Recently I participated in a discussion about modern aspects of some mathematical concepts. Although I found many new and interesting issues there I left with the feeling that we have to be modern without thinking about why: What should be the benefit of introducing a modern view of a well studied mathematical object? Is modernity a consequence of the need for better explanation of the reality or is it just another toy for us? What do you think?
There are some interesting papers on this important subject for mathematicians. Modern mathematics—in the sense the term is used by working mathematicians these days—took shape in the period from 1890 to 1930, mainly in Germany and France. Strikingly new concepts were introduced, new methods were employed, and whole new areas of specialization emerged. At the same time, the nature of mathematical truth and even the consistency of mathematics were put into question, as mathematicians, logicians and philosophers grappled with the subject’s very foundations. Before star using the term modernism, usually reserved for the changes that took place in literature and the arts during the period in question, is better to define what we are going to understand by this concept here. Modernism is defined as an autonomous body of ideas, having little or no outward reference, placing considerable emphasis on formal aspects of the work and maintaining a complicated—indeed, anxious—rather than a naïve relationship with the day-to-day world.
In the following paragraphs a brief description of the main mathematical developments occurred in the period considered are included, according to the view of Jeremy Gray in his book The Modernist Transformation of Mathematics.
The initial high points in the 19th century were the discovery of a consistent non-Euclidean plane geometry and the renewal of work on projective geometry featuring the duality between the notions of point and line. Later, on the road to abstraction, Felix Klein combined these together with Euclidean geometry under the single roof of his famous “Erlangen Program,” in which each geometry is classified according to the transformations leaving its basic notions invariant. Finally, David Hilbert’s on the study of the foundations of geometry in 1899 presented a new and more rigorous axiomatization of Euclidean geometry, for which the independence of various of the axioms—such as the parallel postulate—from the others was established by the construction in each case of a model satisfying all the axioms but the one in question. Later on, Hilbert spoke of the variety of possible interpretations of the basic notions by saying that one could replace points, lines and planes with tables, chairs and beer mugs, respectively, with corresponding relations between them. Hilbert’s view was that mathematical concepts are implicitly defined in structural terms by axiomatic systems, of which all one needs to know is that they are consistent and complete.
Incidentally, the unique ideas of Bernhard Riemann on the treatment of curvature for non-Euclidean manifolds of arbitrary finite dimension, were associated with the developments in geometry before modernism. But the nature of Riemann’s work is of quite a different character, employing as it does concepts from analysis in an essential way. Far ahead of its time, it would eventually provide the underlying mathematics for the general theory of relativity and lead to the modern subject of differential geometry.
The existence of a number of non-Euclidean geometries and the increased distancing of geometry from reality raised problems for the philosophers, especially the post-Kantians. The question was how to accommodate Immanuel Kant’s view that space and time are fixed in the intrinsic human structuring of experience and that this is what makes Euclidean geometry (and arithmetic) true. Also, according to Kant, mathematics, which proceeds by constructions in intuition, constitutes synthetic a priori knowledge. An attack on that general idea came from a different direction, in spirit going back to Gottfried Wilhelm Leibniz, via Gottlob Frege’s attempt to show that arithmetic, at least, is analytic in the philosophical sense (that is, its statements are true solely in virtue of the meaning of the concepts involved) through its reduction to logic. Although Bertrand Russell discovered a fatal contradiction in Frege’s system, his elaborate effort with Alfred North Whitehead to repair this approach to the foundations of mathematics in their Principia Mathematica proved to be enormously influential for the subsequent development of mathematical logic in the 20th century, even though it too was beset with problems.
In support of Gray modernist thesis in algebra, he takes special note of the attacks by Ernst Kummer on Fermat’s Last Theorem using factorization in certain classes of “ideal” complex numbers; this was to have a distinctive set-theoretical turn in the hands of Richard Dedekind later in the century. As another example, the abstract concept of a group originated in the work of Évariste Galois on the relations between permutations of the roots of a polynomial equation and its solvability or unsolvability by radicals. But then finite groups in general were studied for their own sake, with the investigation of all possible such systems meeting certain special conditions, such as being commutative, “simple,” solvable and so forth.
In analysis, The most important issue is the replacement by Henri Lebesgue of the concept of integration—intuitively conceived of as determining the area under a curve or the volume under a surface—with the concept of the measure of a set of points, controlled by four abstract axiomatic conditions. Another example is the introduction of concepts of distance between functions in the study of their approximation to minimal solutions of variational problems and then moved on to deal with these in terms of a general concept of metric spaces. Hardly mentioned is the combination of this idea with the algebraic concept of vector space in the modern subject of functional analysis, at the hands of John von Neumann, to an abstract mathematical framework for the interpretation of quantum mechanics.
In the first decade of the 20th century, Ernst Zermelo isolated the controversial axiom of choice needed to establish basic properties of Cantor’s transfinite numbers and showed that it had been used implicitly by mathematicians for many arguments elsewhere, especially in analysis.
Philosophically, the justification for the axiom system of set theory that Zermelo introduced requires a Platonistic account of the nature of mathematics, according to which the objects of mathematics exist in an abstract realm outside of space and time independent of human ideas and constructions; concomitantly, mathematical truths hold whether or not they can be established by human beings. This accords with mathematical practice to the extent that mathematicians believe their work is a matter of discovery and verification, not of invention.
Dear Demetris: I find your question extremely interesting. I must from the beginning state, that my view is an accumulative one, and not the rejection of the old for the new.
Now at your question: For me the whole modern era (1789-1989), is characterised by the Law of Excluded middle. Because of this everything should be black or white. This is substantiated in Cantor's Set Theory+Hilbert's program+Bourbakism. Modern mathematics are important and interesting. However, let us take e.g. Intuitionism in the form of Brouwer's original intuitionism. since proofs by contradiction is prohibited then we loose ~2/3 of mathematics. This is not harmonious with my principle of accumulation of knowledge. It is clear that intuitionism is not "modern mathematics". If however we take intuitionism as a generalisation, i.e. imagine a circle the inside of which contains objects, black or white. In this circle classical mathematics can be applied. If we extend this circle in order to contain also some "gray" objects. Then this kind of intuitionism contains "modern" classicla mathematics plus some new theorems which are impossible to prove using only classical mathematics.
Having said that, i would like to state that postmodern era is characterised by the introduction of the "middle", the gray. Thus the postmodern mathematics are characterised by the introduction of Many-valued logics, fuzzy logics, Topos theory in which the internal logic is intuitionistic, and in general non-Cantorian mathematics. Pluralism is also a characteristic which is out of modern sense. The fact that are many set theoretic universes bothers a lot the modernists.
The proponents of postmodernism unfortunately, then did not use at all mathematics in expressing their theory. So ther is a need to clear up the mess which is taken as postmodernism. I will stop here and i mght come back.
Dear Costas, thanx for a very thought provoking answer. My first reaction is why 1989 as a boundary for "modernity"? If you define modernity as post Bourbaki and fuzzy sets than
the two are more like 60s. Or did somebody introduced the word post modern at that time?
Very good question! I agree with Professor Drossos that your question is very interesting. I entirely agree that there are grey developments (partly classical, partly new) in recent development of mathematics. This is as it should be. For example, Euclid geometry is just as important nowadays as it was when it was introduced. But then, after many struggles with Euclid's postulates, especially his 5th postulate, we now have many beautiful new forms of geometry such as Riemannian geometry and differential geometry.
There are some landmark events that characterise modernity in mathematics. For me, three of these events are
1. Frechet's introduction of metric spaces in his doctoral thesis in1906. Just as in the case of Euclid, many variations and extensions of metric spaces have occurred since 1906, starting with Hausdorff (distance between a point and a set) in 1914, Cech (distance between a pair of sets) in his 1936-1939 seminar, and Lowen (approach spaces) in 1989. Hausdorff, Cech and Lowen distances have been recently extended (see papers by me and Professor Surabh Tiwari on my RG page) and my new book Topoiogy of Digital Images, Springer, 1914.
2. Hausdorff's introduction of topological spaces in 1914. Hausdorff's landmark topology has led to many extensions and variations, especially in terms of work on algebraic topology.
3. Efremovic's introduction of proximity spaces during the first part of the 1930s. Efremovic's work was not published until 1952, simultaneous with the publication of a series of important papers on proximity spaces by Smirnov. The work by Efremovic and Smirnov have led to many extensions and variations, starting with Leader's 1959 paper on clustering, Naimpally's seminal 1970 CUP book that brings together many threads in the story concerning proximity, and more recent work on descriptive proximity spaces (see my recent papers, especially my Notices of the Amer. Math. Soc. paper coauthored with Professor Naimpally in 2012).
There are some interesting papers on this important subject for mathematicians. Modern mathematics—in the sense the term is used by working mathematicians these days—took shape in the period from 1890 to 1930, mainly in Germany and France. Strikingly new concepts were introduced, new methods were employed, and whole new areas of specialization emerged. At the same time, the nature of mathematical truth and even the consistency of mathematics were put into question, as mathematicians, logicians and philosophers grappled with the subject’s very foundations. Before star using the term modernism, usually reserved for the changes that took place in literature and the arts during the period in question, is better to define what we are going to understand by this concept here. Modernism is defined as an autonomous body of ideas, having little or no outward reference, placing considerable emphasis on formal aspects of the work and maintaining a complicated—indeed, anxious—rather than a naïve relationship with the day-to-day world.
In the following paragraphs a brief description of the main mathematical developments occurred in the period considered are included, according to the view of Jeremy Gray in his book The Modernist Transformation of Mathematics.
The initial high points in the 19th century were the discovery of a consistent non-Euclidean plane geometry and the renewal of work on projective geometry featuring the duality between the notions of point and line. Later, on the road to abstraction, Felix Klein combined these together with Euclidean geometry under the single roof of his famous “Erlangen Program,” in which each geometry is classified according to the transformations leaving its basic notions invariant. Finally, David Hilbert’s on the study of the foundations of geometry in 1899 presented a new and more rigorous axiomatization of Euclidean geometry, for which the independence of various of the axioms—such as the parallel postulate—from the others was established by the construction in each case of a model satisfying all the axioms but the one in question. Later on, Hilbert spoke of the variety of possible interpretations of the basic notions by saying that one could replace points, lines and planes with tables, chairs and beer mugs, respectively, with corresponding relations between them. Hilbert’s view was that mathematical concepts are implicitly defined in structural terms by axiomatic systems, of which all one needs to know is that they are consistent and complete.
Incidentally, the unique ideas of Bernhard Riemann on the treatment of curvature for non-Euclidean manifolds of arbitrary finite dimension, were associated with the developments in geometry before modernism. But the nature of Riemann’s work is of quite a different character, employing as it does concepts from analysis in an essential way. Far ahead of its time, it would eventually provide the underlying mathematics for the general theory of relativity and lead to the modern subject of differential geometry.
The existence of a number of non-Euclidean geometries and the increased distancing of geometry from reality raised problems for the philosophers, especially the post-Kantians. The question was how to accommodate Immanuel Kant’s view that space and time are fixed in the intrinsic human structuring of experience and that this is what makes Euclidean geometry (and arithmetic) true. Also, according to Kant, mathematics, which proceeds by constructions in intuition, constitutes synthetic a priori knowledge. An attack on that general idea came from a different direction, in spirit going back to Gottfried Wilhelm Leibniz, via Gottlob Frege’s attempt to show that arithmetic, at least, is analytic in the philosophical sense (that is, its statements are true solely in virtue of the meaning of the concepts involved) through its reduction to logic. Although Bertrand Russell discovered a fatal contradiction in Frege’s system, his elaborate effort with Alfred North Whitehead to repair this approach to the foundations of mathematics in their Principia Mathematica proved to be enormously influential for the subsequent development of mathematical logic in the 20th century, even though it too was beset with problems.
In support of Gray modernist thesis in algebra, he takes special note of the attacks by Ernst Kummer on Fermat’s Last Theorem using factorization in certain classes of “ideal” complex numbers; this was to have a distinctive set-theoretical turn in the hands of Richard Dedekind later in the century. As another example, the abstract concept of a group originated in the work of Évariste Galois on the relations between permutations of the roots of a polynomial equation and its solvability or unsolvability by radicals. But then finite groups in general were studied for their own sake, with the investigation of all possible such systems meeting certain special conditions, such as being commutative, “simple,” solvable and so forth.
In analysis, The most important issue is the replacement by Henri Lebesgue of the concept of integration—intuitively conceived of as determining the area under a curve or the volume under a surface—with the concept of the measure of a set of points, controlled by four abstract axiomatic conditions. Another example is the introduction of concepts of distance between functions in the study of their approximation to minimal solutions of variational problems and then moved on to deal with these in terms of a general concept of metric spaces. Hardly mentioned is the combination of this idea with the algebraic concept of vector space in the modern subject of functional analysis, at the hands of John von Neumann, to an abstract mathematical framework for the interpretation of quantum mechanics.
In the first decade of the 20th century, Ernst Zermelo isolated the controversial axiom of choice needed to establish basic properties of Cantor’s transfinite numbers and showed that it had been used implicitly by mathematicians for many arguments elsewhere, especially in analysis.
Philosophically, the justification for the axiom system of set theory that Zermelo introduced requires a Platonistic account of the nature of mathematics, according to which the objects of mathematics exist in an abstract realm outside of space and time independent of human ideas and constructions; concomitantly, mathematical truths hold whether or not they can be established by human beings. This accords with mathematical practice to the extent that mathematicians believe their work is a matter of discovery and verification, not of invention.
In the name of modernizing mathematics, we must not forget that physical significance is needed to be associated with it. To explain a reality mathematically, one must not bring in an axiom that does not have any physical significance.
Jorge,
Excellent, thoughtful, interesting, profound post! The issue whether mathematics is either discovered (Platonist view) or invented (intuitionist view) or both is unresolved.
Consider, for example, Einstein's observation: "Invention occurs here as a constructive act. This does not, therefore, constitute what is essentially original in the matter, but the creation of a method of thought to arrive at a logically coherent system… the really valuable factor is intuition!"
For more about this, see the attached pdf file.
@Patrick Solé : Dear Patrick, the period 1789-1989, is of course arbitrary, but I think that includes the period of classical capitalism. after 1989, the fall of the Berlin wall, we essentially enter into globalisation and information society. The boundaries of the periods are very vague. For example artists are talking about postmodernism since forties. I like your term "post-Bourbaki". But let take as an example "Category Theory". Neither Eilenberg nor MacLane construe Category Theory as an example of Non-Cantorian mathematics. The down to earth understanding of this theory took some decades. Now we observe that there are a lot of applications of the theory to Psychology, neuroscience, etc. Especially in mathematics took a long until the starting of this thinking. I want also to note that although probability theory has "gray objects", it is not considered as postmodern, since the Law of Excluded Middle holds with probability one! (P(A U \neg A)=1).
You can download the book by V. Tacic Mathematics and the Roots of Postmodern Thought from
http://m.friendfeed-media.com/4b9836b6e823ae59b15bf2afa4fc1530556a1c40
I attached some papers that are helpful As for the description of modern mathematics, the article by Atiyah: Mathematics In 20th' Century is perfect source.
Jorge Pedraza's post is really good. At the risk of over-simplifying, what is "modern" in "modern mathematics" is its abstraction, and focus on structural relationships. Category theory has to be the pinnacle of this. But most recent mathematics tends to be rather more "post-Bourbaki" than "modern" in this sense. For example, there seems (to me) to be more emphasis on problem solving rather than theory/structure building. The recent developments in number theory (e.g., Yitang Zhang's work on separation of primes) and topology (e.g., Grigory Perelman's proof of the Poincare conjecture in dimension 3) have involved more classical analysis ideas than structural mathematics. (I'm not sure where Andrew Wile's proof of Fermat's Last Theorem stands in this sense.) But there does seem to be more of a problem-solving sense to more recent mathematical developments.
Demetris, fellows, thank you for the very interesting question and answers!
I think , in particular, that Jorge's and David's points suggest something interesting: MODERN can have the meanings of both making reference to modernity, and to "modern math" (whichcame to be taught even in schools for the last decades or so).
Costas: I like your points about post-modernity. It presents interesting possibilities for RE-constructing math (as in Tasic, posted by you), and (re-)thinking the history and the philosophy of math, as well as math's relations to other sciences and, in general, knowledge. From an applied mathematician's point of view it is the relation of math to other sciences and, in general, to human cognitive capabilities that seems the most fascinating aspect of discussing postmodernity and math.
A side-comment on the concept of post-modernity: Tasic seems quite right in starting with the idea that no one really knows what postmodernity is. It is possible though - again, from an applied-instrumentalist point of view - that a good starting point might be to admit Lyotard's doubts about something like total (scientific?) knowledge (La condition postmoderne, Rapport sur le savoir, 1979). I think this may be instrumental not as a TOTAL relativizing program (which might be incompatible with our profession), but as a way to open doors to a variety of hypotheses and perspectives (which is perhaps the essence of our method). Of course, Lyotard's "no to totality" is a well-known, if not consensual, starting point; I just wanted to comment on its instrumental aspects.
@Demetris, if Your thread was about modern control theory, I would response with large answer. But, as we are speaking in term of modernity in mathematics, I find this book "A History of Mathematics: From Mesopotamia to Modernity" very appropriate for many reasons! There are many examples on modernity issues, such as Eurocentrism dealing with Greek paradigm and, later on, modern Western mathematics. Nice reading!
http://books.google.rs/books?hl=en&lr=&id=nSO5iMujRUYC&oi=fnd&pg=PR13&ots=8G5u2pz4Ta&sig=dM0dkn5z4i89eyAwQc27U6bIKC0&redir_esc=y#v=onepage&q&f=false
I think that the level of answers is one of the highest I have ever seen during the small time that I am participating in RG discussions. I, honestly, cannot distinguish between the contributors scientists, everybody has given an extensive view.
Some key points, as I have understood till now:
1)From binary logic (black & white, 0 or 1) we are moving to gray zone logic, whatever is the name of it. I have a question for this fact: "Does this modern view came due to the quantum mechanical interpretation of modern Physics or it was just a self evolutionary journey of Maths alone?"
2)I can explain the Riemann's formulation of Differential Geometry, based on Calculus that was introduced by Newton & Leibniz. What is the motivation for the modern thoughts that have been described since now in this thread? Can we wait that such a change in paradigm, such a shift, will increase the accuracy of measurements in Applied Mathematical Theories or it is a matter of 'internal completeness' only?
In my opinion the answer to the first question is that the modern view is self evolutionary of the own Mathematics science. This view can also be influenced by the need to support the development of modern Physics, particularly the need to demonstrate analytically the outcome of some experimental research in the field of Quantum Mechanical, among others.
Regarding your second questions, I have the opinion that the Riemann's formulation of Differential Geometry can be explained using Calculus but also with the use of some other modern tools such as the Theory of Groups, Geodesics, covariant derivates, among others.
The modern view of Mathematics of course has a positive impact in the development of the Mathematical Science, including Applied Mathematics.
Finally, the real question of Demetris was about modernity in mathematics. The Law of the excluded Middle, the rationalism, and the admiration of science, was characteristics of modern era in general. However those characteristics hold true fοr mathematics at least since Euclid. Thus there is a need to specialise the meaning of the term "modern" for mathematics. I insist that the modern era of mathematics started with Cantor’s introduction of Set Theory and Dedekind’s introduction of structuralism. Then we have Hilbert and finally Bourbaki.
Modern thinking has its roots to Enlightenment: “It has been claimed that the entire project of the Enlightenment had as its goal achieving the clarity of mathematics everywhere by employing the method known as "analytic thinking,” whose origins are traceable to mathematics.” (Tacic). Whereas postmodernism has its roots to Romanticism: “I have in mind nineteenth-century romanticism. Its philosophical contributions were, for the most part, separate from mathematics and were opposed to the ideal of formal reasoning that mathematics represented. Romanticist
rebellion, sometimes called "the counter-enlightenment," is known for its critiques of science and reason.” (Tacic)
In mathematics the deconstruction of modernism started with Goedel’s Incompleteness Theorems, that seems to introduced the “middle”. If a statement is undecided in two-valued logic, it might be something like 0< p
I think that I agree with the critical point that Goedel’s Incompleteness Theorems played for the creation of modernity in Mathematics. As for the Physics, we are not sure that Heisenberg's view is such a critical one (although it is over-promoting), since it is a direct corollary of Fourier Analysis for the matter-energy waves. Probably de Broglie was the pioneer of the new era in Physics.
Demetris,
concerning the enquire you made yesterday - "'Can we wait that such a change in PARADIGM, such a SHIFT, will increase the ACCURACY of measurements in Applied Mathematical Theories or it is a matter of 'internal completeness' only?" - allow me to make a slightly off-topic comment: we may have been living a paradigm shift affecting all natural sciences and math at least since the mid XIX c., which may have been epitomized by Prigogine's proclamation of the "end of certainty" a couple of decades ago, and is reflected in most of the previous postings. It may seem weird to conceive of a paradigm SHIFT affecting all natural sciences, However, if we conceive of determininsm as a grand, overall 'paradigm' shared by all natural sciences, we may talk about a paradigm shift, which has to do with Costas' idea of a grey area, I think. Of course, this end of determinism has been cooked up for a long period, and, still, it is not consensual in the entire scientific field. Yet, many problems like the n-body problem, QM, instability, complex systems, are real scientific problems that distance themselves from modern deterministic 'believes'. Maybe this can help situate the question concerning math.
What seems really amazing, and that is also considered in your yesterday's enquire, is that the entire edifice of "normal" science may have hugely benefited from an enormous progress in technical means to "increase accuracy in measurements". So "normal" science is perhaps doing well in many fields; yet, with the advent of instable and complex systems research, and 'complex' cross-disciplinary problems, we may have effected an overall SHIFT, if not in the limits of most "normal" specialities, in the way we see our relationship to explaining the world.
Sorry, this was long for a 'slightly off-topic comment' :-(
Dear Diogenes,
There aren't off topic comments in such general discussions. The same opinion has also me, I agree with Costas' definition for the gray area. I don't know if this is a 'mortal injure' to the determinism or it just a redefinition of it: We could be in principle arrange the areas where determinism holds, the areas where gray logic or fuzzy logic or ... logic holds and finally the full chaotic cases (without even a strange attractor).
As I have learnt from the interconnection between Mathematics & Physics, although each one states that it has a lonely journey, without any influences from the other side, the fact is that science is a holistic object:
Επίσταμαι: Ίσταμαι επί--> I know everything (Epistamai: Istame epi)
Aristotles was one of the first scientists and his paradigm followed many modern scientists like Newton.
So, the demand for a practical benefit from the modern turning in Maths is not a useless one and that's one of the reasons why I asked the question.
But, I am not such an expert in History of Science, so every opinion is welcome.
Being a conservator in my nature, I always protest against any change prima facie. However, mathematics is a tool for development of many sciences. Therefore, the progress in sciences demands development of new and perfection of the existing tools, i.e. mathematical methods. Remember, please, how informatics has changed the mathematical world. Social and economic problems have perfected statistic and probabilistic analysis (the chaos theory has arisen). I believe that astrophysical problems will lead to perfection and further development of topology methods, especially, perhaps, of multidimensional topology.
GK. - allow me to say that, for a conservatist (?) you presented us quite a dialectical, non-platonian view of mathematics. I guess I share that view with you, and I wish it could help widen both the field of applications and the philosophy of math.
To Diogenes Alves:
Thanks, indeed, for your interest to my humble opinion. My conservatism is conditioned, e.g., by my impression of “evolution” of the school course of mathematics (in USSR-Russia). I remember how I liked the classical courses of mathematics in my school years. Further, during a short time of pedagogical practice in University, I have paid attention that mat courses became worse, less determined, and less clear. At last, during my son school years the mat courses became incoherent in full (by my opinion).
Because I am physicist, my dialectics is conditioned by new and new fields of higher mathematics which are developed or even arise when they are demanded by other fields of science. For instance, new geometries of Minkowski and Friedmann, theory of symmetries (Wigner), information theory and algorithmic complexity theory (Kolmogorov), theory of chaos (Poincaré, Kolmogorov, Arnold, …), attractor of Lorenz, dissipative structures (I. Prigogine), …etc.) I believe, that any mathematician could find much more examples of new fields of mathematics, and, all the more, one could remember further perfections in theory of groups, vector analysis, matrix and tensor calculus, theory of differential equations, statistic analysis (Monte Carlo method!), topology, …etc.
Unfortunately, I don’t know well the history of mathematics, and especially, its philosophy. I would be very grateful if you’ll share your knowledge with me.
G.K. (does G. stand for Galina ?): I think you have the privilege to be from Kolmogorov's, and Arnold's (and Lobotchewski and Tsialkowsky ...) country with all its solid math and phys-math tradition. Pls do not take this as a superficial compliment, but, rather, as a bowing reference to a long history, with all its up and downs.
In relation to your question, perhaps the journal Historia Mathematica would provide a couple of good reviews to start with. If you are not familiar with it, may I suggest that you download Aubin & Dalmedico's "Writing the history of dynamical systems and chaos" from HM 29th (2002) volume?
Best, D.
Thank you, Diogenes, I have downloaded successively that interesting paper. I hope to overcome it sometime.
My acquaintance with chaos based on several fine books of Prigogine (fortunately in Russian):
Prigogine Ilya (1980). From Being To Becoming.
Prigogine I.; Stengers, I. (1984). Order out of Chaos: Man's new dialogue with nature.
Prigogine I.; Stengers I. (1997) The End of Certainty: Time, Chaos and the New Laws of Nature.
It would be very attractive to work in the field, but many other problems are very interesting too.
Galina
I agree, Galina, chaos and catastrophes are fun, but so are many other things.
Otherwise, what I think was great in Prigogine, if only I understand him right, was his inclination to look at uncertainty instead of chaos or catastrophes; perhaps this leaves us room to face contingence without fatalism, I'm not sure, but I hope so. This has do to with the meaning of modernity for math and in math, too, I think :)
Greetings from the turbulent tropics.
As Physics is evolving Mathematics are forced to evolve also in order to serve new or modified old theories for explaining new phenomena. So, us our approach of our universe is getting more complicated, new theories have to arise, otherwise we end up with 'paradoxes' and other 'exceptions from the rule', where we are just lazy and don't want to change our mathematical view. We can argue that the time series of new math theories are co-evolute with the time series of new and unexplained physical experiments. But, the main problem now in Physics is the domination of quantum mechanics interpretation based on discrete particle assumption', although there exist experiments like the 'double slit' which strongly indicate the wave nature of material world. Here, the maths exist but the inertia of Physicists has an enormous value...
Thus the number one is not the old or modern view, but the overcoming of scientific egotism.
Modern mathematics began when Bool showed that formal logics is isomorph to set theory. Before, even mathematical reasoning like a syllogism were surrounded by a metaphysical haze. For Euclid, his postulate couldn't be an axiom or a theorem, but should be an absolute truth, like in natural philosophy. Bool algebra was also the begining of computer science, it all has been a real revolution in the history of thought.
Dear Claude, have you ever been wondering why almost all important concepts in our scientific life can be explained form a little set of Algebra, like isomorphisms? I always wonder about such fact...
Dear @Claude, the most of my work is related to the application of Boolean algebra in the field of automation! I like your point of view regarding the influence of Boolean algebra!
@Demetris, I do like isomorphisms, which I have applied a lot in graph theory and applied graphs! For example, in the field of system simplification!
I think that even Plato would agree today with us that there exists an isomorphsim between the World of Ideas and our real world!
Dear Demetris,
Try to define the two terms you are using: "the world of Ideas", and "the Real world". The try to define what are the structures on the two entities, and if they are of the same type, you may be talk about "isomorphism"! These might be impossible!
Dear Costas, to be more accurate, the isomorphism is only for a small subset of the World of Ideas, see Figure 2 of next article:
https://www.researchgate.net/publication/262817990_The_Line_The_Cave_from_The_Republic_of_Plato_and_the_The_World_of_Universals_from_Bertrand_Russell?ev=prf_pub
where you have to compare the upper line with the lower one.
PS I have sent you an email at your university address, did you received it?
Thank you.
Article The Line, The Cave from The Republic of Plato and the The Wo...
When we talk about "modern aspects," that means that the link with the real world is lost. Arithmetics is no longer counting real objects, but studying the properties of a set defined by the axioms of Peano. Then a function from the set of sets to this set is defined and it is proven that it is unique. Geometry is no longer measuring the earth, but studying the transformations of a space (isometries) that preserve a function of the cartesian square of this space to the real line that has certain properties. Isomorphisms exist only between mathematical objects, which are most of the time collected into equivalence classes.
The real world isn't written in the language of mathematics like Galileo believed, but mathematics is the essential tool of physics. It is a idealisation using the notion of infinity that doesn't exist in the real world, so mathematical models are only as good as possible approximations. Theories that are mathematically incompatible like quantum mechanics and general relativity are nonetheless valid in their respective domain of application.
Dear Claude, I don't know if the real world is written in some language, probably it does not need something like this. We are those who need a tool (a language) to explain and if possible to predict the real world. For the time I cannot find any other reliable tool, except Mathematics.
Dear Demetris, I have not receive any email of you! Send it again, use this time
To Dimitris and Costas
I think that meet each other quite often about the same issues. To keep it simple you can just have a look to my lattest paper, it is short.and addresses the questions of what is science, and if there are common structures in scientific theories.
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