-Up to now we usually use the classical mathematics the origin of which is at the end of the 19th century and/or at the beginning of the 20th century. Even the contemporary quantum physics, astrophysics, and AI of the 21st century are still using that classical mathematics! In von Neumann's quantum mathematics there is no any anomaly whatsoever in Thomas Kuhn's 'The Structure of Scientific Revolutions': why?
-Thanks for your answers! Marc
These days we can evaluate integrals and even solve differential equations using Monte Carlo simulation techniques with going into the details of the form of the integrals and differential equations. This is also a form of non-classical mathematics.
Thank you, B.K., for your added answer. However, are those 'evaluation of integrals and solution of differential equations using Monte Carlo simulation techniques... etc.' really not based upon the classical mathematics? You mention: "This is also (cursive is mine) a form of non-classical mathematics." My further question: What are exactly the features of the non-classical mathematics? Please your answer, if possible in Carnapian sort of 'explication'. Thanks, Marc
To know where something is (like your "non classical mathematics"), you must first define precisely what it is and I don't get it from your question...
Could you elaborate more on what this "non classical mathematics" should be?
Thank you, Bruno, for your added answer. This answer implies two questions:
[1] to give a precise definition of the "non classical mathematics".
Let me immediately answer this question: How can I give you a precise definition for something that is unknown to me? That is exactly the reason of my question! (see the description). A more simple description might be as follows: If there is non-classical physics (quantum physics and Einsteinian astrophysics), then where is non-classical mathematics?
[2] to elaborate more on what this "non classical mathematics" should be. Again, see [1]. Thanks anyway for your time...
" How can I give you a precise definition for something that is unknown to me? That is exactly the reason of my question!"
If that is true, you should have formulated your question more like: "Does the equivalent of non-classical physics exists in mathematics?" By asking WHERE is the non-classical mathematics, you are implying, voluntarily or not, that it already exists...
Even if physicists recognize a somehow fuzzy frontier between classical and modern physics, why would that imply that the same exists for mathematics?
Also, classical and modern (i.e. non-classical) distinction in physics is more contextual and historical than a clear fact. See:
https://en.wikipedia.org/wiki/Classical_physics
Thank you, Bruno, for your nice answer.The only thing I can do for you is: please read the description (of my question) and my added answer more carefully.
Dear Marc ~
When I think of the mathematics of “classical” physics (hydrodynamics, properties of materials, Maxwell’s electromagnetism, Einstein’s gravitational theory, etc) I see that it is predominantly based on the concept of continuity. Space and time are thought of as continuous variables and physical phenomena are desribed by continuous “fields”. The appropriate mathematical tools are differential equations. “Discreteness” rather than continuity entered physics with Planck’s “quantum” concept, which led to the “non-classical” physics of quantum theory. By analogy, I would identify “classical” mathematics as the mathematics of continuity; "non-classical" mathematics would then be the mathematical study of discrete structures. But those branches of mathematics already exist, so I admit to being rather puzzled by the question "Where is the 'non-classical mathematics'?"
Dear Marc Cavallo,
Having read your question and the answers, I found many aspects, and perhaps you could specify more exactly.
If non-classical means so new that it is unknown, then there is need for definitions and postulates within disciplines; linguistic and mathematical.
One of the answer suggests the smallest volume of the universe, as a task. I mean, that the texture of such a part would be more realistic to define, and a possible start would be to practise by applying mathematics to a cell.
It is very hard to distinguish between classical and non classical mathematics.
Even for continuum and discrete opposition question, there is an ancient logical binding.
In a square it is impossibile to allineate a natural number of atoms on the diagonal and an other natural number of atoms on the side of a square and calculate thei ratio. So mathematics doesn’t describe nature or it doesn’t exist reality.
Godel demonstrated that mathematics is a experimental science ; so it underlies to Yin Yang dynamics.
In every classical aspect of mathematics, like continuous, there is something of non classical and viceversa.
And every aspect is classical for something but non classical for something else
https://www.researchgate.net/project/Mathematical-Concept-of-B-numbers-and-their-Representations-in-Rigged-Hilbert-Space
is an attempt to set up a new type of 'fuzzy' numbers which can be used for formulating physical theories.
The basic idea is to consider sequences of rational numbers which have several accumulation points which are located in a small interval, and to define equivalence classes of such particular sequences, and to define arithmetic operations for these equivalence classes. The distribution of the accumulation points of such a sequence is described by compressed Gauss-like functions, to which a complex-valued phase factor is attached which is specified by the number of accumulation points of the sequence. Thereby, mappings are defined from the set of equivalence classes, i.e. the set of the so-called "B-numbers", into the set of complex-valued square-integrable functions and generalized functions, i.e. a separable Hilbert space and it’s nuclear companion, i.e. a rigged Hilbert space.
Dear all,
Since my added answer to Bruno Martin, a couple of days ago, I never expect such an avalanche of reactions from all of you. Thank you mathematicians! Let me start sequentially with:
[1] Eric Lord, Dear Eric,
(a) continuity and continous are for me rather vague and ambigous concepts. Mathematics (I mean the 'classical' one) is by definition based upon discreteness! Both, geometry and arithmetic, are discrete. In other words, without discreteness there is no mathematics!
(b) even in quantum physics there is, according to me, only particle (which can be numbered) not wave (which cannot be numbered; see also my previous question: what if singularity is wave?). Maybe there is even no 'particle-wave duality! Because the nature of particle is solid and that of wave is fluid! So, essentially different! Thanks anyhow for your feedback!
[2] Lena J.T. Stromberg, Dear Lena,
(a) the smallest volume of the universe is, according to me, the singularity. See above [1] sub (b).
[3] Francesco Marchetti, Dear Francesco,
(a) there is more than opposition between continuum and discrete; what I understand from physicists and mathematicians: this a question of 'matter' which structurally is trinitary: (i) solid (= discrete); (ii) fluid (=continous); (iii) gaseous (= airy).
(b) a question that I would like to ask you (and any other): If the factual reality consists of matter, space, and time (see e.g. Wigner): what then is the nature and origin of this three? (Maybe this is a cosmic question, but see above ad [1] and [2]).
(c) the connection between Goedel's mathematics and Yin Yang dynamics is not clear to me.
In addition, why only experimental science? and not science as experience?
[4] Peter Breur, Dear Peter,
(a) intuitionist mathematics does not believe in Boolean logic; me too! because Boolean logic is exclusively predicated upon discreteness; this is not the whole factual reality! (see again [3] sub (b)).
(b) about topos theory, category theory, etc.. please contact Dana Scott (who is at this ResearchGate).
(c) that there is an infinite set is logically a contradiction...
(d) your use of the term/concept constructivism is not clear to me; please explain it to me. Thanks, Marc
(d) you mention classical fuzzy set theory; my question (for you): is there any non-classical fuzzy set theory?
(e) I am very interested on your writing about a non-standard prologue setup script for Haskell.
[5] Sydney Ernest Grimm, Dear Sydney,
(a) as you see, I am at work, it is now 1.02 hour!
(b) to write down foundational mathematical (and physical) question is not stupid; it is exactly what a philosopher of science ought to do!
(c) for probability and possibility, see my previous question.
[6] Peter Breur, Dear Peter, (again you):
(a) "Where is the non-classical mathematics?" is not a question of "pseudo science"; for you cannot prove something that does not (yet) exsist!
(b) let me confine myself to fuzzy set theory; my question to you: is fuzzy set theory a good example of non-classical mathematics? (see again (a));
(c) again a question of mine: is a point nothing but 'a' particle? and consequently not wave, and you cannot say 'a' wave because wave is not numberable! (see my preceding question: what if singularity is wave?).
[7]Sydney Ernest Grimm, Dear Sydney,
(a) 'non-locality' is an absurd term/concept, because the factual reality consists of matter, space and time, and 'locality' refers to space! A better term/concept would be 'hicceity' (proposed by Stalnaker). However this proposal is not unproblematic: is 'space' qua gender 'male'? nobody knows...
(b) which description is, according to you, a "new-age belief"?
(c) your use of 'continuum' is taken for granted, viz. gratuitous; please explain it explicitly!
[8] Peter Breur, Dear Peter,
(a) has physics nothing to do with mathematics? wow, this is a real novelty for me!
(b) your "there is an interesting move afoot towards a very finitistic non-classical mathematics" is very flattering to me, thank you, Peter! however, I still do not stand on the shoulders of giants like Laplace...
Dear all, thank you for your feedbacks! I appreciate that very much! Marc
It is still an open question whether or not it is possible to describe some 'properties' of the 'smallest volume in the universe' in an objective and determinable manner at all, because any experimental attempt to attach a property or a number to a physical object at most lowest scale might violate the alleged properties of the physical object. Hence, the concept and process of attaching properties and numbers should reflect this non-objectivity and non-decidability. However, non-objectivity and non-decidability is not considered in traditional mathematics. Even it is comfortable to believe in the infinity of 'natural numbers' (Peano axioms) and in the construction of the completeness of 'real numbers' and to use the resulting calculus for setting up some physical theories, it is not clear whether or not this kind of mathematics is appropriate for all branches of physics, in particular for physics at the most lowest scale. To my point of view, one way out of this dilemma is to setup a reasonable set of fuzzy-like numbers which carry an inherent statistical character which is supposed to be appropriate for situations in physics where non-objectivity and non-decidability and 'uncertainty' must not be ignored.
dear marc
i suggest you to know Bipolar Quantum Linear Algebra of W.R. Zhang
and his nonlinear operators.
The kernel is to substitute the true false based logic with equilibrium based logic, closer to fuzzy logic
Thank you all for your valuable contributions.
I'm still need more clarification that will differentiate the classical and non-classical mathematics.
Regards,
Jamilu Sabi'u
You can check out these sites until you find the right answer for your question
http://www2.cs.cas.cz/~ncm/
Article The (Non-)classicality of (Non-)classical Mathematics
Thank
Rana Z Al-Kawaz
Interesting discussion on non-classical mathematics.... As an industrial engineer, I have observed that deterministic problems are often solved using classical mathematics. In reality, real life problems are nondeterministic. Industrial problems involving complex systems with nondeterministic inputs are better solved using metaheuristic techniques.
Fuzzy, rough and soft logic and their applications are considered as a non classical mathematics.
Dear Tareq,
But fuzzy logic is at the latest instance based on the same classical principles upon which classical mathematics is based. I am looking for a genuine non-classical mathematics or non-classical logic (if you like) that transgresses the boundaries of the classical mathematics/logic which we inherit from the 19th century and harks back to the Aristotelian logic. Please think with me about this problem. Cheers, Marc
I suggest you to know the BDL (dynamic bipolar logic) of Wen Ran Zhang. It is equilibrium based logic, beyond true-false logic and it recovers causality that probability lost.
perhaps BDL manages to overcome the limitation of Lao Tzu "the tao that is said is not the eternal tao, the beautiful words are not true - the true words are not beautiful" that has prevented the dynamic logic yin yang to have a scientific language.
Thank you both, Francesco and Modestus! I am going to read the logic and mathematics of Wen Rang Zhang. Can I draw the conclusion that the truth/false logic/mathematics, and the rest of our up to now known science, viz. the s.c. 'normal' logic/mathematics is exactly based upon limitation? This is a foundational question. Please, your answers! Thanks, Marc
Still on Non-Classical Mathematics........
Thank you Marc Carvallo! we cannot jump into conclusion that the truth/false logic/mathematics, and the rest... is exactly based upon limitation. Mathematical logic explores the applications of formal logic to mathematics. It has close connections to meta-mathematics, the foundations of mathematics, though in industrial engineering we often use meta-heuristics. I am trying to source for a book by Noson Yanofsky titled "The Outer Limits of Reason" . Though I read through the abstract which looked at what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, logic, and our own thought processes. He also describes simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve; perfectly formed English sentences that make no sense; different levels of infinity; the bizarre world of the quantum; the relevance of relativity theory; the causes of chaos theory; math problems that cannot be solved by normal means; and statements that are true but cannot be proven. He explains the limitations of our intuitions about the world—our ideas about space, time, and motion, and the complex relationship between the knower and the known. After looking at this book and some other interesting books on meta-mathematics. I will have something interesting to say. Thank you.
Regards, Modestus.
Thank you Modestus for your extensive answer. Maybe I 'll have a look at Noson Yanofsky's book, if it is present in our university's library. My search about that limitation regards the nature and origin of this. A very short answer might be as follows: we know that numbers are limited, particles and propositions of formal logic are id. So, we are afflicted with this limitation. What then could be the nature and origin of this limitation? Maybe you can help me with any answer. Thanks, Marc.
Thank you Marc Carvallo. Your question seems like philosophy of mathematics. Maybe answers will come after reading the logic and mathematics of Wen Rang Zhang and the book by Noson Yanofsky titled "The Outer Limits of Reason" . Please I will appreciate it if you can help me with soft copy of the two books. Thank you.
Regards,
Modestus.
You can find something of Zhang work on GesearchGate
or buy "Yin Yang Bipolar Relativity a unifyng theory of Nature Agents and Causality" It is a little expensive but complete.
Dear Marc
To answer your question we can adopt for instance the point of view of Sydney (historical period), of Eric (conection with physics) or others.
In mathematics the term it is related to "classical logics", based for instance on intuitionist and constructivist renderings of set theory, arithmetic, analysis, and so on.
In this context, non-classical mathematics is based for instance on relevant, paraconsistent, contraction-free, modal, and other non-classical logical frameworks. In short, non-classical means a study of mathematics that can be formalized in some logic other than classical logic.
In general there are many topics involving the transition between classical and non-classical mathematics. The border line not allways is clear.
I collect here topics related to both sides and to this transition:
Intuitionistic, constructive, and predicative mathematics: Heyting arithmetic, intuitionistic set theory, topos-theoretical foundations of mathematics, constructive or predicative set and type theories, pointfree topology.
Substructural and fuzzy mathematics: relevant arithmetic, contraction-free naïve set theories, axiomatic fuzzy set theories, fuzzy arithmetic, etc.
Inconsistent mathematics: calculi of infinitesimals, inconsistent set theories, etc.
Modal mathematics: arithmetic or set theory with epistemic, alethic, or other modalities, modal comprehension principles, modal treatments of vague objects, modal structuralism, etc.
Non-monotonic mathematics: non-monotonic solutions to set-theoretical paradoxes, adaptive set theory, etc.
Alternative classical mathematics: alternative foundational theories over classical logic, categorial foundations of mathematics, non-standard analysis, etc.
Topics related to non-classical mathematics: meta-mathematics of non-classical or alternative mathematical theories, their relative interpretability, first- or higher-order non-classical logics, etc.
Dear Cesar,
Thank you for the extensive answer. After more than 5 ages (I count this from the origin of modern science, beginning from Galileo in the 17th century) we are steeped in some sort of prejudice called classical mathematics, for itself it is not that serious but for a philosopher of math it is a serious one. From your extensive answer I take one as a possible candidate for the non-classical mathematics, viz. that of the 'paraconsistent logic' (if you allow me to use 'logic' and 'mathematics' interchangeably). I know very little about that 'paraconsistent logic' except the works of Graham Priest. Maybe you can help me with other authors to be read.
But again, we have another problem: that of 'nothingness' (the other way around that of 'infinity'), see e.g. Russel vs. that prof. of the University of Graez. Anyhow, thanks! Marc.
Hi Marc
In the link below you will find several publications on the topic of paraconsistent logics and some links on other non-classical math topics. Best regards
https://philarchive.org/
A better link Marc
https://philarchive.org/s/paraconsistent logics
Dear Marc, I also agree with the main consensus among dialecticians that this branch of mathematics does not violate the law of contradiction of formal logics. I am a physicist but I like philosophy and of course math. Paraconsistent logics go in my view on the opposite direction. Quantum mechanics for instance "belongs" to dialetics no doubt.
Dear Cesar and Wulf,
Thank you both for the links and extensive review. I allow myself vulnerable for asking questions to all of you and others well-seasoned in mathematics. Thanks, I am going to read every link you give. Marc.
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