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

Dear Jonathan,

Thank you for extracting most important questions from in what I disagree with common theory.

Meanwhile I found Book 7 of Euclid's Elements at

http://aleph0.clarku.edu/~djoyce/Java/elements/bookVII/bookVII.html

Definition 1:

A unit is that by virtue of which each of the things that exist is called one.

Definition 2:

A number is a multitude composed of units.

Compare this with the wrong notion of numbers like points on the real line.

Compare it also with other definitions and notions of numbers, e.g.:

1) Pythagoras: Anything is number

2) Albert of Saxony: the number of points of a wooden bar

3) Kronecker: Natural numbers were made by god ...

4) Dedekind: Numbers are creation of the human mind ...

5) Peano's axioms

6) Frege's logical definition of numbers

7) Hilbert's brutal number axioms I-IV

8) Weyl: continuous sauce between Euclidean numbers

9) Russell: number = class of all classes that are similar to a given class

10) v. Neumann: start with the empty set

I conclude: Numbers should no longer be considered like points but as already Euclid correctly understood like measures. The word incommensurable does not exclude that a real number can be a measure.

Regards,

Eckard

Dear Jonathan,

Thank you for extracting in what I disagree with common theory.

Meanwhile I found Book 7 of Euclid's Elements at

http://aleph0.clarku.edu/~djoyce/Java/elements/bookVII/bookVII.html

Definition 1:

A unit is that by virtue of which each of the things that exist is called one.

Definition 2:

A number is a multitude composed of units.

Compare this with the wrong notion of numbers like points on the real line.

Compare it also with other definitions and notions of numbers, e.g.:

1) Pythagoras: Anything is number

2) Albert of Saxony: The numbers a wooden bar consists of

3) Kronecker: The natural numbers were made by god ...

4) Dedekind: Numbers are creations of the human mind ...

5) Frege's logical definition

6) Peano's axioms

7) Russell:... the class of all classes that are similar to a given class.

8) Hilbert's brutal axioms I-IV

9) Weyl's continuous sauce between Euclid's numbers

10) v. Neumann's start at the empty set

We should admit that numbers correspond to measures instead of points. Why not allow a measure to be strictly speaking incommensurable? Incommensurable does not mean not measurable at all.

Well said Eckard,

I like the idea of starting from nothing, when talking about number. There is an increase in comprehension when children are taught to count 0,1,2.. instead of 1,2,3... But when I was that age, the latter formalism was the most common or conventional. If we take a constructivist view of this matter, it forces us to find a minimal set of assumptions which allow counting and/or measuring. I talk about this in Quantum Biosystems Vol 1 no. 1, but briefly - measurement proceeds by procedural steps of observation, change, and comparison. If we repeat 'observe, move, compare' enough times a sense of size, distance, and proportion becomes apparent. So no one observation can give the total picture, but an understanding arises as the number of observations increases. But when talking about a number of objects, things are a little different.

You see; when we see a single image containing a small number of similar objects (beans or oranges perhaps), we can immediately distinguish between a field of view with three - and a field with four or five objects. On the other hand, a planet-sized ball cannot be easily distinguished from a marble, unless our field of view contains both objects at once - enabling us to make a comparison. So size is relative, in the sense where the scale is determined by a size comparison with other objects. But the number of objects in a given space is in a sense more absolute. However, a scale change in the field of view may admit quite a few more oranges or beans to enter the picture. So there is a distinct relation between the two dissimilar concepts.

So this invokes or involves a set-theoretic view, where the boundary is arguably a topological distinction between objects in the set (within the field of view under consideration) and those outside it. Taking this view allows us to represent oranges on the screen with points lying within a circle. But of course; a real orange has size and might be half in our field of view, with the rest outside it. So at some stage of any real determination of number and measure, the sense of proportion arising from the concept of size comes into play.

Anyhow; I greatly enjoyed your recounting the various definitions of number past thinkers have brought us. To a degree, the concept of number is natural - or arises naturally from observation and comparison. However; there is a lot of definition that we accept on faith, or reject at our own peril. But I think that's what Isham is talking about when he says we need to revamp Math in a major way, because it is exactly this kind of definition by hidden assumptions that must be re-examined. And I think the constructivists had the right idea, but until now have lacked some of the tools that category theory offers. Using various topoi allows us to approach the concept of number differently, or variously, it would seem.

But I really like what Rudy Rucker has written about the subject of numbers and numeration. I would heartily recommend 'Mind Tools' or 'Infinity and the Mind' if you can find them, but I know he has covered the same topic elsewhere too.

All the Best,

Jonathan

For Georgina,

I had not been previously aware of Sweetser's work, but if I had been I would likely have given it a mention in my contest essay where I explicitly talk about energy-space as a universal dynamic worthy of consideration. And I love the fact that Quaternions offer significant improvement and simplification over the multi-vector approach - for various types of calculation within Engineering, Computer Graphics, and Physics. Nonetheless, I agree with Lawrence for the most part, in the assessment that Sweetser attempts to make Quaternions do things they don't do so well - making some of his theory a bit over complicated.

My thought is that we want to use the right Math for the job. And it is evident that Quaternion Math was slighted, once multi-vector analysis came on the scene, because there are things it really does better - or handles more easily and simply. I had the idea that Real numbers are good for Classical mechanics and conventional Gravity, Complex numbers are required for describing Wavelike phenomena and Electromagnetism, Quaternions are useful when we include Spin or Rotation - which should mean the Weak Force would be modeled well, and Octonions would then be required where there are still more degrees of freedom (or dimensions) as in the sub-atomic realm - and for the Strong force.

I will have to look more deeply into what Sweetser is saying, however, as I think there will be a few gems.

All the Best,

Jonathan

Jonathan,

Do we really "need to revamp Math", i.e. to try and improve it and hide its faults? To me the verb revamp shows disapproval. It is definitely not just better but as I am convinced necessary to honestly reveal the faults like sets of points instead of measures, aleph_2, and all the other ridiculous nonsense. Set theory based topology is not even in position to separate a piece into two equal halves. No add on will ever remedy what is foundational wrong. I hope FQXi is the place where such attitude will be taken seriously. I do not have decades to wait and wait for the LHC and other effort if it is impossible to find SUSY.

What about my criticism of too much abstraction, I prefer the most adequate descriptions like e.g. measured data belonging to only positive elapsed time for many reasons. Well, it will hurt those who enjoy speculating almost without limit.

I will search for Rudy Rucker.

Problems with

I apologize for my problems with breakdowns of connection.

Eckard

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 -----

How to prove string theory in one easy lesson.

Add both sides of the blackboard to get string theory to balance.

and add the five string theories.

1/3 APPLE 1/3 ORANGE 1/3 ORANGE= 1 APPLE/ORANGE.

Add the equatons three at a time selecting the equations intelligently to be added.

Then keep adding until you have just one equation for all string theory.

Then put this equation to the test................

in the real world.

Steve

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