European Mathematical Society - 03-02
https://euro-math-soc.eu/msc-full/03-02
enComputation, Proof, Machine. Mathematics Enters a New Age
https://euro-math-soc.eu/review/computation-proof-machine-mathematics-enters-new-age
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a translation of <em>Les Métamorphoses du calcul</em> (2007) that was awarded the <em>The Grand Prix de Philosphie de l'Académie Française</em>. The main idea is to argue that computation seems to supersede proof by reasoning as a fundamental element in the building of mathematics. These two approaches are placed in their historical relationship with respect to each other. The Greek developed proof by reasoning where computation failed. Nowadays, we seem to evolve in the opposite direction.</p>
<p>
Ever since the Greek laid the foundations of our "Western" mathematics, based on axioms and proving theorems by logical inference, not much changed fundamentally until these foundations were scrutinized and revolutionized in the twentieth century. Computation has been considered as boring and of lesser importance that is happily left to machines. However when computers are being used for proving theorems like was the case with the <em>Four Color Problem</em>, computing starts to be competing with the traditional proof methods. Where the traditional methods set a bound on the length of what is a feasible proof (Wiles' proof of Fermat's last theorem needed over 150 pages), computing may extend considerably what can be proved. It is the thesis of the author that computing will revolutionize mathematics in the coming decades.</p>
<p>
In a first historical part, it is shown how the Greek had their proofs by reasoning, but computation had also old roots in Mesopotamia and of course we have the Euclidean algorithm, Thales computed the height of pyramids and much later calculus entered prominently as a computational tool, and even invaded geometry after Descartes.</p>
<p>
But it is the second part in which the mathematical (r)evolution of the 20th century is sketched that is at the core of the book. Dowek first discusses predicate logic started off by Frege who wanted to define the integers via set theory. As we know, this led to contradictions, so that it had to be revised by Russell and Whitehead. They were the ones who introduced types. These developments were influential on the role of computing in the first half of the 20th century. Early in that century two related theories developed independently: computability and constructibility. To understand the role of computability, consider the Euclidean algorithm. If it computes <em>z</em> as the greatest common divider of <em>x</em> and <em>y</em> then the algorithm enables us to decide that the proposition "<em>z</em> is the gcd of <em>x</em> and <em>y</em>" is true. So the algorithm becomes a way of proving propositions. The next step is to wonder whether any mathematical proposition can be decided, i.e. "proved" by an algorithm with reasonable restrictions? In other words: can any function <em>f(x)</em> be computed in finite time and with finite density of information? Now algorithms and not numbers became the subject of computation. So we arrive at Hilbert's decision problem. It was answered independently by Church and Turing. Church used lambda calculus and Turing his Turing machines. They both found that there is no algorithm to decide whether a proposition is provable in predicate logic. There were alternative instruments to come to this result formulated by Gödel or by Kleene who used recursive functions, all of them equivalent as was shown later. So it became clear that reasoning was not just a method to uncover results that were already implicit in the axioms. Constructivism is inherent in an algorithm since it gives the results explicitly, but non-constructive proofs can be accepted too if it can be shown that some result exists even if we do not know it explicitly. For example if the assumption of non-existence leads to a contradiction. This however relies on a principle of excluded middle: either <em>A</em> or not <em>A</em>. This principle was what Brouwer, as a radical intuitionist, wanted to get rid of, but rejecting that principle of course endangered many results in mathematics. The conflict between constructive and non-constructive lies in the interpretation of "there exists". For the first it means that "it exists and we know it" for the second "it exists even if we do not known it". Dowek elaborates on Church's thesis and lambda calculus in separate chapters and on the constructivism of Brouwer and how it influenced the design of algorithms.</p>
<p>
The third part culminates in Dowek's own thesis: the crisis of the axiomatic method and the dawning supremacy of computation. In the 1970's automatic computer programs for theorem proving were conceived stimulating the fantasy of science fiction authors about machines domination humanity. But disappointing results moved developers to less ambitious proof checking algorithms and correctness proofs of programs or circuits. Also symbolic computation and computer algebra systems entered mathematical research and there was the proof by Appel and Haken of the four color problem in 1976. They had reduced the problem to an analysis of a map with ten regions only. This still required so many cases that checking all of them was done by a computer program. There was a big controversy whether or not to accept this as a proof. Although the program has been verified for correctness by two automatic systems, one can never be sure that there is not a minor flaw left undetected of whether the verifying algorithms themselves were flawless. Time to rethink our concept of proof as more computer assisted proofs started to appear since the 1990's. Anyway since proofs become more complex, in fact too complex and too long to be written out in full requiring hundreds or thousands of papers that no one can ever read or check, one might be forced to accept computer proofs. At least if no shorter proofs exist. Which then triggers the question whether it is provable that no short proof exists fort some problem. Like instruments have long ago changed the development of physical sciences, it is time to acknowledge that computers can and should be used as instruments to do experiments within mathematics, experiments which, if not proving things, can at least show the way or help formulating hypotheses, a practice that is well accepted in natural sciences. Although of course experimenst alone will never replace proofs.</p>
<p>
Dowek ends his book with a program for further research. To name just a few open research questions. Can we show that a problem does not have a short axiomatic proof? Can we replace all axioms by computation rules? If not, in which cases? How much of the efficiency of mathematics, when applied to the physical sciences, can be explained by the Church-Turing thesis? Which branches of mathematics will profit more from the use of computers? What will be the impact of computers on mathematical writing?</p>
<p>
This is not easy reading, but Dowek has done a tremendous job in explaining all the theories, logical and philosophical approaches in an accessible way. The technical details are left out but simple examples such as the Euclidean algorithm do make clear where the, sometimes subtle, differences lay, which may however have far reaching consequences. Whether or not computation will eventually eliminate axioms is still undecided, but from observation, it is clear that computation takes up an ever growing part of the job of a mathematician. If you, as a mathematician, are concerned about the foundations of what is your daily occupation, this is a book you should read, even though it might not influence your own research topic. If your subject is logic or theoretical computer science or philosophy of mathematics, this book is a must read, although you probably may be familiar with it already since it has been around in French for about 8 years now.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a translation of the prize winning <em>Les Métamorphoses du calcul</em> (2007). The main idea is to argue that computation seems to supersede proof by reasoning as a fundamental element in the building of mathematics. Some philosophical thoughts about logic and the foundations of mathematics.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/gilles-dowek" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Gilles Dowek</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/cambridge-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">cambridge university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-0-521-11801-9 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">120 USD</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">160</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/logic-and-foundations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Logic and Foundations</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.cambridge.org/us/academic/subjects/computer-science/computing-general-interest/computation-proof-machine-mathematics-" title="Link to web page">http://www.cambridge.org/us/academic/subjects/computer-science/computing-general-interest/computation-proof-machine-mathematics-</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/03-mathematical-logic-and-foundations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03 Mathematical logic and foundations</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/03-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03-02</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/03fxx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03Fxx</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/03dxx" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03Dxx</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/03b15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03B15</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A30</a></li></ul></span>Mon, 17 Aug 2015 10:06:15 +0000Adhemar Bultheel46360 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/computation-proof-machine-mathematics-enters-new-age#commentsQualitative computing. A Computational Journey into Nonlinearity
https://euro-math-soc.eu/review/qualitative-computing-computational-journey-nonlinearity
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In an historical perspective, the author recalls how <em>paradoxes</em> were the impetus for leaps of improvement in the evolution of mathematics. The $\sqrt{2}$ introduced the irrationals beyond the rationals, 0 was the missing link between positive and negative numbers, $\sqrt{-1}$ introduced the complex numbers beyond the reals, $\infty$ allowed to study divergent series, and the quaternions $\mathbb{H}$ (for Hamilton) related to the space-time computations of relativity theory. The latter can be extended to octonions $\mathbb{G}$ (for Graves) and this is the start of a sequence of Dickson algebras: $\mathbb{R}\subset\mathbb{C}\subset\mathbb{H}\subset\mathbb{G}\subset\cdots$, or more generally a sequence $A_k=A_{k-1}\times A_{k-1}$ of algebras equipped with a recursively multiplication. This definition doubles the dimension in each step so that $A_k$ has dimension $2^k$ (Dickson 1919). As $k$ increases, more and more classical properties of the multiplication are lost: the square may be negative in $A_1=\mathbb{C}$, commutativity is lost in $A_2=\mathbb{H}$, associativity is lost in $A_3=\mathbb{G}$, zero divisors occur, etc. The sequence of Dickson algebras is what Chatelin calls <em>Numberland</em> where hypercomputation takes place. Working in $\mathbb{R}$ is what Chatelin calls <em>thought</em> or one-dimensional thinking, but moving to $\mathbb{C}$, this becomes <em>intuition</em> or two-dimensional thinking. Together with $\infty$ they form <em>Reason</em> $ = \{\mathbb{R},\mathbb{C},\infty\}$.</p>
<p>
The first chapters explore the calculus, i.e., all the computational rules in $A_k$. Leaving the strict computational conventions of familiar grounds gives the freedom to choose on how to define or compute things. Thus loosing properties for increasing k means opening up for many more possibilities. For example classical causality is based on the ordering in $\mathbb{R}$. A new linear concept of causality or derivability is given via a particular linear map (a <em>derivation</em>) in a Dickson algebra and the nonlinear core of the Dickson algebras is the part that is out of reach of all possible derivations.</p>
<p>
The next chapter explores the norm and the singular values decomposition of the multiplication maps $L_x$ and $R_x$, i.e. left or right multiply with an element $x\in A_k$. The different possible definitions of a norm in the Dickson algebras give rise to different geometries. Complexification of a Dickson algebra is the generalisation of $\mathbb{C}=\mathbb{R}+i\mathbb{R}$, i.e. $A_k=A_{k-1}\times 1 \oplus A_{k-1}\times\tilde{1}_k$ where $\tilde{1}_k$ is the hypercomplex unit of $A_k$. It is illustrated and related to the dynamics of Verhulst's logistic equation.</p>
<p>
The Dickson algebras have a dimension that is a power of 2. For the algebra for which the dimension is not a power of 2, one needs to resort to addition instead of multiplication. As an application the spectrum of the perturbed matrix $A(t)=A+tE$ is investigated for varying $t\in\mathbb{C}$.</p>
<p>
When Dickson algebras are defined over the integers or in $\mathbb{Z}_r$ (in particular r = 2), several possible applications open up like number theoretic problems, floating point representation (the probability of the first digit in the representation, known as the <em>Borel-Newcomb paradox</em>), <em>Sharkovski's theorem</em> and the ordering of the natural numbers, etc.</p>
<p>
More number theoretic applications are possible in the first four of the Dickson division algebras mentioned above ($A_0=\mathbb{R},\ldots,A_3 \mathbb{G}$), because they have no other zerodivisor than zero. When these algebras are considered as rings (addition and multiplication), then as an application, number theoretic theorems of (2, 4, and 8) squares can be analysed, i.e., which natural numbers can be written as a sum of 2,4, or 8 squares. This results in a quest for possibilities of 8-dimensional arithmetic.</p>
<p>
Besides the discrete/continuous dichotomy, there is also the real/complex dichotomy. How these different dichotomies interact in computation is illustrated in the next chapter which analyses two applications. The first is about the relativity of the concept of inclusion. Think of fuzzy sets, but also about the dynamics of chaotic systems. The second one is about Fourier analysis and complex signals.</p>
<p>
The computation of e.g. an SVD, which we know as a concept in linear algebra, leads to paradoxes when it is applied in a nonlinear environment of nonassociative Dickson algebras ($k \ge 3$). Classical logic is deductive and tries to avoid any paradox (Russel, Turing). Chatelin however sees these paradoxes as an opportunity to leave the classical deductive logic and escape to a more organic logic. That is a logic that allows to reason about hypercomputing. An alternative (organic) representation of complex numbers and higher dimensional complex vectors is given and it is illustrated how these are used in computation.</p>
<p>
The concluding chapter is about Euler's $\eta$ function. This is explored as a tool to give weight, or meaning, or depth to hypercomplex numbers, or as Chatelin calls it, <em>organic intelligence</em>.</p>
<p>
This is a book unlike anything I have read before. The potential reader who is looking for philosophical aspects should be warned that there are hard mathematics involved, but the mathematician should be warned as well, that he/she should be willing to abandon familiar grounds and follow the ideas and philosophy behind the mathematical exposition. This book is almost a paradox in itself. The reader is guided around some of the phenomena at the boundaries of Numberland which is much like an experience Alice must have had when she explored Wonderland. I do not think the book will become the computational bible of the future, but as an exercise in out-of-the-box-thinking it has overwhelmingly succeeded. It is far from giving a solution to all problems posed by nonlinear computational problems. It is not even giving a definitive solution to the most elementary partial problems. As Chatelin writes herself, the right choice to make among the many possible choices that can be made in higher dimensional Dickson algebras, can only be validated by experience.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">A. Bultheel</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">KU Leuven</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
With this book, the author wants to sketch a framework to lift logic and computation beyond the traditional framework of the 20th century. This entails hypercomputing beyond linear algebra, which means the introduction of calculus in Dickson algebras, which is a nested sequence of algebras $A_k$ of dimension $2^k$ where traditional properties of multiplication are gradually given up as k increases (commutativity, associativity, alternativity, etc.) but providing much more possible choices that can be made and hence opening up a wide unexplored area of new paradigms for computing. Also the traditional logic based on the sequence of natural numbers is left for a new organic logic. The book is very algebraic, but at the same time it includes many epistemological sections, it is philosophical, treats aspects of logic, and sketches the historical evolution of the ideas.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/fran%C3%A7oise-chatelin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">françoise chatelin</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/world-scientific" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">world scientific</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2012</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-981-4322-92-8</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">127 £ (net)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">600</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.worldscientific.com/worldscibooks/10.1142/7904" title="Link to web page">http://www.worldscientific.com/worldscibooks/10.1142/7904</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/03-mathematical-logic-and-foundations" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03 Mathematical logic and foundations</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/03-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">03-02</a></li></ul></span>Wed, 21 Nov 2012 09:10:22 +0000Adhemar Bultheel45470 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/qualitative-computing-computational-journey-nonlinearity#comments