Readers' Choice: True Lies: Why Mathematics is an Illusion

Jonathan,

I looked at what I found about the man who is a descendant of Hegel. No, I am not interested in science fiction, set theory and other mysticism.

I prefer Archimedes, Euclid, Galilei, and Ren.

Eckard

Thank you Eckard,

For what it is worth, I greatly appreciated the link to Euclid's Elements, once I corrected the erroneous capital J. It is welcome that we can have this discussion, despite our difference of opinion on the value of Rucker's 'mysticism.'

As to whether Math can or should be revamped, I think there is at least a matter of its being somewhat inconsistent in usage of terms or definitions of concepts for various branches. But Isham's contention that Math needs to be replaced or re-worked entirely is suspect. It does seem like grandstanding.

Nonetheless; I continue to believe we will see a ripple effect across the whole of Mathematics, which results from insights coming out of category theory, topos theory, and other related work. If nothing else, it has promise for showing how the various kinds of Math are functionally related, and this itself is exciting!

All the Best,

Jonathan

Dear Jonathan,

When I asked why not considering measures instead of points basic to numbers, Terence Tao reacted as follows: He deleted my request from his public discussion and pointed to a lesson on fundamentals by his colleague Jim Ralston and Folland's book on Real Analysis.

I did not yet ask someone else for the permission to make his reply to essentially the same question of mine public. However, his more proficient reply caused me to ask myself why ancient mathematics was not able to develop further after Euclid.

I still do not agree on that Euclid's unity-based notion of number is too restricted. I rather consider it the only correct while open for extension basis of mathematics. I am not an admirer of Euclid as a genius but I consider him someone who summarized the best knowledge of ancient mathematics. Remarkably, he did not include the Pythagorean idea of numbers like pattern pebbles to be recognized.

Later more.

Regards,

Eckard

Once again, Euclid was obviously correct when he based the notion number on the notion unit alias element.

Of course, we may extend his definition 2.

Of course, we may count points. However, the notion number line is somewhat misleading. According to a contribution in Mathforum@Drexel: Historia-Mathematica there is evidence for the use of number line about 5000 years ago in Egypt.

Everybody interprets the number X attributed to a point P on the number line as a measure alias relation to the measure one, namely the distance between a reference point zero and a particular point P in relation to the unit 1, regardless whether this bracket is a larger or a smaller measure than the reference unit 1.

Any measure is based on comparison. Any process of counting is based on its unit one as a reference measure. We may add in the sense of combining measures that refer to different units of reference, e.g. circumference c and diameter d of a circle, so called incommensurable measures, which were well known to Euclid. Since there is no restriction to the resolution of rational measures(= rational numbers), it is even possible, in principle, to decide whether any d-based number is smaller or larger than c.

Could we modify Dedekind's cut as a "bra-cut"? I think so. Let's ask for consequences. If we understood it as a measure alias bracket, a positive number would always be given as the limit from the left. The "number" zero has the measure zero, i.e., it quite naturally requires particular care instead of arbitrarily remedying bans.

Perhaps all my primary objections against present mathematics would disappear. Mathematics would not suffer but perhaps benefit if loosing sets that are no measures. Admittedly, replacing the understanding of numbers as sets of points by the original Euclidean understanding as measures would give rise to, let's say, scrutinize Cantor's naive beliefs and his belonging putative evidences for good. Having carefully read Fraenkel 1923, I consider them untenable anyway.

I also dealt with the somewhat related question why was nearly a seeming standstill in the development of mathematics during about a millennium between Euclid (325-275) and his translator Johannes Campanus of Novarra (1260). My conclusions are not yet reliable.

Eckard

The timespan between -275 and +1260 is 1535 years.

A rapid development of mathematics begun not before stimulation in particular by Columbus, Copernicus, Kepler, and Galilei when printed books were available in the 17th century. Calculus has roots already published by Fermat in 1629. Calculus was not based on points. In his Geometria indivisibilus continuorum, Bonaventura Cavalieri wrote in 1635 correctly: Indivisible of lines are lines (not points).

In the middle ages, after the crusaders conquered Byzantium in 1204, contributions to mathematics and philosophy came often from bishops and from the first universities:

Campanus, see above

William of Ockham (1300-1349)

Johannes Buridan, U of Paris (Buridan's ass is to be found much later)

Albert of Saxony (1316-1390)

Nicole of Oresme (1323-1382):

graphic representation of functions, rotation of earth

Nicolaus Cusanus (1401-1464): endless universe without a center

Francoise Vieta (1540-1603): infinite product

Eckard

Hello again,

I thank you Eckard for making me think. What you are saying makes a lot of sense, but the focus seems to be on a historical progression. While I feel there is an evolution of Mathematics, and that discovering some higher Math depends upon the conceptual basis resulting from earlier work, it is amazing what has been lost and found again - over the years. Take Archimedes, for example, who is now known to have developed the rudiments of integral calculus. Then it wasn't re-discovered until many years later. And, as you seem to be implying, mathematicians have also picked up a lot of baggage along the way or made things over-complicated sometimes.

I argue that there is a more sensible way to relate to all of this, by the conceptual relationship of ideas resulting from their innate dependencies. That is; from a constructivist and process-theoretic view, all possible operations have a basis. Set theory is founded on the concept of interiority/exteriority. Without topological (hard) distinctions to define boundaries, the ideas of Set theory are not possible to construct, in their conventional form. But, if we generalize on the interiority/exteriority concept a bit - we get near/far or proximal/distal, which takes us into the realm of measurement and/or geometry. So there is a certain inter-relatedness to conceptual bases of ideas.

If I am not mistaken, the ideas found in category theory allow us to construct such mappings, or to discover the ways in which mathematical frameworks and concepts are related. If they are right, Isham and Döring will discover that a unique topos characterizes the view and structure of various 'positions' within the subject of Math, and can potentially show a kind of relativity between logical frameworks - upleveling the idea that there is only one correct logic, and replacing that with a concept more like the idea of a Multiverse in Physics.

Now; the fact the possibility exists doesn't mean they are right, but the territory they are exploring is probably useful regardless.

All the Best,

Jonathan

Jonathan,

Let me try and say it as simply as possible: Any measure, no matter whether length, area, wight, or something else can be represented by an abstract quantum called number. As soon as we have chosen a unit measure one, we may imagine this number a multiple/part of a continuum that can be repeated and split without limitation.

The unit has two ideal ends: zero and one, the neutrals of addition and of division. Unfortunately, mathematics calls these ideal points elements. This led to a nonsensical distinction between open and closed measures. Recall Euclid and Peirce: A point has no parts, but each part of a continuum has parts. If we admit the ideal infinite accuracy of "real" numbers then there is no difference between open and closed. The measure x of concern extends from zero to the limit from the left: ]=(=[0, x[. An adjacent measure from x (limit from the right) to y (limit from the left) does not let a gap. The gap has the measure zero.

I prefer |sign(0)|=1, not 0.

In physics we learned where is one body there must not be an other one.

Points without the belonging reference to zero are no correct correlates of numbers.

Regards,

Eckard

Let me add what nonsense results from Dedekind's imprecise and intentional style of thinking: I found Wikipedia/complex numbers: "stress and rotate points". I would like calling the complex numbers, which are represented in complex plane, phasors or maybe vectors rather than points. Points cannot be manipulated at all because they do not have parts.

Eckard Blumschein

I'm going to have to get better at take-aways before I can understand these comments.

http://steve-jeffreystheoryofeverything.blogspot.com/2010/02/jeffreys-theory-of-everything.html

----- Original Message -----

Add sting theory so that you have both sides of the blackboard balanced as it was originally.

And then add the physics equations 1 ODD 1 EVEN= 2 ODD.

And 1 ODD 1 ODD= 2 EVEN and 2 ODD 2 EVEN= 4 EVEN.

By adding the string theory equations at random in a spreadhseet..

THIS IS A TURING MACHINE AND WORTHY OF THE PRIZE.

It can print continuusly forever until you say stop.

And can come up with a new E=MC^2 every day.

Steve

Having read some books on history of mathematics, I feel obliged to blame already Gauss 1831 rather than his pupil Dedekind 1872 for giving rise to the still ongoing confusion between phasor and point with far reaching implications.

On the other hand, Gauss used to utter realistic views, e.g.: "We must humbly acknowledge that if number is only a product of our minds, space has a reality even outside our minds, to which we a priori cannot completely prescribe its laws."

Eckard

Dear Isham and Doring,

Please see the mathematical equation representing our self and the universe we live in.

zero = i = infinity .

If 0 x 0 = 0 is true, then 0 / 0 = 0 is also true

If 0 x 1 = 0 is true, then 0 / 0 = 1 is also true

If 0 x 2 = 0 is true, then 0 / 0 = 2 is also true

If 0 x i = 0 is true, then 0 / 0 = i is also true

If 0 x ~ = 0 is true, then 0 / 0 = ~ is also true

It seems that mathematics, the universal language, is also pointing to the absolute truth that 0 = 1 = 2 = i = ~, where "i" can be any number from zero "0" to infinity "~". We have been looking at only first half of the if true statements in the relative world. As we can see it is not complete with out the then true statements whic are equally true. As all numbers are equal mathematically, so is all creation equal "absolutely".

This proves that 0 = i = ~ or in words "absolutely" nothing = "relatively" everything or everything is absolutely equal. Singularity is not only relative infinity but also absolute equality. There is only one singularity or infinity in the relativistic universe and there is only singularity or equality in the absolute universe and we are all in it.

Love,

Sridattadev.

there is only 1 true number and it is 1

all other numbers are tricks to make the equations meet observations

CHRIS ISHAM'S VERY INTERESTING TOPOS-THEORY MATHEMATIZATION OF PHYSICS MAY HAVE ORIGINATED WHEN HE DROVE EDWARD SIEGEL[J. NONCRYSTALINE SOLIDS 40, 453(1980)], ONCE AT QUEEN MARY COLLEGE/UNIVERSITY OF LONDON AND INPE/CNPq, SAO JOSE DOS CAMPOS, BRAZIL, FROM IMPERIAL COLLEGE TO A CONFERENCE AT OXFORD CIRCA 1990 AND SEEMS TO FOLLOW ON EDWARD MACKINNON'S CALL FOR AND JACK AND IAN STEWART'S[THE COLLAPSE OF CHAOS: DISCOVERING SIMPLICITY IN A "COMPLEX" WORLD, PENGUIN(1994)] CALL FOR WITHOUT IMPLIMENTATION OF "COMPLI-CITY"(BOTTOM UP MANY TO FEW INDUCTION) AND "SIMPLE-XITY"(TOP-DOWM FEW TO MANY DEDUCTION) BOTH SIMULTANEOUSLY AUTOMATICALLY VIA EDWARD SIEGEL'S[SYMPOSIUM ON FRACTALS..., MRS FALL MEETING, BOSTON(1989)-5-SEMINAL-PAPERS!!!-ATTACHED HEREIN] FUZZYICS=CATEGORYICS=PRAGMATYICS("Son of 'TRIZ'")/CATEGORY-SEMANTICS COGNITION(AKA SYNERGETICS PARADIGM AND DICHOTOMY) ANALYTICS DISCOVERY, IN AND UNIFYING PHYSICS TO "PURE"-MATHEMATICS, OF ARISTOTLE'S "SQUARE-OF-OPPOSITION" WITH PLATO'S "FORMS" AND HEGEL'S "BECOMING" BY A SYNTHESIS OF WILLIAM LAWVERE CATEGORY-THEORY/TOPOI WITH KARL MENGER[DIMENSIONTHEORIE, TEUBNER (1929] WITH WIERZBICKA-LANGACKER-LAKOFF COGNITIVE-SEMANTICS WITH PAWLAK-ZIARKO-SLOWINSKI ROUGH-SETS THEORY.

SIEGEL'S RESULT IS A (2+1)-DIMENSIONAL MATRIX(LOOKING LIKE A TIC-TAC-TOE DIAGRAM) OF SYNONYMS/FUNCTORS VERSUS ANTONYMS/MORPHISMS VERSUS ANALOGY/METAPHOR.

AMAZINGLY NEUROIMAGING SCIENTIST JAN WEDEEN(HARVARD MEDICAL SCHOOL) RECENTLY DISCOVERED THIS EXACT STRUCTURE IN HUMAN BRAIN IMAGING!!!

SIEGEL'S FUZZYICS=CATEGORYICS=PRAGMATYICS("Son of 'TRIZ'")/CATEGORY-SEMANTICS COGNITION(AKA SYNERGETICS PARADIGM AND DICHOTOMY)ANALYTICS, BY PARSIMONY ELIMINATION VIA DISAMBIGUATION OF JARGONIAL-OBFUSCATION("LOTSS FANCY SHMANCY LINGO TO SNOW THE RUBES")THUS PROGRESSES UP THE STIKELEATHER "HIERARCHY-OF-THINKING"(HoT)WHERE EACH LOGIC-LEVEL IS META VIA "MINING" THE NEXT-LOWEST LOGIC-LEVEL:

DATA -> INFORMATION -> KNOWLEDGE -> UNDERSTANDING -> MEANING -> WISDOM -> INSPIRATION -> "SPIRITUALITY" -> UNIVERSALITY -> SIMPLICITY ->??? EVOLVING TO VIA PHILOSOPHY OF PARSIMONY TO "K.I.S.S.(KEEP IT SIMPLE STUPIDS)!!!Attachment #1: 2_FULL_PAPER_COMPLEX_QUANTUM-STATISTICS_IN_FRACTAL-DIMENSIONS.pdfAttachment #2: STRATIFIED_FUZZYICS__HoT_HIERARCHY-of-THINKING_from_ASKING_ORTHOGONAL_QUESTIONS_WHAT_WHEN__WHERE__WHY.doc

Tom,

Do you agree with Feynman? I know your great love for the mathematical...

"Physics is not mathematics, and mathematics is not physics. One helps the other. But in physics you have to have an understanding of the connection of words with the real world. It is necessary at the end to translate what you have figured out into English, into the world, into the blocks of copper and glass that you are going to do the experiments with. Only in that way can you find out whether the consequences are true. This is a problem which is not a problem of mathematics at all." (Feynman, 1965, p. 49).

"Mathematicians are only dealing with the structure of reasoning and they do not

really care what they are talking about. They do not even need to know what they

are talking about, or, as they themselves say, whether what they say is true." (Id.)

"In other words, mathematicians prepare abstract reasoning ready to be used if

you have a set of axioms about the real world. But the physicist has meaning to

all his phrases." (Id.)

"Mathematicians like to make their reasoning as general as possible,...[whereas]

the physicist is always interested in the special case." (Feynman, 1950, p. 50)

"[T]he poor mathematician translates [the special case] into equations, and as the symbols do not mean anything to him he has no guide but precise mathematical

rigour and care in the argument." (Id.)

Therefore, it is up to the physicist to narrow the scope of the problem and define what is required of the mathematician in rather specific terms.

Strangely enough, Feynman concluded his discussions on the relation of mathematics to physics with the following observation: "the mathematical rigour of great precision is not very useful in physics." (Id., pp. 50 - 51) The reason is that great precision can dampen or limit the intuition and creative imagination of the physicist, which he needs in order to modify his original ideas or guess at new solutions. (Id., p.51) An approximate mathematical conclusion is often more helpful.

Regards,

Akinbo

Hi Akinbo,

Sure, I agree. I have said repeatedly that unless it can be shown that a mathematical model is independent of its physical result (i.e., that meaning is independent of language), there can be no demonstrated correspondence between the theory and the physics, and hence no rational -- no truly objective -- theory.

The main way in which special and general relativity differs from quantum mechanics is that relativity is written in mathematical language that allows exact solutions to the equations that incorporate the theory (this is true of all classical physics). The reason that quantum theory is subject to the many interpretations that we hear, is that it is not based in equations that incorporate the theory, it is based in the phenomenology of quantum mechanics -- as Feynman said, it's all about explaining "the experiment with the two holes."

Quantum theorists from Bohr, Feynman, Bell and to the present day, have struggled to develop from scratch a quantum foundations theory independent of the experiment; they have not been successful. The tendency among theorists now is to simply accept that nature does not obey a strict mathematical structure, that events are random at foundation.

Just to add, the reason that I don't often respond any more to Pentcho or to Peter J, is that they have a conviction that relativity can -- like quantum mechanics -- be "interpreted" starting with physical results rather than from what the theory says. Nothing could be further from the truth, and such discussions go nowhere. Rational theories are true by correspondence, not by interpretation.

Best,

Tom

Thanks Tom. In your short reply 'truth' appears at least thrice. Now are there two truths, one mathematical and one physical? If there are two types of truth, which one are you looking for?

Mathematically, if something arrives later than it used to under a given condition, it can be interpreted that clocks run slower or time is dilated OR the speed of propagation is reduced/ affected. All are mathematically equivalent.

But in physics, they may not be all equivalent. Some of the mathematically correct choice of interpretation may lead to riddles, paradoxes and absurdities. In that case, the philosopher is likely to prefer the interpretation free of these. What is mathematically false, cannot be physically correct. What is physically false, cannot be mathematically correct.

Your point is however important. As someone said, you serve as a Quality Controller.

Regards,

Akinbo

*Regarding my statement in italics, a line having length and of zero width is physically impossible, same with a surface of zero thickness. Mathematicians should therefore take a look at this conflict.

I used the word truth only once, Akinbo, and not in the context you ascribe to it. My usage of "truly" means "really," and the use of "true statement" means "logically coherent." In the final case, my use of the word truth means "facts." I am not concerned with truth as you use it -- my concern is for rational, objective knowledge.

You write: "Mathematically, if something arrives later than it used to under a given condition, it can be interpreted that clocks run slower or time is dilated OR the speed of propagation is reduced/ affected. All are mathematically equivalent."

No. You are assuming an ideal clock that doesn't exist.

"What is mathematically false, cannot be physically correct. What is physically false, cannot be mathematically correct."

Nothing is ever physically false. There are only physically false interpretations of phenomena. That is why a mathematically complete theory that corresponds with the physical phenomena it incorporates and describes, is not open to interpretation.

"*Regarding my statement in italics, a line having length and of zero width is physically impossible, same with a surface of zero thickness. Mathematicians should therefore take a look at this conflict."

There is no conflict. The ideal line is a metric of 1 dimension, and the ideal surface is a plane of 2 dimensions. Your confusion results from thinking that because we live in 3 dimensions we cannot describe existence in any other terms. Language itself, however, is dimensionless.