibniz (1646-1716)

One could argue with the repertoire, and the Dutchman L. Brouwer vigorously argued against Hilbert's choice, but instead of arguing what is the "right" repertoire, one can turn the concept of proof from an absolute concept to one relative to the repertoire of admitted manipulations. In passing we note that, having abolished the notion of "the true repertoire", we still are free to have our preferences.

A major shortcoming of the latter view is that it gives no technical guidance for proof design and makes it very difficult to teach that kind of mathematics. Yet, or perhaps for this reason, many of the more conservative mathematicians still cling to this form of consensus mathematics; they will even vigorously defend their informality.

Here I must admit that I am a poor historian or a poor psychologist, for I see two things that I cannot reconcile with each other. On the one hand Hilbert, generally recognized as the greatest mathematician of his generation, has stated very explicitly that rigour in proof is not the enemy of simplicity, and, also, that the effort for rigour leads us to the discovery of simpler and more general methods of proof. He confirms the prediction Leibniz made, that "the symbols would direct reason".

On the other hand the world of mathematics seems to continue as if these things had never been said.

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They don't even take the trouble of explaining why Hilbert was wrong or misguided, it looks more as if he is just ignored. Davis and Reuben Hersch: after "Heinlein" comes "Hofstadter". Hilbert is not mentioned. I should add that the blurb characterizes the book as "A passionate plea against the use of formal mathematical reasoning as a method for solving mankind's problems Sure, mathematicians manipulated formulae prior to Hilbert, but with the exception of the most familiar cases —such as doing arithmetic—, people manipulated interpreted formulae, i.

follow link Hilbert the formalist showed that such interpretation was superfluous because which manipulations were permissible could be defined in terms of the symbols themselves. By leaving the formulae uninterpreted, their manipulations becomes much simpler and safer. As my initial protest against the "meaningless ballet of symbols" illustrates, the preferences of the formalists are an acquired taste, but since Hilbert we know that the taste is worth acquiring.

During Hilbert's life, Leibniz's Dream, by and large, just stayed a dream. People viewed formal proofs as an interesting theoretical possibility or an unrealistic idealization, and they would regard their own proof as a usually sufficient sketch of a formal argument. They would even assure you that, if you insisted, they could formalize their informal argument, but how often that claim was valid is anybody's guess.

In the 2nd half of the 20th Century, things shifted with the advent of computers, as more and more people began to adopt formal techniques. Parts of Leibniz's Dream became reality, and it is quite understandable that this happened mostly in Departments of Computing Science, rather than in Departments of Mathemetics. Firstly, the computing scientists were in more urgent need of such calculational techniques because, by virtue of its mechanical interpretability, each programming language is eo ipso a formal system to start with.

Secondly, for the manipulation of uninterpreted formulae, the world of computing provided a most sympathetic environment because we are so used to it: it is what compilers and theorem provers do all the time! And, finally, when the symbol manipulation would become too labour-intensitive, computing science could provide the tools for mechanical assistance. In short, the world of computing became Leibniz's home; that it was my home as well was my luck. Most circumstances conspired to make me very fortunate. Braben, gave a grant titled "The Taming of Complexity", from which Netty van Gasteren's work could be supported; a subsequent grant for "The Streamlining of the Mathematical Argument" would enable the young British logician L.

Wallen to visit us in Austin as a "post doc". In criticizing the Cartesian formulation of the laws of motion, known as mechanics , Leibniz became, in , the founder of a new formulation, known as dynamics , which substituted kinetic energy for the conservation of movement. At the same time, beginning with the principle that light follows the path of least resistance, he believed that he could demonstrate the ordering of nature toward a final goal or cause. Gottfried Wilhelm Leibniz.

Article Media. Info Print Print. Table Of Contents. Submit Feedback. Thank you for your feedback. Alternative Title: Gottfried Wilhelm von Leibniz. Top Questions. Leibniz was born on June 21 July 1, New Style ,