1st May, 2020

The Pedagogical University of Cracow

Question

Asked 30th Apr, 2018

-Up to now we usually use the *classical mathematics* the origin of which is at the end of the 19^{th} century and/or at the beginning of the 20^{th} century. Even the contemporary quantum physics, astrophysics, and AI of the 21^{st} 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

Maybe I refer to the title of your question, but I say something on **classical** mathematics.

Namely, there in the paper: T. J. Stępień, Ł. T. Stępień, "Atomic Entailment and Atomic Inconsistency and Classical Entailment", *Journal of Mathematics and System Science , *vol. 5, 60-71 (2015); arXiv:1603.06621 , a new solution of the well-known problem of entailment was given, i.e. so-called atomic entailment was defined and so-called Atomic Logic was formulated (the abstracts related to this paper: T. Stepien, „Logic based on atomic entailment”, *Bulletin of the Section of Logic*, vol. 14 (2), 65 – 71 (1985) ; T. J. Stepien and L. T. Stepien, „Atomic Entailment and Classical Entailment”, *The Bulletin of Symbolic Logic*, vol. 17, 317 – 318 (2011)).

In the next paper: T. J. Stepien and L. T. Stepien, „The Formalization of The Arithmetic System on The Ground of The Atomic Logic”,* Journal of Mathematics and System Science*, vol. 5, 364 – 368 (2015); arXiv:1603.09334*, *it has been proved that the classical Arithmetic can be based on the Atomic Logic (presented in the paper: T. J. Stępień, Ł. T. Stępień, "Atomic Entailment and Atomic Inconsistency and Classical Entailment") - the abstract of this paper: T. J. Stepien and L. T. Stepien, "The formalization of the arithmetic system on the ground of the atomic logic" , * The Bulletin of Symbolic Logic * 22 , No. 3, 434 - 435 (2016).

Please to read the paper: B. Buldt, "The Scope of Godel's First Incompleteness Theorem", *Log. Univers.* vol. 8, 499–552 (2014) - this paper had been published 1 year before the paper T. J. Stępień, Ł. T. Stępień, „The Formalization of The Arithmetic System on The Ground of The Atomic Logic” . Namely, in the paper B. Buldt, *Log. Univers.* vol. 8, 499–552 (2014), it has been written among others, that (page 531): "As such, relevant arithmetic will be weaker than classical arithmetic [...]".

Atomic Logic (which is a non-classical logic), is sufficient to formalize Classical Arithmetic System.

Lukasz

Dear Marc Carvallo,

The term “non-classical” is related to a period. Personal I don’t think that this is a meaningful differentiation between branches of mathematics.

Most mathematics is developed with the help of the phenomenological point of view. Thus we can use it to describe phenomena in physics too. Unfortunately, our universe shows to be non-local and phenomena are created by an underlying continuum.

That’s why I have the opinion that “non-classical mathematics” must be capable to describe the new insights. Probably there is no need for “non-classical mathematics” because there is an enormous amount of different mathematical concepts. Maybe we only have to restructure all the mathematics. (For example, set theory cannot be foundational because it is related to the phenomenological point of view.)

However, this is an answer that's not really "exiting". So I give you a mathematical problem that enables you to create all the "non-classical mathematics".

The question is: "**What are the mathematical properties of the smallest volume in the universe**?"

With kind regards, Sydney

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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.

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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

Dear Marc Carvallo,

The term “non-classical” is related to a period. Personal I don’t think that this is a meaningful differentiation between branches of mathematics.

Most mathematics is developed with the help of the phenomenological point of view. Thus we can use it to describe phenomena in physics too. Unfortunately, our universe shows to be non-local and phenomena are created by an underlying continuum.

That’s why I have the opinion that “non-classical mathematics” must be capable to describe the new insights. Probably there is no need for “non-classical mathematics” because there is an enormous amount of different mathematical concepts. Maybe we only have to restructure all the mathematics. (For example, set theory cannot be foundational because it is related to the phenomenological point of view.)

However, this is an answer that's not really "exiting". So I give you a mathematical problem that enables you to create all the "non-classical mathematics".

The question is: "**What are the mathematical properties of the smallest volume in the universe**?"

With kind regards, Sydney

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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?

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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:

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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'?*"

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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

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Dear Lena J-T Strömberg,

It wasn’t my intention to get Marc Carvallo “at work” ;-))

May be it was a bit stupid to write down this foundational mathematical question. However, mathematics is not scale invariant thus I couldn’t stop myself to type the question.

“Where is the non-classical mathematics?” suggests an unspoken belief in the scale variance of mathematics. What means in practise that we are convinced that we can use mathematical probability in spite of the fact that it doesn’t exist at the quantum level. So we are creating our own conceptual problems (e.g. like Schrödinger’s cat).

With kind regards, Sydney

Dear Peter Breuer,

The non-local universe as a whole creates internal configurations and because we have a certain size we can only reflect the configurations we belong to (are part of). Fortunately we have instruments to observe beyond the capacities of our senses so “we” have enlarged the configurations we are aware of.

We can say that the description above is some kind of a “new-age belief”. But it becomes reality when we arrive at the origin of phenomenological physics (the microcosm). We can throw dices to discover the principles of probability but these principles originates from the underlying reality at the microcosm. In fact this is the structure of the basic quantum fields, everywhere existent in the universe. This continuum is creating the known phenomena and if I want to describe the underlying reality with the help of mathematical probability that is obtained from macroscopic phenomenological observations there is something wrong. At least in relation to our concept of quantum reality.

With kind regards, Sydney

2nd May, 2018

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 can**not** 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

Marc, I am out of bed now and I have to get some coffee first before I start an answer! ;-))

So, you asked for “non-classical mathematics” but it exists already. The ancient Greek philosophers had the opinion that our universe is a mathematical universe. And it was Parmenides who worked out the concept of an underlying reality that creates all the phenomena in the universe. He stated that there exist no emptiness in our universe. Aristotle extended the concept because he added dynamics and “non-locality”. The underlying reality is in rest and everything else – all the phenomena – are moved by the underlying reality (the unmoved mover). By the way, I am not a philosopher so this is not my pet ;-))

If you analyse the general structure of theories in physics and mathematics you will discover that most of it represents a reduced or simplified reality. In physics we have a name for it: phenomenological physics. In fact, your question could be: “Where is the non-phenomenological mathematics?” Because mathematics and physics have the same foundations (it was Max Tegmark who brought the issue back to live with his paper “The mathematical universe”).

The ancient Greek philosophers had the concept of the existence of “atoms”. Unfortunately we don’t know what they had in mind. It is a bit vague and of course we have claimed the word for our own purpose: the building blocks of chemical bonds. But the ancient Greek philosophers knew about the relation between reality and structure and some texts can be interpreted that atoms represent the structure of the underlying reality.

In modern physics we are convinced that reality is created by the underlying quantum fields (basic fields because the phenomena represent composed fields, fields – mathematical properties – that are transferred within the basic structure). There is a constant of alteration, Planck’s constant, and every alteration have the same velocity in relation to Planck’s constant: the constant speed of light. Moreover, all the alterations – quanta transfer – are limited by invariance: the law of conservation of energy.

The concepts in modern physics must be like paradise for mathematicians. Everywhere there is structure and everything is related. Thus the only thing you have to do is to describe the mathematical properties of the smallest volume in the universe. Because all the phenomena are composed/created by the smallest volumes of the spatial structure. It is a mathematical description so you will describe and prove the foundations of mathematics (the second prove is the direct relation between the description and observable reality).

By the way, singularities exist but are part of the smallest volume. A singularity has no boundary.

With kind regards, Sydney

3rd May, 2018

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

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Dear Karl G. Kreuzer,

Maybe it is not so complicated. We don’t have to observe the “phenomenological” properties of the underlying structure, we have to observe the general properties of the universe. For example, Heisenberg’s uncertainty principle and Planck’s constant are related to the interactions between phenomena. It isn’t by accident that the minimal resolution of Heisenberg’s uncertainty is Planck’s constant. If the mathematical description (foundations of mathematics) shows the appearance of both “anomalies” it will be quite convincing that the mathematical description must be right. Invariance, a constant alteration of the mathematical properties of every volume and a constant transfer of mathematical properties between all the volumes will complete the “physics part” of the proof. I suppose the “mathematical part” of the proof is more difficult. Because it is subject to the acceptance by the mathematical community. In physics we have “mother nature” we can consult. But who is the arbitrary authority in mathematics? Maybe they will look with one eye in the direction of the theoretical physicists… ;-))

With kind regards, Sydney

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

Thank

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.

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 19^{th} century and harks back to the Aristotelian logic. Please think with me about this problem. Cheers, Marc

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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

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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.

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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.

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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.

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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 17^{th} 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.

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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

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.

This thread has named several candidates for “non-classical” mathematics, often based on alternative types of logic(s). Most of these proposals have been shot down by one or the other of the participants, for good reasons. Any type of radical innovation (I try to avoid the overused term “paradigm shift”) has had lingering undercurrents for a while before coming out and adding to the mainstream. For instance, the “not-yet-classical” concept of “set” had a forerunner in Bolzano (probably even earlier), and was picked up in a somewhat naive form by Cantor. Dedeking played around with sets graphically, and finally the concept crystallized into Zermelo-Fraenkel or other proper “set theory.”

Sometimes it’s not that sweeping, but rather a non-classical theorem or method. Take Cantor’s “diagonal method” of defining certain decimal expansions, which proved that the cardinality of the real numbers is higher than the cardinality of the rationals. That was a radical new non-classical idea and is still considered, by some mathematicians, a controversial method, even though it has become a standard technique, for instance in Turing’s proof of the undecidability of First Order Logic, to quote a rather hefty example.

Somewhat under the radar has been a recent attempt to add non-classical new methods and results to mathematics. The hero here is Jacob Lurie of Harvard. He won the 2014 MacArthur Award. As the Harvard Gazette put it: Lurie is “rewriting large swaths of mathematics from a fresh point of view, while also working to apply his foundational ideas to prove important new theorems in other areas.” His non-classical new mathematics is nothing for the faint-hearted and pretty radical – and very abstract (topos theory).

But rather than add to the catalog of non-classical mathematics, I’d like to point out a **non-classical new approach of ***doing* mathematics – collectively, in large distributed teams, on-line, openly.

Of course, there has always been a degree of collaboration among scientists. But especially in mathematics, the loner in his study has more often worked in jealous secrecy. Newton didn’t share his insights, he guarded his theories and theorems carefully, first from his rival Hooke, later from Leibniz. Quite recently: Andrew Wiles worked on Fermat’s Last Theorem alone, sharing almost nothing until he smelled success. Perelman proved the Poincare Conjecture in complete secrecy in his mother’s apartment in Saint Petersburg.

Well, the non-classical counterculture is sweeping aside this obsession with secrecy. One of the earliest champions was, as is well known by now, Terence Tao with his Polymath projects, each with many threads, all at a high professional level, sharing and not hiding results, doubts, and critique. If you haven’t already, take a look: https://terrytao.wordpress.com/ .

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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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Maybe I refer to the title of your question, but I say something on **classical** mathematics.

Namely, there in the paper: T. J. Stępień, Ł. T. Stępień, "Atomic Entailment and Atomic Inconsistency and Classical Entailment", *Journal of Mathematics and System Science , *vol. 5, 60-71 (2015); arXiv:1603.06621 , a new solution of the well-known problem of entailment was given, i.e. so-called atomic entailment was defined and so-called Atomic Logic was formulated (the abstracts related to this paper: T. Stepien, „Logic based on atomic entailment”, *Bulletin of the Section of Logic*, vol. 14 (2), 65 – 71 (1985) ; T. J. Stepien and L. T. Stepien, „Atomic Entailment and Classical Entailment”, *The Bulletin of Symbolic Logic*, vol. 17, 317 – 318 (2011)).

In the next paper: T. J. Stepien and L. T. Stepien, „The Formalization of The Arithmetic System on The Ground of The Atomic Logic”,* Journal of Mathematics and System Science*, vol. 5, 364 – 368 (2015); arXiv:1603.09334*, *it has been proved that the classical Arithmetic can be based on the Atomic Logic (presented in the paper: T. J. Stępień, Ł. T. Stępień, "Atomic Entailment and Atomic Inconsistency and Classical Entailment") - the abstract of this paper: T. J. Stepien and L. T. Stepien, "The formalization of the arithmetic system on the ground of the atomic logic" , * The Bulletin of Symbolic Logic * 22 , No. 3, 434 - 435 (2016).

Please to read the paper: B. Buldt, "The Scope of Godel's First Incompleteness Theorem", *Log. Univers.* vol. 8, 499–552 (2014) - this paper had been published 1 year before the paper T. J. Stępień, Ł. T. Stępień, „The Formalization of The Arithmetic System on The Ground of The Atomic Logic” . Namely, in the paper B. Buldt, *Log. Univers.* vol. 8, 499–552 (2014), it has been written among others, that (page 531): "As such, relevant arithmetic will be weaker than classical arithmetic [...]".

Atomic Logic (which is a non-classical logic), is sufficient to formalize Classical Arithmetic System.

Lukasz

Linearizing the time-dependent Schrödinger equation not possible without c?

Question

8 answers

- Asked 4th Oct, 2017

- Oliver Tennert

In https://projecteuclid.org/download/pdf_1/euclid.cmp/1103840281 J.-M. Lévy-Leblond has shown that linearizing the (full, time-dependent) Schrödinger equation leads to a spinor equation. The mathematics is straightforward, no issue with that. My issue is: it is argued that the resulting linearized equation would be Galilei covariant. Yet, as I see it, the key equation (which is (25) on p. 293 in the cited link above) is just the non-relativistic limit of the Dirac equation, with c=1. This is not a miracle to me, as linearizing the Schrödinger equation by means of dimensionless matrices (that actually turn out to be the Dirac matrices) is only possible by introducing a velocity scale. In the paper above this is not transparent from the beginning, as the natural unit convention "c=1" is used.

The pure fact that linearizing the Schrödinger equation is a means to get a spinor equation is rarely discussed in the literature anyway. But where it is, it is argued that the resulting equation is non-relativistic and thus Galilei covariant. I argue that it can't be, and that it is neither Galilei- nor Lorentz-covariant! The former it can't because it has the velocity scale "c" in it (which, in the original paper or anywhere else is omitted due to the unit convention), the latter it can't because it is only the non-relativistic limit of the Dirac equation.

Do I have some error in my reasoning?

Looking forward to some replies.

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- Aleksandar Janjic

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