# At PBS: Puzzle of mathematics is more complex than we sometimes think

Astrophysicist Mario Livio shares some thoughts: Math: Discovered, Invented, or Both?

The puzzle of the power of mathematics is in fact even more complex than the above examples from electromagnetism might suggest. There are actually two facets to the “unreasonable effectiveness,” one that I call active and another that I dub passive. The active facet refers to the fact that when scientists attempt to light their way through the labyrinth of natural phenomena, they use mathematics as their torch. In other words, at least some of the laws of nature are formulated in directly applicable mathematical terms. The mathematical entities, relations, and equations used in those laws were developed for a specific application. Newton, for instance, formulated the branch of mathematics known as calculus because he needed this tool for capturing motion and change, breaking them up into tiny frame-by-frame sequences. Similarly, string theorists today often develop the mathematical machinery they need.

Passive effectiveness, on the other hand, refers to cases in which mathematicians developed abstract branches of mathematics with absolutely no applications in mind; yet decades, or sometimes centuries later, physicists discovered that those theories provided necessary mathematical underpinnings for physical phenomena. Examples of passive effectiveness abound. Mathematician Bernhard Riemann, for example, discussed in the 1850s new types of geometries that you would encounter on surfaces curved like a sphere or a saddle (instead of the flat plane geometry that we learn in school). Then, when Einstein formulated his theory of General Relativity (in 1915), Riemann’s geometries turned out to be precisely the tool he needed! More.

## 119 Replies to “At PBS: Puzzle of mathematics is more complex than we sometimes think”

1. 1
bornagain77 says:

a few related notes:

In what I consider an absolutely fascinating discovery, Einstein’s General Relativity has shown that 4-dimensional (4D) space-time, along with all energy and matter, was created in the ‘Big Bang’ and continues to ‘expand equally in all places’:

There is no centre of the universe! According to the standard theories of cosmology, the universe started with a “Big Bang” about 14 thousand million years ago and has been expanding ever since. Yet there is no centre to the expansion; it is the same everywhere. The Big Bang should not be visualized as an ordinary explosion. The universe is not expanding out from a centre into space; rather, the whole universe is expanding and it is doing so equally at all places, as far as we can tell. [11]
Philip Gibbs

Thus from a 3-dimensional (3D) perspective, any particular 3D spot in the universe is to be considered just as ‘center of the universe’ as any other particular spot in the universe is to be considered ‘center of the universe’. This centrality found for any 3D place in the universe is because the universe is a 4D expanding hypersphere, analogous in 3D to the surface of an expanding balloon. All points on the surface are moving away from each other, and every point is central, no matter where you live in the universe. And as such, it may now be possible for the Earth to be, once again, considered ‘central in the universe’.

“People need to be aware that there is a range of models that could explain the observations… For instance, I can construct you a spherically symmetrical universe with Earth at its center, and you cannot disprove it based on observations… You can only exclude it on philosophical grounds… What I want to bring into the open is the fact that we are using philosophical criteria in choosing our models. A lot of cosmology tries to hide that.” [11a,b,c]
– George Ellis

That every 3-Dimensional place within the universe may be considered central to the expansion of the universe may seem very counter-intuitive to most people, but that is exactly what has now been shown.

In regards to trying to understand this counter-intuitive finding, it is helpful to note that ‘higher dimensional’ mathematics was developed by Reimann, (who was a devout Christian),, and that that development in higher dimensional mathematics was necessary before Einstein could elucidate the 4-D spacetime of General Relativity, (or even before Quantum Mechanics could be elucidated);

The Mathematics Of Higher Dimensionality – Gauss & Riemann
https://vimeo.com/98188985

I find the best way to get this ‘centrality of the Earth in the universe” point across is to visualize it first hand. The first few minutes of this following video clearly get this ‘centrality in the universe’ point across:

Centrality of The Earth Within The 4-Dimensional Space-Time of General Relativity – video
https://vimeo.com/98189061

Moreover, this ‘circle’ of the CMBR that is found by modern science to encompass the Earth, from the remnant of the creation event that brought the entire universe instantaneously into being, was actually predicted in the Bible centuries earlier:

Proverbs 8:27 (King James Version)
“When he prepared the heavens, I was there: when he drew a circle upon the face of the depth:”

Proverbs 8:27 (New International Version)
“I was there when he set the heavens in place, when he marked out the horizon on the face of the deep,”

Job 26:10
He has inscribed a circle on the face of the waters at the boundary between light and darkness.

Moreover, 4-D space-time is not the only place ‘centrality’ is found, because of advances in Quantum Mechanics, the argument for God from consciousness can now be framed like this:

1. Consciousness either preceded all of material reality or is a ‘epi-phenomena’ of material reality.
2. If consciousness is a ‘epi-phenomena’ of material reality then consciousness will be found to have no special position within material reality. Whereas conversely, if consciousness precedes material reality then consciousness will be found to have a special position within material reality.
3. Consciousness is found to have a special, even a central, position within material reality. [14]
4. Therefore, consciousness is found to precede material reality.

I find it extremely interesting, and strange, that quantum mechanics tells us that instantaneous quantum wave collapse to its ‘uncertain’ 3D state is centered on each individual conscious observer in the universe, whereas, 4D space-time cosmology (General Relativity) tells us each 3D point in the universe is central to the expansion of the universe. These findings of modern science are pretty much exactly what we would expect to see if this universe were indeed created, and sustained, from a higher dimension by a omniscient, omnipotent, omnipresent, eternal Being who knows everything that is happening everywhere in the universe at the same time. These findings certainly seem to go to the very heart of the age old question asked of many parents by their children, “How can God hear everybody’s prayers at the same time?”,,, i.e. Why should the expansion of the universe, or the quantum wave collapse of the entire universe, even care that you or I, or anyone else, should exist? Only Theism, Christian Theism in particular, offers a rational explanation as to why you or I, or anyone else, should have such undeserved significance in such a vast universe. [15]

Psalm 33:13-15
The LORD looks from heaven; He sees all the sons of men. From the place of His dwelling He looks on all the inhabitants of the earth; He fashions their hearts individually; He considers all their works.

Moreover, from a slightly different angle, ‘Life’, with a capital L, is also found to be central to the universe in that Christ’s resurrection from the dead provides a very credible reconciliation to the most profound enigma in modern science. Namely the unification of General Relativity with Quantum Mechanics/Special Relativity (Quantum Electrodynamics) into a ‘Theory of Everything’:

The Center Of The Universe Is Life (Jesus Christ) – General Relativity, Quantum Mechanics, Entropy and The Shroud Of Turin – video
http://vimeo.com/34084462

The Galileo Affair and “Life/Consciousness” (Jesus) as the true “Center of the Universe”

2. 2
daveS says:

BA77,

That every 3-Dimensional place within the universe may be considered central to the expansion of the universe may seem very counter-intuitive to most people, but that is exactly what has now been shown.

I’m not quite understanding your argument here. It seems like you are saying that because there is no unique center to the universe, the Earth is at the center of the universe. Obviously that must be a misreading, but I don’t know how else to construe your point.

3. 3
bornagain77 says:

Oh wait daveS, the excerpted quote and your objection are not even on the same page of thought.

But then again, just to be objecting was your main point in the first place was it not?

There is far more to the argument, but I got the main point of higher dimensional math across, and I’ll pass elucidating further seeing as it is disingenuous you doing the objecting.

i. e. I’ve got much better things to do today than watch you chase your atheistic tail in a circle.

4. 4
Mapou says:

Why is the effectiveness of math so unreasonable to some? Both math and the universe are discrete and super-symmetrical. That is to say, we live in a yin-yang, discrete universe. I fail to see the great mystery.

5. 5
daveS says:

BA77,

Oh wait daveS, the excerpted quote and your objection are not even on the same page of thought.

I disagree. As you explained, there is no distinguished “center” in the universe, just as there is no distinguished “center” on the 2-dimensional surface of a balloon.

So AFAICS, it makes no sense to consider the Earth to be “central in the universe” any more than some random point in space halfway to the Andromeda galaxy.

6. 6

From further on in the article:

Personally, I believe that by asking simply whether mathematics is discovered or invented, we forget the possibility that mathematics is an intricate combination of inventions and discoveries. Indeed, I posit that humans invent the mathematical concepts—numbers, shapes, sets, lines, and so on—by abstracting them from the world around them. They then go on to discover the complex connections among the concepts that they had invented; these are the so-called theorems of mathematics.

7. 7
Robert Byers says:

At first glance it seems there is no such thing as math. Its just a language for reality. Yet its not real itself. The universe is laid out, as the bible says, but God doesn’t use math.
Its just us dividing things up and the great order allows the divisions to always come together to the original fixed state.
math might be like pushing a ball under the water and poping up again. Yet its got nothing to do with the ball but other laws.
I see math as a special case dealing with a organized universe.
Its not actually there. Its mere distances.
Math is almnost irrelevant to scientific insights.
Newton was wrong to emphasise math. his stuff was from insights and math just confirmed it.
Just thinking.

8. 8
kairosfocus says:

I think a fairer conclusion is that we recognise, through a process of intellectual discovery and clarification, the reality of certain abstract entities such as two-ness. That one comes up so soon as we realise self vs not-self.

It turns out that two-ness is a necessary property of any possible world as can be seen from doing the chain

{} –> 0

(0) –> 1

{0, 1} –> 2

etc.

Shapes, lines, points and so are likewise recognised from experience of reality.

We then proceed to create various functions and the like.

Then, along comes an Euler, and BANG!

0 = 1 + e^i*pi

That sort of utterly shocking coherence strongly suggests exploration and discovery much moreso than mere arbitrary invention. That sort of irreducibly complex coherence is far too neat for it to be the product of arbitrary action . . . contrast the confusion of mismatch that reigns in the world of car parts, which are arbitrarily specified, model by model.

Engineering produces functional systems, but getting coherence far and wide takes higher level co-ordination. Mathematical realities are far too coherent to be just engineered models.

Multiply that by the astonishing effectiveness of mathematics (and linked logic) in capturing and predicting aspects of reality.

Methinks, echoing was it Boyle or Kepler, we are thinking Someone’s thoughts after him.

KF

9. 9
daveS says:

KF,

Then, along comes an Euler, and BANG!

0 = 1 + e^i*pi

While I might agree with some of your other points, I don’t read much into Euler’s identity myself. After all, it is a bit of a contrivance. You can of course express it more compactly (and more sensibly) as e^i*pi = -1. Does it then lose some of its magic?

10. 10
bFast says:

Is mathematics discovered or invented? Wrong question! The real question is, “Is invention discovery or creation”?

While there is a creative component in invention — the consideration to attempt to produce a certain something, there is also a fundamental discovery component to invention: figuring out how to do the something that creativity imagined.

As a multiply-patented inventor, I frequently hear people tell me “what would be nice is …” This is human creativity at work. However, just because Hollywood-style hover-boards would be nice, making such a thing is not possible without an (as yet unmet) discovery.

Consider Thomas Edison working to invent the light bulb. He is famous for claiming that he tried about 1,000 things that didn’t work before finally finding a method that did work. Why didn’t these other methods work? Oh yea, because though they were surely creative, they did not fit within the framework of reality. And when Edison finally found something that worked, what did he utter? “Eurika, I found it!”? Probably something like that. His utterance was very similar to what a gold prospector utters when he discovers the mother lode. His utterance is similar, his methods — strategic searching — is similar as well.

The answer to the question “is mathematics discovery or invention” then, is “Yes!”

11. 11
kairosfocus says:

DS, only at the price of obscuring the relationships between a nexus of numbers, relationships and operations. The five most prominent numbers in one astonishingly coherent framework tied to oscillations and waves [thus frequency space]. Huge insights beckon when the classic form is rearranged . . . a common enough thing in strategic synthesis, the moment of lateral insight and eye-opening synthesis. Likewise for me a real breakthrough in economic insight was to think in terms of the dynamics of aggregate supply & demand, with saturation as resources tighten up highlighted leading to insights on overheating an economy, stagflation, shocks and Schumpeter’s creative destruction . . . here GDP fell 50% when the volcano hit. Just one little diagram or equation, but then comes the moment of insightful synthesis. But then, I am a moderate constructivist in education philosophy. KF

12. 12
bFast says:

DaveS, “I don’t read much into Euler’s identity myself.”

Benjamin Peirce … stated that the identity “is absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth”. (From wikipedia entry of Euler’s identity.)

Hmmm, is DaveS blind, or is Benjamin Pierce hallucinating?

DaveS, “You can of course express it more compactly (and more sensibly) as e^i*pi = -1. Does it then lose some of its magic?” Actually, yours is the more commonly expressed form of Euler’s identity. It doesn’t loose magic in this format — it increases in magic.

BTW: invention is discovery.

13. 13
stenosemella says:

KF@18: “DS, only at the price of obscuring the relationships between a nexus of numbers, relationships and operations. The five most prominent numbers in one astonishingly coherent framework tied to oscillations and waves [thus frequency space]. Huge insights beckon when the classic form is rearranged . . . a common enough thing in strategic synthesis, the moment of lateral insight and eye-opening synthesis. Likewise for me a real breakthrough in economic insight was to think in terms of the dynamics of aggregate supply & demand, with saturation as resources tighten up highlighted leading to insights on overheating an economy, stagflation, shocks and Schumpeter’s creative destruction . . . here GDP fell 50% when the volcano hit. Just one little diagram or equation, but then comes the moment of insightful synthesis. But then, I am a moderate constructivist in education philosophy. KF”

Is this really supposed to mean anything? Be profound? Could you please rephrase this in a fashion that can be comprehended by a human being?

14. 14
daveS says:

Hi bFast,

Benjamin Peirce … stated that the identity “is absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth”. (From wikipedia entry of Euler’s identity.)

Hmmm, is DaveS blind, or is Benjamin Pierce hallucinating?

That’s an interesting quote, given that he was a Harvard professor. It’s a result that is accessible and understandable to calculus students, so I’m not sure why he thinks it’s so mysterious.

Actually, yours is the more commonly expressed form of Euler’s identity. It doesn’t loose magic in this format — it increases in magic.

I’ve generally (maybe always?) seen it expressed as e^i*pi + 1 = 0. If you do a google image search for Euler’s identity, I think you’ll see more versions like that than the simplified one. That version includes the five important numbers 0, 1, e, i, and pi, which is a major reason why the identity is famous.

15. 15
idnet.com.au says:

If there is no spacial center of the universe, there can still be an informational or even a relational center of the universe. If most of the interesting organizational complexity in the universe that we know of is here, then Earth has special status. If God visited Earth then I think Earth should qualify for special status.

16. 16
Don Pedro says:

If you understand the connection between the exponential function and the trigonometric functions, there’s nothing particularly mysterious about Euler’s identity. And if you don’t understand the connection, it means that relatively simple things like complex numbers, Taylor series, and linear differential equations are enshrouded in total mystery for you. In other words, you are mathematically illiterate.

17. 17
bornagain77 says:

A DEFENSE OF THE (Divine) REVELATION AGAINST THE OBJECTIONS OF FREETHINKERS, BY MR. EULER
Excerpt: “The freethinkers (atheists) have yet to produce any objections that have not long been refuted most thoroughly. But since they are not motivated by the love of truth, and since they have an entirely different point of view, we should not be surprised that the best refutations count for nothing and that the weakest and most ridiculous reasoning, which has so often been shown to be baseless, is continuously repeated. If these people maintained the slightest rigor, the slightest taste for the truth, it would be quite easy to steer them away from their errors; but their tendency towards stubbornness makes this completely impossible.”
http://www.math.dartmouth.edu/.....2trans.pdf

18. 18
kairosfocus says:

Steno,

Pardon, we here face fairly esoteric mathematics being discussed at short hand.

Have you done complex algebra, the power series forms for sin and cos, the exponential e^i*theta form, and phasors (an extension of vectors into rotation) through theta = omega*t?

If not, very little will make sense, pardon.

Worse, beyond lurk Fourier series, Fourier transforms and even Laplace Transforms, with stability and z transforms next in line.

Wiki has a first phase discussion:

http://en.wikipedia.org/wiki/Euler%27s_formula

Then, phase 2 is here:

http://en.wikipedia.org/wiki/Euler%27s_identity

Pardon, this is fairly technical.

I will clip the Wiki discussion of the famous identity:

Euler’s identity is often cited as an example of deep mathematical beauty.[3] Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:[4]

The number 0, the additive identity.
The number 1, the multiplicative identity.
The number pi, which is ubiquitous in the geometry of Euclidean space and analytical mathematics (pi = 3.14159265…)
The number e, the base of natural logarithms, which occurs widely in mathematical analysis (e = 2.718281828…).
The number i, the imaginary unit of the complex numbers, a field of numbers that contains the roots of all polynomials (that are not constants), and whose study leads to deeper insights into many areas of algebra and calculus.

(Both pi and e are transcendental numbers.)

Furthermore, the equation is given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.

Stanford University mathematics professor Keith Devlin has said, “like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence”.[5] And Paul Nahin, a professor emeritus at the University of New Hampshire, who has written a book dedicated to Euler’s formula and its applications in Fourier analysis, describes Euler’s identity as being “of exquisite beauty”.[6]

The mathematics writer Constance Reid has opined that Euler’s identity is “the most famous formula in all mathematics”.[7] And Benjamin Peirce, a noted American 19th-century philosopher, mathematician, and professor at Harvard University, after proving Euler’s identity during a lecture, stated that the identity “is absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth”.[8]

A poll of readers conducted by The Mathematical Intelligencer in 1990 named Euler’s identity as the “most beautiful theorem in mathematics”.[9] In another poll of readers that was conducted by Physics World in 2004, Euler’s identity tied with Maxwell’s equations (of electromagnetism) as the “greatest equation ever”.[10]

I will add that this is a point of convergence and coherence across entire fields of Math that gives pause. Something deeply unifying is there beneath all the diversity and a huge swath of intellectual history.

That is why that formula has a mystical power, it points straight to the issue of the one and the many, which inter alia includes how beauty is the pleasing, elegant unification of diversity into a powerful, inspiring whole.

(And yes, beauty goes far beyond being merely in the eye of the beholder.)
KF

19. 19
stenosemella says:

KF: “Have you done complex algebra, the power series forms for sin and cos, the exponential e^i*theta form, and phasors (an extension of vectors into rotation) through theta = omega*t?”

Yes. And it can be explained in English. I have no idea what it was you were speaking. That is why I asked for the translation.

20. 20
Alicia Renard says:

@ stenosemella at comment 13

KF lives on a small island in the Caribbean devastated by a volcanic eruption that, after emigrations, has reduced the population to not much more than 5,000.

The island received £324m between 1997-98 and 2011-12, half of the total given [by UK government] to overseas territories in this period.

Montserrat premier presses for more UK aid

21. 21
Don Pedro says:

KF

Pardon, we here face fairly esoteric mathematics being discussed at short hand.

Esoteric? Bloody hell! In my time, they taught this magic stuff in secondary school (and I didn’t go to Hogwarts).

22. 22
kairosfocus says:

PS: Went looking for a good survey on the one and the many, tough searching. I will say that in the Sistine Chapel, Plato points up to the one and Aristotle down with spread fingers to the many. That, is how big and central and pervasive this one is.

23. 23
kairosfocus says:

DP, Yes, the first level is advanced HS Math, where the basic math that gets you to the expression lies. The rest is well beyond, that allows you to see how much is converging into that expression, and how powerful it is. KF

24. 24
kairosfocus says:

AR, I wonder how the plight of volcano devastated Montserrat is relevant to this thread, other than in what you suggest by juxtaposition and dismissive tone. I will say, devastation and 2/3 loss of land area, loss of main infrastructure including power plant, sea port, air port, also 50% of housing, 2/3 population (due not to mere emigration but forced involuntary evacuation under natural disaster) and coming on 2 dozen lives. Compression of economy by 50%, and need for recovery and transformation, in a context of 350 years of history, including enslavement, blood, toil and tears under the lash of one of the great crimes of history, the triangular trade that built so much of the capital base for the Industrial Revolution and more, leading to of course the issue of a debt of honour. But also, the growth of a British identity that led to a list of names here and all over the Caribbean of those who fought and fell serving under the Union Jack by free choice in two World Wars — well, do I recall the cenotaph at my old High School in Barbados and the solemn ceremonies of Remembrance Day; and that roll of honour paid in Caribbean blood continues right on up to the ongoing wars post 9/11. Indeed, one of those Firemen who fell in the WTC Towers was son of a Fire Chief here, who was serving in NYFD; and, the last name on the November 11 Remembrance Day list for this small British nation — as British as the other, better known nations under the Union Jack — is that for one who fell in Iraq. As the IDC of House of Commons stated in the 1998 report on M/rat, para 101, “Our responsibilities to Dependent Territory citizens are of a greater and different order to our more general humanitarian responsibilities to the developing world and involve different priorities.” That, given 350 years of history, is not merely due to UN Charter Art 73 obligations. Something the FCO 2012 White Paper rightly and frankly acknowledges. A transformational investment is indicated, to move from year to year sustenance to an economy that will again stand on its feet for one of the minority peoples of Britain. So, just for the record, a little better balance and perspective would be reasonable. KF

25. 25
stenosemella says:

KF, I am still looking for a translation of your previous post that does not require the Rosetta Stone to interpret. It can’t be that difficult. Baffling with BS is not a legitimate argument. It just makes you look like a fool.

26. 26
kairosfocus says:

Steno, given your resort just above I suggest to you, for record, that I have already pointed to a partial list of the many themes and domains in Mathematics that converge in the Euler expression and show it to be a point of surprisingly deep coherence in Mathematics. That sort of surprising point of unity is a manifestation of one of the deepest broadest challenges in philosophy, the issue of the one and the many. It is also the source of the perception of deep mathematical beauty in the expression, as beauty is the elegant synthesis of unity amidst diversity. Moreover, that the logical-structural-quantitative patterns of the cosmos show this kind of astonishing unity as hinted at and outlined is itself a sign of deep unity in the deepest basis of reality. You may not notice or be moved by it, but others, of the greatest calibre, do notice and are moved. The significance and power of Euler’s expression are not on trial, your ability to recognise and acknowledge it, may well be. And, your resort to veiled vulgarity to dismiss does not speak well of the likely verdict. KF

27. 27
stenosemella says:

KF, again you try to use bafflegab to answer a very simple question. You asked me if I had done complex algebra, blah, blah, blah. My answer was yes. Now please explain the point you were trying to make at comment 11 in plain English. Is it really that difficult for you to do so?

For the record, you are not impressing anyone other than yourself with this verbose nonsense. I seriously want to know what you were trying to say in that comment. I assure you that nobody else was able to parse that nonsense. Please do better.

28. 28
kairosfocus says:

Steno

I point to two appreciations that just may help open your eyes — if you are willing. There is much more:

http://www.science4all.org/le-.....-identity/

and:

http://www.wired.com/2014/11/eulers-identity/

(I suspect your core problem is that, as with so many today, you lack familiarity with and are dismissive towards the theme and challenge, the one and the many, universals and particulars in a coherent world; one of the biggest, deepest and richest of all problems. I repeat my last comment, as it seems apt: >> given your resort just above I suggest to you, for record, that I have already pointed to a partial list of the many themes and domains in Mathematics that converge in the Euler expression and show it to be a point of surprisingly deep coherence in Mathematics. That sort of surprising point of unity is a manifestation of one of the deepest broadest challenges in philosophy, the issue of the one and the many. It is also the source of the perception of deep mathematical beauty in the expression, as beauty is the elegant synthesis of unity amidst diversity. Moreover, that the logical-structural-quantitative patterns of the cosmos show this kind of astonishing unity as hinted at and outlined is itself a sign of deep unity in the deepest basis of reality. You may not notice or be moved by it, but others, of the greatest calibre, do notice and are moved. The significance and power of Euler’s expression are not on trial, your ability to recognise and acknowledge it, may well be. And, your resort to veiled vulgarity to dismiss does not speak well of the likely verdict. >>)

KF

29. 29
bornagain77 says:

KF, here is a interesting piece of trivia that you might be interested in:

From your second reference, Euler’s is graphed out as a right handed spiral

The Baffling and Beautiful Wormhole Between Branches of Math
picture
http://www.wired.com/wp-conten.....uler_f.jpg
gif of Euler’s
http://www.songho.ca/math/eule.....r_ex01.gif

What is intersting to the fact that Euler’s graphs to the right is that DNA is also a right hand spiral.

In the following article, Adam Rutherford takes exception to the many incorrect examples of left handed DNA spirals he finds at many reputable institutions:

DNA’s twist to the right is not to be meddled with, so let’s lose the lefties – Adam Rutherford – 30 April 2013
http://www.theguardian.com/sci.....t-to-right
Correct DNA spiral – image
http://images.wisegeek.com/dna-close-up.jpg

of related interest to the right handed Euler’s spiral and DNA spiral, is the fact that information can be encoded onto a photon by taking advantage of the photon’s spin and orbital angular momentum. The method in which this encoding of information onto a photon is accomplished is by means of a ‘spiral phase plate’

pdf slides
page 18 and 19: image of how to encode information onto a photon by taking advantage of its spin and orbital angular momentum
http://www.bostonphotonics.org....._RBoyd.pdf

Ultra-Dense Optical Storage – on One Photon
Excerpt: Researchers at the University of Rochester have made an optics breakthrough that allows them to encode an entire image’s worth of data into a photon, slow the image down for storage, and then retrieve the image intact.,,, As a wave, it passed through all parts of the stencil at once,,,
http://www.physorg.com/news88439430.html

“By its conventional definition, a photon is one unit of excitation of a mode of the electromagnetic field. The modes of the electromagnetic field constitute a countably infinite set of basis functions, and in this sense the amount of information that can be impressed onto an individual photon is unlimited.”
Robert W. Boyd – The Enabling Technology for Quantum Information Science 2013 – University of Rochester, Rochester, NY – lead researcher of the experiment which encoded information in a photon in 2010
http://www.bostonphotonics.org.....eminar=202

You may well ask, “What does encoding information onto a photon with a ‘spiral phase plate’ have to do with DNA?”
Well, it turns out that, DNA, besides communicating with the cell by molecular means, also communicates with the the cell using light:

Are humans really beings of light?
Excerpt: A particularly gifted student talked him into another experiment.,,, He also found that DNA could send out a wide range of frequencies, some of which seemed to be linked to certain functions. If DNA stored this light, it would naturally emit more light on being unzipped. These and other studies proved to Popp that one of the most essential sources of light and biophoton emissions was DNA.
http://viewzone2.com/dna.html

Biophotons – The Light In Our Cells – Marco Bischof – March 2005
Excerpt page 2: The Coherence of Biophotons: ,,, Biophotons consist of light with a high degree of order, in other words, biological laser light. Such light is very quiet and shows an extremely stable intensity, without the fluctuations normally observed in light. Because of their stable field strength, its waves can superimpose, and by virtue of this, interference effects become possible that do not occur in ordinary light. Because of the high degree of order, the biological laser light is able to generate and keep order and to transmit information in the organism.
http://www.international-light.....hotons.pdf

Pretty neat huh?

30. 30
stenosemella says:

KF, all I asked is that you explain your comment at 11 in a way that we can all make sense of. If you can’t do that, it is no shame to admit it. Cheers.

31. 31
bornagain77 says:

stenosemella,

KF has done more than his fair share trying to make Euler’s more accessible for you.

If you can’t grasp the overall point of ‘unity where there should be none’ there is no shame in admitting it.

But then your point was never really to try to understand anything of significance in the first place was it? but was only to be as obtuse as possible to the point of distraction!

Fruit flies have the same effect of distraction! 🙂

I think you should set your sights higher than being a fruit fly of distraction on a blog.

32. 32
Querius says:

Lol, bornagain77! Well said.

-Q

33. 33
stenosemella says:

BA77, I understand Euler. I just don’t understand KF. If you understand him, please summarize what he was trying to say at 11. Please use your own words and not a bunch of cut and paste.

Seriously, I would love someone to tell me what KF was trying to say at 11. Querius, feel free to attempt an interpretation as well. You are both quick to jump to KF’s defence but you don’t understand anything he says any better than I do. The only difference is that I admit it. It speaks volumes.

34. 34
daveS says:

I inadvertently overlooked KF’s #11. I wager that’s the first time ever that Euler’s identity has been linked to ‘stagflation’. 🙂

35. 35
Querius says:

bornagain77,

Have you seen this panel discussion of the Holographic Principle? Start at 1:13:00 – 1:13:30 to hear what Princeton physicist Herman Verlinde has to say about information.

. . . this development will be that we start actually with information, so information is going to be our starting point, and space-time is not something that we start with, but we forget about what space is and time, and then somehow the information by thinking about how much information is . . . what information is doing, and then the space-time will be what we call emergent—it will come out of a bunch of zeros and ones.

“And God said, ‘Let there be light,’ and there was light.” Genesis 1:3 NIV

“In the beginning was the Word (logos), and the Word was with God, and the Word was God.” John 1:1 NIV

“The Son is the radiance of God’s glory and the exact representation of his being, sustaining all things by his powerful Word.” Hebrews 1:3 NIV

Out of the mouth of prophets and physicists.

-Q

36. 36
Don Pedro says:

Stenosemella,

Some of the local fauna use the Sea Cucumber Defence: if they feel threatened, they eject their entrails together with some toxic sticky threads to ensnare their enemy.

37. 37
kairosfocus says:

Steno:

The tone, vulgarity and dismissiveness you resorted to above (not to mention unresponsiveness to two appreciations that SHOULD give you pause) give the lie to the Parthian shot on that’s all.

To draw out and list the many connexions and convergences that surround that amazing expression is indeed hard to do; esp. in the ambit of a blog combox that does not normally allow images etc, and makes symbols particularly cumbersome. That’s why in part I pointed and invited exploration. And, knowing that it is a lot quicker to link, I did a bit of searching. The appreciations were hard to find, but what is instructive is that the issue of the one and the many is suspiciously thin on the Internet . . . as is the discussion of beauty as a manifestation of it.

Did you see the way I highlighted the sheer beauty and powerful impact thereof? As in, beauty can be understood as the elegant balance of unity amidst diversity. (And BTW, the whole world of phi, the golden ratio and beauty in proportion is ALSO directly connected to this stuff . . . this one is like a golden chalice of magic wine that just keeps flowing with ever so many choice vintages.)

The thought that hits me is that of sitting in a library among the labour of the books and glancing up to see a world class, un-self-conscious stunning beauty walk up to your table, smile and sit across from you. And, you notice, astonishingly, that the others sitting in your study group seem astonishingly blind to the jaw-dropping sheer impact of the young lady now asking to join the circle . . . she is a world class Mathematician, too.

You say, yes, Ma’am. She then powerfully ties together ever so many things in a sweeping, coherent symmetry that opens up vistas, worlds even.

You are dumbstruck.

This would be a woman to walk with in life and beyond.

She is, of course Solomon’s Sophia of old, come to visit.

But others are unappreciative and want to be rid of her with her silly nonsense.

And, suddenly, you realise there is a horrible blindness sitting all around you.

Sophie shakes her head, and says to you Boethius sends greetings.

Graciously, she thanks the circle for the five minutes of time and leaves, a fragrant perfume of the most exquisite composition being in her wake.

You are now in a completely different world, and know, just know, that Sophie has just utterly redefined your reference for beauty, insight and more. She is going to haunt your dreams forever, as your elder sister and teacher. And companion and consoler in distress.

And you simply cannot believe the blind, crude grumbling dismissiveness of your study circle.

Kairos has come, and many have loved darkness rather than light.

KF

PS: Let me again excerpt the Wiki survey clipped long since in 18 above, as just a faint outline:

Euler’s identity is often cited as an example of deep mathematical beauty.[3] Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:[4]

The number 0, the additive identity.
The number 1, the multiplicative identity.
The number pi, which is ubiquitous in the geometry of Euclidean space and analytical mathematics (pi = 3.14159265…)
The number e, the base of natural logarithms, which occurs widely in mathematical analysis (e = 2.718281828…).
The number i, the imaginary unit of the complex numbers, a field of numbers that contains the roots of all polynomials (that are not constants), and whose study leads to deeper insights into many areas of algebra and calculus.

(Both pi and e are transcendental numbers.)

Furthermore, the equation is given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.

Stanford University mathematics professor Keith Devlin has said, “like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence”.[5] And Paul Nahin, a professor emeritus at the University of New Hampshire, who has written a book dedicated to Euler’s formula and its applications in Fourier analysis, describes Euler’s identity as being “of exquisite beauty”.[6] [–> that extends to Laplace and Z transforms and has implications in Quantum Physics also]

The mathematics writer Constance Reid has opined that Euler’s identity is “the most famous formula in all mathematics”.[7] And Benjamin Peirce, a noted American 19th-century philosopher, mathematician, and professor at Harvard University, after proving Euler’s identity during a lecture, stated that the identity “is absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth”.[8]

A poll of readers conducted by The Mathematical Intelligencer in 1990 named Euler’s identity as the “most beautiful theorem in mathematics”.[9] In another poll of readers that was conducted by Physics World in 2004, Euler’s identity tied with Maxwell’s equations (of electromagnetism) as the “greatest equation ever”.[10]

Maybe, some of what calls out to me here is that I am an applied Physicist, and have seen the range and power of that sweep.

38. 38
kairosfocus says:

DS, The analytical connexion is real. In the early-mid 1970’s my Dad, then a leading Macroeconomist in our homeland, used Fourier analysis to make connexions on impacts of proposed Fiscal policy and to predict likely outcomes and so to issue cautious advice. Sadly, accurately. KF

39. 39
kairosfocus says:

F/N: Opening of Consolation:

_________

But I, because my sight was dimmed with much weeping, and I could not tell who was this woman of authority so commanding—I was dumfoundered, and, with my gaze fastened on the earth, continued silently to await what she might do next. Then she drew near me and sat on the edge of my couch, and, looking into my face all heavy with grief and fixed in sadness on the ground, she bewailed in these words the disorder of my mind:

Alas! in what abyss his mind Is plunged, how wildly tossed! Still, still towards the outer night She sinks, her true light lost, As oft as, lashed tumultuously By earth-born blasts, care’s waves rise high.
Yet once he ranged the open heavens, The sun’s bright pathway tracked; Watched how the cold moon waxed and waned; Nor rested, till there lacked To his wide ken no star that steers Amid the maze of circling spheres.
The causes why the blusterous winds Vex ocean’s tranquil face, Whose hand doth turn the stable globe, Or why his even race From out the ruddy east the sun Unto the western waves doth run:
What is it tempers cunningly The placid hours of spring, So that it blossoms with the rose For earth’s engarlanding: Who loads the year’s maturer prime With clustered grapes in autumn time:
All this he knew—thus ever strove Deep Nature’s lore to guess. Now, reft of reason’s light, he lies, And bonds his neck oppress; While by the heavy load constrained, His eyes to this dull earth are chained.
II.

‘But the time,’ said she, ‘calls rather for healing than for lamentation.’ Then, with her eyes bent full upon me, ‘Art thou that man,’ she cries, ‘who, erstwhile fed with the milk and reared upon the nourishment which is mine to give, had grown up to the full vigour of a manly spirit? And yet I had bestowed such armour on thee as would have proved an invincible defence, hadst thou not first cast it away. Dost thou know me? Why art thou silent? Is it shame or amazement that hath struck thee dumb? Would it were shame; but, as I see, a stupor hath seized upon thee.’ Then, when she saw me not only answering nothing, but mute and utterly incapable of speech, she gently touched my breast with her hand, and said: ‘There is no danger; these are the symptoms of lethargy, the usual sickness of deluded minds. For awhile he has forgotten himself; he will easily recover his memory, if only he first recognises me. And that he may do so, let me now wipe his eyes that are clouded with a mist of mortal things.’ Thereat, with a fold of her robe, she dried my eyes all swimming with tears.
SONG III.
The Mists dispelled.
Then the gloom of night was scattered, Sight returned unto mine eyes. So, when haply rainy Caurus Rolls the storm-clouds through the skies, Hidden is the sun; all heaven Is obscured in starless night. But if, in wild onset sweeping, Boreas frees day’s prisoned light, All suddenly the radiant god outstreams, And strikes our dazzled eyesight with his beams.
III.

Even so the clouds of my melancholy were broken up. I saw the clear sky, and regained the power to recognise the face of my physician. Accordingly, when I had lifted my eyes and fixed my gaze upon her, I beheld my nurse, Philosophy, whose halls I had frequented from my youth up.

‘Ah! why,’ I cried, ‘mistress of all excellence, hast thou come down from on high, and entered the solitude of this my exile? Is it that thou, too, even as I, mayst be persecuted with false accusations?’

‘Could I desert thee, child,’ said she, ‘and not lighten the burden which thou hast taken upon thee through the hatred of my name, by sharing this trouble? Even forgetting that it were not lawful for Philosophy to leave companionless the way of the innocent, should I, thinkest thou, fear to incur reproach, or shrink from it, as though some strange new thing had befallen? Thinkest thou that now, for the first time in an evil age, Wisdom hath been assailed by peril? Did I not often in days of old, before my servant Plato lived, wage stern warfare with the rashness of folly? In his lifetime, too, Socrates, his master, won with my aid the victory of an unjust death. And when, one after the other, the Epicurean herd, the Stoic, and the rest, each of them as far as in them lay, went about to seize the heritage he left, and were dragging me off protesting and resisting, as their booty, they tore in pieces the garment which I had woven with my own hands, and, clutching the torn pieces, went off, believing that the whole of me had passed into their possession. And some of them, because some traces of my vesture were seen upon them, were destroyed through the mistake of the lewd multitude, who falsely deemed them to be my disciples. It may be thou knowest not of the banishment of Anaxagoras, of the poison draught of Socrates, nor of Zeno’s torturing, because these things happened in a distant country; yet mightest thou have learnt the fate of Arrius, of Seneca, of Soranus, whose stories are neither old nor unknown to fame. These men were brought to destruction for no other reason than that, settled as they were in my principles, their lives were a manifest contrast to the ways of the wicked. So there is nothing thou shouldst wonder at, if on the seas of this life we are tossed by storm-blasts, seeing that we have made it our chiefest aim to refuse compliance with evil-doers. And though, maybe, the host of the wicked is many in number, yet is it contemptible, since it is under no leadership, but is hurried hither and thither at the blind driving of mad error. And if at times and seasons they set in array against us, and fall on in overwhelming strength, our leader draws off her forces into the citadel while they are busy plundering the useless baggage. But we from our vantage ground, safe from all this wild work, laugh to see them making prize of the most valueless of things, protected by a bulwark which aggressive folly may not aspire to reach.’ . . . >>
_____________

There is ever so much more.

Here, again is Sophia, giving grim warning in the joint voice of David and Solomon:

>>Prov 1:20 Wisdom cries aloud in the street,
in the markets she raises her voice;
21
at the head of the noisy streets she cries out;
at the entrance of the city gates she speaks:
22
“How long, O simple ones, will you love being simple?
How long will scoffers delight in their scoffing
and fools hate knowledge?
23
If you turn at my reproof,[a]
behold, I will pour out my spirit to you;
I will make my words known to you.
24
Because I have called and you refused to listen,
have stretched out my hand and no one has heeded,
25
because you have ignored all my counsel
and would have none of my reproof,
26
I also will laugh at your calamity;
I will mock when terror strikes you,
27
when terror strikes you like a storm
and your calamity comes like a whirlwind,
when distress and anguish come upon you.
28
Then they will call upon me, but I will not answer;
they will seek me diligently but will not find me.
29
Because they hated knowledge
and did not choose the fear of the Lord,
30
would have none of my counsel
and despised all my reproof,
31
therefore they shall eat the fruit of their way,
and have their fill of their own devices.
32
For the simple are killed by their turning away,
and the complacency of fools destroys them;
33
but whoever listens to me will dwell secure
and will be at ease, without dread of disaster.” >>
>>>>>>

Kairos is come, let us turn from darkness to light.

KF

PS: Steno, fasten your eyes on the clip from Wiki again. If that summary which I provided at 18 above and which you so dismissively brushed aside is not enough, what short of a monograph would have satisfied you? And, would you have patience to read and ponder such if that powerful little summary just bounced off your carapace? Do, answer me.

40. 40
kairosfocus says:

Q, yes indeed. Heb 1:1 – 3 and several parallels are rich, beautiful and exceedingly powerful with insights. KF

41. 41
kairosfocus says:

BA, never made the connexion to THAT helix, but you are right, it is there, the helix is at the core of life. And extending the circle cross-time as the rotation proceeds gives a helix. KF

42. 42
kairosfocus says:

Don Pedro, I trust the above overnight will prove adequate. KF

43. 43
daveS says:

KF,

DS, The analytical connexion is real. In the early-mid 1970?s my Dad, then a leading Macroeconomist in our homeland, used Fourier analysis to make connexions on impacts of proposed Fiscal policy and to predict likely outcomes and so to issue cautious advice. Sadly, accurately. KF

No doubt there is a connection. I just found it jarring to see, in the middle of this discussion of a beautiful equation, a word with such a differing level of beauty.

And while I don’t want to downplay all these connections, I think it is possible to exaggerate their significance and lapse into a mathematical version of Six Degrees of Kevin Bacon.

44. 44
kairosfocus says:

DS, Kev Bacon, I know not from the breakfast food. But this, I know, the Euler eqn is perhaps the most striking commonly encountered case of a microcosm of the issue of the one and the many I know, one that opens up a gateway to a new world of insight. For those willing to enter therein. Mathematics is in it, universals and particulars, infinity is in it, natural numbers, reals and transcendentals, complex numbers, underlying issues of sequence and series [including that decimal notation is an infinite series in disguise], opening up the whole realm that the frequency-time domain relationship opens up via Fourier and Laplace going on to Z transforms and digital signal processing in a digital world, cybernetics too, and of course in Q-mech wave functions beckon. Exponentials underlie growth and decay, pointing also to the wonder of the eigenfunction. In science, the search for the unifying principle is a classic quest. As well, the very concept of beauty as elegant unity in the midst of diversity lies in this. And, amidst it all, sits Sophie, calling us to sup at her magic chalice of wisdom. Where, breakthrough moments of insight have in them the power of epipany . . . or perhaps even, Theophany. For, in him we live and move and have our being. And, which is where this begins, we are invited to think his thoughts after him, here seeing in one expression a shocking unification pointing to a root Mind underlying the mathematical order in reality — don’t forget the contrast to car parts. And yes, that is scary, shocking. But beautiful. KF

45. 45
daveS says:

Or possibly, in the words of Theodoric of York, “Naaaahh”.

46. 46
daveS says:

KF,

I guess my main point is that it’s hard to tell where the significant connections end and where the free association begins, so I don’t find this exercise very compelling. YMMV.

47. 47
kairosfocus says:

DS, naaaah to what? To the significance of the problem of the one and the many — unity and diversity in a coherent common world? To the point that epiphany is real? To the point that the Euler eqn has in it a unifying microcosm that poses the tip of an iceberg and beckons us to explore beneath its surface? That, in so doing we will see an unexpected but powerful unification of domains of mathematical thought directly connected to ever so much of empirical reality? That this shows governing power of such logical-mathematical principles in so much of said reality, giving rise to its astonishing effectiveness as Wigner remarked? That such contrasts sharply to, say, the way that car parts do not exhibit such unification precisely because of want of unifying mind, leading to the notorious parts search challenge? Or, naaah to what else? KF

48. 48
kairosfocus says:

DS, I suggest that in mathematics, there are but few free associations; but instead a logical structure. Further to this, the issue of the one and the many is an underlying issue for things that unify the diversity of phenomena and in this case, that holds to the extent of microcosm. Going further, that is directly relevant to understanding how the question of beauty applies to a mathematical expression. So, I must disagree with the dismissal by assertion, “free association.” KF

49. 49
daveS says:

KF,

DS, naaaah to what?

The part beginning with theophany and thinking God’s thoughts. It would be hard to determine whether one is simply fooling oneself.

50. 50
kairosfocus says:

DS, I would suggest to yourself that one can fool oneself on many things. That is why there is a challenge of worldview grounding on comparative difficulties. In the case of the reality of God and the possibility of life changing revelatory insight, I suggest that a glance at ontological and moral issues may open up a fresh look. KF

51. 51
stenosemella says:

DP@36: “Some of the local fauna use the Sea Cucumber Defence: if they feel threatened, they eject their entrails together with some toxic sticky threads to ensnare their enemy.”

Your comparison of KF to a sea cucumber is more appropriate than you initially intended. Especially considering the orifice that the sea cucumber uses for respiration.

52. 52
kairosfocus says:

Steno, do you have any substantial contribution, say, as informed by 37 – 39 above and as further informed by 11 & 18? KF

53. 53
daveS says:

KF,

DS, I would suggest to yourself that one can fool oneself on many things. That is why there is a challenge of worldview grounding on comparative difficulties. In the case of the reality of God and the possibility of life changing revelatory insight, I suggest that a glance at ontological and moral issues may open up a fresh look. KF

I guess I’m just much more pessimistic about the power of human reason than you are. It’s clearly within our capacity to count the number of balls in an urn or derive Euler’s formula by manipulation of infinite series. However, I wouldn’t bet my lunch money on arguments involving “maximally great beings” and the like.

54. 54
kairosfocus says:

DS,

it seems then the issue pivots on reason. We err and error exists. But that is in itself an undeniable, knowable truth. Knowable to certainty as to try to deny is to inadvertently demonstrate. So, reason, though limited by error is not thereby dismissible on non-mathematical matters.

I will say, that on major issues, it often takes centuries to come to a settled pattern; so much the worse for the modern tendency to try to tell truth by the clock or calendar.

As to necessity of being, that is readily seen once one examines modes of being, and on a maximally great being and root of reality (which will be necessary) such will be a serious candidate and will be either impossible as a square circle is, or else it will be actual. Where, it is quite certain that if there had ever been an utter nothing, there would forever be non-being as such has no causal power.

Thus, something always was, the issue is its nature.

That nature can be discerned from our experience of being under moral government. For, to project general delusion on this is to be in self referential incoherence on mind and to destroy not only mathematics but rationality.

We are left to see that there must be a world-foundational IS that grounds OUGHT.

There is — after many centuries of debate — just one serious candidate: the inherently good creator God, a necessary and maximally great being, worthy of service by doing the good.

Where, as God is a serious candidate necessary being, those who object to his existence should at minimum be prepared to show cause for thinking such a reality-root being to be impossible. Let us have done with dismissive rubbish on how believing in God is akin to superstition, fairy tales, delusion or even child abuse or the like.

The time for such village atheist arrogance is long since past. But such attitudes are all too common, and are often loudly and rudely announced without regard to basic respect.

That, I hold, is a reasonable view of reality and its root, as well as of alternatives.

And, I suggest that the God thus on the table is willing to answer sincere prayer.

But then, this runs far afield of mathematics and the particular case in view, how the Euler expression shows a powerful and deeply relevant unity across vast realms of mathematics.

Which, admittedly, is a candidate sign of unifying mind behind reality, mind who invites us through things like Mathematics, to think his thoughts after him.

KF

55. 55
daveS says:

Well, given that people can’t even come to a consensus on the Maximally Great Star Trek captain, I’m not sure there’s much point in speaking of a MGB in the abstract. But you are right, this is getting rather far from Euler’s identity.

56. 56
kairosfocus says:

DS, a Star Trek Captain would be a contingent being, not a serious candidate to be a necessary being, a common fate for ever so many parodies of a necessary and maximally great being. Anything within the material cosmos or anything composite (emerging in capacity from interaction of parts) fails instantly. Going beyond, the logic is the same logic we use in Math and Science, deduction and induction, the issues are the strength of the logic and the truth of the premises or facts adduced. The modal ontological frame of reasoning is readily shown to be strictly valid per propositional calculus, so the issue pivots on premises and in particular the premise that such a being is possible. As you will observe I have posed the claim that if one wishes to reject that possibility, one needs to have good cause to infer impossibility, as if possible then in some possible world and therefore in the actual one as such NB candidates are embedded in the substructure of any world if possible. The issue on the table still is that Euler’s expression manifests great unifying power across whole domains of Mathematics, pointing to an underlying deep coherence not to be expected of a mere artifact, so there is a lot more of discovery in Math than many would give credit, discovery that points to underlying mind behind the cosmos, as the same order is expressed in ever so many domains of reality leading to Wigner’s remarked on astonishing effectiveness. Discovery that includes not only zero and negative numbers but also the vector structure that lies underneath complex numbers. In that sense, even i is likely a discovery rather than an invention. KF

PS: Consensus is not a criterion of truth by itself, though many truths are a matter of consensus. Warrant is the issue: on what basis do/should we accept a given A as true, an accurate description of reality?

57. 57
daveS says:

KF,

Yes, of course I was being somewhat facetious in speaking of a MGSTC.

The issue on the table still is that Euler’s expression manifests great unifying power across whole domains of Mathematics, pointing to an underlying deep coherence not to be expected of a mere artifact, so there is a lot more of discovery in Math than many would give credit, discovery that points to underlying mind behind the cosmos, as the same order is expressed in ever so many domains of reality leading to Wigner’s remarked on astonishing effectiveness.

Well, I don’t know. What would you expect mathematics to look like in the absence of an underlying mind behind the cosmos?

58. 58
kairosfocus says:

DS, the alternate model on the table is that Math is invented, a special type of modelling. In its way, comparable to diverse car models . . . where I started days back in no 8 above to Jerad (who seems to have dropped out). And while cars tend to have general similarities, we see exactly the lack of overall unity in the abundance of inconsistent parts. Even, among vehicles of the same general type from the same manufacturer. I have seen astonishing differences between an 1800 cc and a 2000 cc of the alleged same model and year. We are NOT seeing that, but instead astonishing convergences and deep interconnexions; in this case injecting i into a series problem dropped out amazing results and opened up gateways for more and more. Which then bleed directly over into precisely mapping to physical reality in ever so many ways as Wigner highlighted. That strongly makes a unifying, logical-mathematical mind look like the explanatory candidate to beat. KF

59. 59
daveS says:

KF,

DS, the alternate model on the table is that Math is invented, a special type of modelling. In its way, comparable to diverse car models . . . where I started days back in no 8 above to Jerad (who seems to have dropped out). And while cars tend to have general similarities, we see exactly the lack of overall unity in the abundance of inconsistent parts. Even, among vehicles of the same general type from the same manufacturer. I have seen astonishing differences between an 1800 cc and a 2000 cc of the alleged same model and year. We are NOT seeing that, but instead astonishing convergences and deep interconnexions; in this case injecting i into a series problem dropped out amazing results and opened up gateways for more and more. Which then bleed directly over into precisely mapping to physical reality in ever so many ways as Wigner highlighted. That strongly makes a unifying, logical-mathematical mind look like the explanatory candidate to beat. KF

Does this mean that if math was an invented thing, then the axioms and/or rules of logic we would be using would be different than those we currently have? If so, how would they differ?

60. 60
kairosfocus says:

DS, again, logic is discovered, at least so far as the core principles of reason tied to distinct identity and sufficient reason are concerned — ponder a bright red ball on a table and the world partition it imposes. There are odd sets of logic axioms out there that arguably boil down to modelling and they do lead off in odd and potentially incoherent directions; e.g. try blending partial set memberships and distinct identity some time. The key point is, that apart from a very deep unity, it would be passing strange for things like the imaginary arbitrary number sqrt(-1) to lock in with exponentials and logs, esp. the value of x such that the area under y = 1/x between 1 and x is 1, and the ratio of diameter to circumference of a circle to interlock in the astonishing way we see 0 = 1 + e^i*pi. Such does call out for explanation, as does the onward extension into Fourier, Laplace and Z transform analysis, system behaviour (differential and difference eqns . . . transfer functions, block diagram algebra and more ) and ever so much more. It is the candidate to beat, that we are seeing coherent order that we are discovering because it was built in from the outset. KF

61. 61
Querius says:

kairosfocus,

Very nicely described, but apparently some of your audience is preoccupied with maintaining the darkness and cursing the light.

But I enjoyed reading the pearls your posted and hope to cast some myself. 😉

-Q

62. 62
Querius says:

DaveS,

Does this mean that if math was an invented thing, then the axioms and/or rules of logic we would be using would be different than those we currently have? If so, how would they differ?

You’ve never taken a class in non-Euclidean geometries, right?

-Q

63. 63
daveS says:

KF,

Could you give specific examples of these axiom systems based on modeling?

Here’s what I’m wondering: If we have the same sorts of modeling in mind, these axioms should also lead to the natural numbers and integers.

Once you have the integers, it’s natural to construct its quotient field, and there’s only one way to do that, resulting in the rational numbers.

Of course then one considers the completion of Q (thought of as a metric space); again, this completion is unique, and we get R. If we skip this step, then we have problems. For example, in Q^2, the line y = x does not intersect the unit circle centered at the origin.

Finally, the algebraic closure of R is essentially unique, so we arrive at C.

You can probably see my point. How can we tinker with our current system, in an attempt at simulating “invented” mathematics, and still arrive at a system which is useful for modeling and where Euler’s formula is destroyed?

64. 64
daveS says:

Hi Querius,

You’ve never taken a class in non-Euclidean geometries, right?

-Q

I have. But you may have misinterpreted my position. I’m just trying to explore how we know that our current mathematical system(s) are much richer in connections than “invented” mathematics.

65. 65
kairosfocus says:

DS, I already did. Partial set memberships are a part of a different approach, which in many cases will contrast with distinct identity. it is possible to hold partial membership in Hot, Warm and Cold sets (to fractions that need not add up to 100%) per a Fuzzy set model sometimes used in control systems; e.g. in Air Conditioners. Membership here takes a quite different meaning from distinct identity and there can obviously be very different results . . . try plotting a Venn diagram and assigning location under such circumstances; the issue of divergence will be dramatically shown. In their time, irrationals, negatives and imaginary numbers were most distinctly not obvious nor a natural progression. That they turn out to come together as in the Euler expression then open up vistas, was a great surprise. Still is. KF

66. 66
kairosfocus says:

Q, let’s hear from you. It will help return things to an even keel. I confess my astonishment at the refusal to acknowledge the rather obvious nexus status of the Euler expression, duly appreciated in context. Which is close to the heart of the themes Wigner sounded as pointed to in the OP. For just one example I remember my amazement at seeing how the Laplace transform turned differential eqns into a visualisable framework, and then I felt like a kid in a sweetie shop when I learned about the heavy rubber sheet representation with poles and zeros and how frequency responses could be read off from that. Later, my students loved it, and years later were telling me how they were still pole spotting from t-domain behaviours esp damped oscillations. All, tying right back to what lies in the context of that one eqn. Microcosm and gateway, even wormhole. KF

67. 67
kairosfocus says:

F/N: I note for record that this morning I invited Steno to respond to my remarks at 37 ff:

http://www.uncommondescent.com.....ent-560307

The lack of response is duly noted.

KF

68. 68
daveS says:

KF,

DS, I already did. Partial set memberships are a part of a different approach, which in many cases will contrast with distinct identity. it is possible to hold partial membership in Hot, Warm and Cold sets (to fractions that need not add up to 100%) per a Fuzzy set model sometimes used in control systems; e.g. in Air Conditioners. Membership here takes a quite different meaning from distinct identity and there can obviously be very different results . . . try plotting a Venn diagram and assigning location under such circumstances; the issue of divergence will be dramatically shown.

Divergence in what sense?

To be clear, what I’m interested in seeing is a complete axiom system for an alternative mathematics, which is as good as our current system is at modeling specifically. I would expect it to support simple tasks such as counting as well as Euclidean geometry and classical mechanics at a minimum (Otherwise, I would expect the mathematicians to keep inventing until they make all this work).

It should also be demonstrably less rich in connections than our current system(s).

Obviously that’s a tall order, and I can’t expect a complete answer here. But if this can’t be at least sketched out, I think it presents a problem for your argument.

In their time, irrationals, negatives and imaginary numbers were most distinctly not obvious nor a natural progression. That they turn out to come together as in the Euler expression then open up vistas, was a great surprise. Still is. KF

Sure, but I’m saying that as soon as you commit yourself to the integers, you are going to arrive at the complex numbers eventually. The fundamental notion of distance then leads to the metric topology on C and hence convergent series.

69. 69
daveS says:

KF,

Q, let’s hear from you. It will help return things to an even keel. I confess my astonishment at the refusal to acknowledge the rather obvious nexus status of the Euler expression, duly appreciated in context.

If you’re referring to me, I obviously accept that Euler’s formula has connections to a wide range of mathematical fields. I don’t even object to the notion that some mathematical concepts are “discovered”, in that I wouldn’t be surprised if any sufficiently advanced community of life forms eventually forms concepts like our “integer”, “rational number”, and so forth.

The specific part of your argument that I question is whether and how we know that our mathematical system is especially rich in connections, and how this is evidence for God.

70. 70
stenosemella says:

Mr. Mullings: “F/N: I note for record that this morning I invited Steno to respond to my remarks at 37 ff:”

I am still awaiting a translation of your comment at 11. My time is too valuable to try to make sense of any of your sermons.

71. 71
Querius says:

Steno,

What’s pretty obvious to me is your unwillingness to comprehend the simple description from kairosfocus about anticipating a paradigm shift in science. He suggests the potential for Euler’s equation, considering it’s fundamental nature, to be applied to the fundamental nature of the universe. No big stretch for most of us. He then used a paradigm shift in his perception of economics as an analogy.

If you were really interested in understanding what kairosfocus was thinking, you’d have come up with something a better than a question that would be considered rude in any college classroom, especially the bit about your time being too valuable.

-Q

72. 72
Querius says:

DaveS wrote,

I’m just trying to explore how we know that our current mathematical system(s) are much richer in connections than “invented” mathematics.

So in relation to your question, what did you conclude from your non-Euclidean geometry class(es)?

-Q

73. 73
Querius says:

kairosfocus,

First off, my first reaction to Mario Livio’s observation was that he’s describing the divide between theoretical (aka pure) mathematics and applied mathematics.

As many mathematicians in the past have done, you expressed a sense of appreciation and astonishment at the assemblage and relationships in Euler’s equation—that compact “mathematical jewel” with representative facets of addition, multiplication, exponentiation, e, pi, i, integers, irrational numbers, imaginary numbers, and a playful hint at deeper relationships within. What’s not to be amazed at?

But I noticed the different reactions.

On one side, people had a sense of delight, astonishment, and curiosity. On the other, there was dismissiveness, suppression, and reduction.

To me the second group is what I’d expect from academic arrogants, bureaucrats of books and bells who crush their students’ curiosity, creativity, and enthusiasm under the weight of their need to dominate, and who leave a trail of injured survivors in their wakes. What a pity. You give people like this some poetry, and they’ll alphabetize the words, count the letters, and announce that they’ve seen nothing new.

In contrast, I once had the privilege of attending some lectures on physical chemistry many years ago. The presenter was so enthusiastic about his subject, it was infectious! His explanations and illustrations were simple, clear, and delightful. What a difference!

-Q

74. 74
kairosfocus says:

Steno,

First, you doubtless know why I use my consultancy persona in online discussions — it reduces spam issues and ID theft opportunities. Your disrespectful and rude insistence on violating that speaks volumes, volumes that fit in with your general attitude as has been on display.

An attitude that is further manifest from your playing the you no speaka da Inglish rite tactic, despite the fact that in 18, there was an expansion of what was said in 11 . . . with an expanded summary using an apt wiki clip, subsequently, links to two appreciations out there.

My response at this point is that I do not believe you in your rhetorical stance, for more than enough has been given for one who is genuinely interested to understand.

From me and from others.

Q’s comment speaks volumes, which you should heed.

(Q, thanks.)

I think the truth is, this case is one that speaks powerfully to how some major features of Mathematics open up vast vistas, showing an inner logical architecture of reality that invites reflection on where you evidently are desperate never to go: there is a divine mind’s shadow on the doorstep.

In Mathematics, of all places.

But then, Mathematics is by nature an abstract, defiantly non-science discipline on the power of pure logic studying quantity, structure and relationships that somehow seem to govern so much of reality. A discipline that turns out to be a major necessity in Science. With it’s daughter-discipline, computing, alongside.

In short, your response speaks to this being one of the many chinks in the armour of today’s dominant (and too often domineering) evolutionary materialist scientism and its fellow travellers. But, such from the outset, cannot coherently account for mind and its powers of knowledge, insight, reason and purpose, so that is not news.

I think that the very same Euler, a Christian, has somewhat to say. I think something was clipped above, but a fuller cite is appropriate, from a letter to a Princess on the so-called free thinkers of his time:

The apostles and a multitude of Christians [–> 500+ eye witnesses, recorded about 25 years after the event while most of these were still alive with an implied invitation to check them, cf here on] unanimously agree not only that Jesus Christ rose from the dead, but also that they have seen him with their own eyes since the resurrection and that they even communicated with Him. If one has paid attention to the doctrine and to the constancy with which it been maintained [–> i.e. in the teeth of not only objections but dungeon, fire and sword or worse], one cannot say with any semblance of truth that one has believed nothing of what has been said in this regard and that it is thus an obvious lie. One would be even less likely to say that the apostles were seduced by false imagination and that their facts were nothing but an illusion. Either that or we will be forced to state that God had miraculously blinded them all at the same time in order to propagate a false doctrine . . . .

The resurrection of Jesus Christ is . . . an incontestable fact, and since such a miracle can only be the work of God alone, it is thus impossible to doubt the divinity of the Savior’s mission. Consequently, the doctrine of Christ and his apostles is divine, and since its goal is our true happiness, we can be most assured of our belief in all the promises that the Gospel has made to us, both for this life and the one to come, and we can regard the Christian religion as a work of God who is tied to our salvation. It is not necessary to expand any further on these reflections, since it is impossible for anyone, once they are convinced of the resurrection of Jesus Christ, to retain the slightest doubt about the divinity of the Holy Scripture.

The freethinkers cannot put forward anything plausible against this bedrock on which the divinity of the Holy Scripture firmly rests. When they are forced to turn their attentions to this, they do all they can not to address the root of the question. They resort to all manner of loopholes to change the subject and attack other items, where they claim to find incomprehensible things and even contradictions . . . .

The freethinkers have yet to produce any objections that have not long been refuted most thoroughly. But since they are not motivated by the love of truth, and since they have an entirely different point of view, we should not be surprised that the best refutations count for nothing and that the weakest and most ridiculous reasoning, which has so often been shown to be baseless, is continuously repeated. If these people maintained the slightest rigor, the slightest taste for the truth, it would be quite easy to steer them away from their errors; but their tendency towards stubbornness makes this completely impossible. [From: A DEFENSE OF THE REVELATION AGAINST THE OBJECTIONS OF FREETHINKERS, BY MR. EULER FOLLOWED BY THOUGHTS BY THE AUTHOR ON RELIGION, OMITTED FROM THE LAST EDITION OF HIS LETTERS TO A PRINCESS OF GERMANY, 1805, point XXXIV on. (It is well worth the pause to read the printer’s remarks on how these things came to be excluded from the generally accessible editions of the work.)]

KF

75. 75
kairosfocus says:

DS, I believe I have provided enough for you to follow up if you wish to explore the issue of limited mathematical modelling that does not typically have the sort of deep, powerful, broad, vista-opening coherence and power we see from the Euler expression in context. We do live in an age of Google search. The point I am making, is that certain facets of Mathematics show the sort of coherence, beauty and power that this case we are focussed on as a chief exemplar of Wigner’s remark on the astonishing effectiveness of Mathematics in the physical sciences, exemplifies. It matters not to that point that other areas show more of the modelling approach and divergences that I have compared to how car models have similar general architecture but utterly divergent parts. I think we need to redevelop a sense of wonder and appreciation that acknowledges the sheer, raw beauty and power of such things. You give me reason to appreciate my conversations with former students who spoke of how, long years after studying Control Systems together, they were still finding themselves pole spotting from time domain damped oscillatory behaviour, on the heavy rubber sheet model. Which, BTW is very much connected back to Euler’s work. KF

76. 76
kairosfocus says:

Q, thanks again. I see you have been sitting with Sophie and listening to her. She has much to teach us, if we will but listen. KF

77. 77
Don Pedro says:

So in relation to your question, what did you conclude from your non-Euclidean geometry class(es)?

Non-Euclidean geometry does not represent a different kind of mathematics. It is a generalisation of the notion of “geometry” (working according to the same logic and with some of the same or similar axioms). By the same token, complex numers are a generalisation of the notion of “number”. Nothing new here. Fractions, and negative and irrational numbers (not to mention the concept of “zero”) also had the status of “invented” extensions of classical arithmetic until they became familiar enough to be considered “normal”. Complex numbers have been around for centuries now, and many people no longer regard them as esoteric.

e^({pi}*i)+1=0 is an impressively elegant expression, but it doesn’t mean much by itself and you can’t deduce much from it. It’s just a special instance of Euler’s general formula, with x={pi}. The latter formula is vastly more useful and interesting, but hardly mystical. Mathematics is better served if you try to understand it thoroughly and not just gaze at selected equations in mute awe (or worse still, if you fall into flight of ideas and don’t stay mute).

78. 78
kairosfocus says:

F/N: i will say this, this thread has led me to do a couple of days searching on the issue of the one and the many, which I have always seen as central: unity amidst diversity in an evidently coherent, common world. I am finding an amazing lack of broad and deep reflection on it in current times, reflected in how deep I have had to search to find things of interest and substance. And yet it is one of the earliest concerns of Western thought, and is of deep, enduring significance today. I am thinking that this, too points where ever so many are desperate not to go, and it has become a road mainly not travelled save by the odd theologian-philosopher and one or two others. At least, that is the strong impression I am picking up. It certainly does not come up for the sort of rich and deep reflection I would expect of something that is such a nexus problem. But then, this is the age of the narrow specialist and this problem is just the opposite of the leanings of our day. Methinks De Bono on the power of broad, lateral thinking may have somewhat to say to us. I think I need to book a long, leisurely lunch date with good old Sophie, our elder sister and counsellor. And take along my newly modded 20+ year old Sheaffer fountain pen that now has about a 0.8 mm italic/calligraphy style broad nib by dint of grinding per recipe of pen enthusiasts. It did my heart good to get my fingers dirty with ink and swarf, etc. with an end product that gives me what a classic Parker 51 stub italic nib would do. Joins my new Parker 45 ball pen (I picked that up from Amazon on seeing it is no longer being made . . . lost one a decade back under suspicious circumstances) and longstanding 0.5 mechanical pencil. KF

79. 79
kairosfocus says:

DP, Interesting points. I think the key issue is connexions. Which are pivotal to seeing the microcosm revealed. Just out of curiosity, how do you respond to the Mind/Concept Map idea: http://en.wikipedia.org/wiki/Mind_map . (I think this is a useful test on the sort of nexus thinking issue that underlies this thread. As a hint, I love it as an ideas and connexions visualising tool that is an aid to creativity and to thinking through. In my day job so to speak, I am in the business of multi-level, multi-directional strategic, three moves plus ahead “chess.” That is your move now needs to be shaped by perceived opportunities for various counter-moves and options for your onward move, etc; all, in real time and with heavy stakes on the board.) KF

PS: BTW, I use concept frameworks and connexions to then guide exploration. As a curriculum designer, my favourite course architecture is a spiral web with key concept and activity anchor points and lines, that uses a spiral of learning activities to introduce and deepen exposure to a subject loop by loop that deepens insight on the key points as development proceeds. While, making use of key, nexus case studies. And, my fav development strategy is the evolutionary spiral http://en.wikipedia.org/wiki/Spiral_model and note US DoD here http://www.dau.mil/pubscats/Pu.....k-ja03.pdf

80. 80
kairosfocus says:

Q, I think you have a point on paradigms, shifts and how there is a gap across the paradigms that leads to living in different mind spaces. The cramping nature of a priori evolutionary materialism is on shocking display. KF

PPS: Here’s some of my thought on economics and policy:

On industrial renewal:

http://www.angelfire.com/pro/k.....ics_ja.htm

Here is thought on education:

81. 81
daveS says:

Querius,

So in relation to your question, what did you conclude from your non-Euclidean geometry class(es)?

Well, I didn’t really attempt to quantify the number of connections corresponding to any particular choice of axioms. However, it’s clear that you can get interesting and useful mathematics even in a non-Euclidean setting.

82. 82
kairosfocus says:

DS, Yes, especially projective, Spherical and of course Riemannian; where the divergences on the parallel lines postulate illustrate the different car parts analogy rather aptly. None of that detracts from the force of the shocking convergence and breakthrough connexions that lurk in the Euler expression. KF

83. 83
daveS says:

KF,

DS, I believe I have provided enough for you to follow up if you wish to explore the issue of limited mathematical modelling that does not typically have the sort of deep, powerful, broad, vista-opening coherence and power we see from the Euler expression in context. We do live in an age of Google search.

And I do appreciate the discussion. In fact I had done some googling, but didn’t find anything relating to the number of connections in a fuzzy logic-based system. I think it’s your responsibility to provide that information. I also think that’s an impossible task—you simply can’t make the kind of comparison necessary to support your claim about the relative impoverishment of “invented” mathematics.

The point I am making, is that certain facets of Mathematics show the sort of coherence, beauty and power that this case we are focussed on as a chief exemplar of Wigner’s remark on the astonishing effectiveness of Mathematics in the physical sciences, exemplifies. It matters not to that point that other areas show more of the modelling approach and divergences that I have compared to how car models have similar general architecture but utterly divergent parts. I think we need to redevelop a sense of wonder and appreciation that acknowledges the sheer, raw beauty and power of such things. You give me reason to appreciate my conversations with former students who spoke of how, long years after studying Control Systems together, they were still finding themselves pole spotting from time domain damped oscillatory behaviour, on the heavy rubber sheet model. Which, BTW is very much connected back to Euler’s work. KF

I share the same sense of wonder. I like mathematics as much as the next person. It’s the coolest thing ever, IMO. In fact, I probably spend more time than I should reading about it. But in the end, I agree with Don Pedro in #77.

84. 84
kairosfocus says:

DS, Pardon me but this takes on the face of a case of successive tangents that ends in pointless distraction. In reply to Jerad, I pointed to a striking case of the convergence and broad sweep of implications of the Euler eqn. I had also pointed out that had Math in general been instead like a collection of different models, you would not be likely to see the sort of deep coherence within math and onwards into experienced reality. Ever since all sorts of tangents, some loaded with toxic points and insinuations. On your request I have provided now two cases of the divergent models effect, fuzzy logic and non Euclidean Geometry which shows sharp divergence precisely on the parallel lines/angle sum triangle axiom. I have shown my point, making a reasonable contrast to two families of the sort of divergence that car parts show. This contrasts with what happened surrounding Euler’s 1748 breakthrough. Now, I see a manufactured, projected “duty” to provide axioms, connections and so forth, doubtless on the assumption that if I fail to produce what would in effect be a collection of monographs, my main point can be dismissed. Not so. I have done what is needed, and have illustrated that a core part of relevant Math reflects a deep unity in the world and in math, which aptly correspond. By contrast, other things do not have that character. My point is made and in that core area there is good reason to infer that we are discovering a core deep coherence in reality that a very good candidate explanation is that said reality comes from a deeply logical-mathematical designing mind. KF

PS: I do not include general logic as an example as it is better seen as pre-math, philosophy. And yes, that also has diverse possible constructed frameworks.

85. 85
daveS says:

Yikes. “Toxic points and insinuations”? I certainly didn’t have anything like that in mind while typing my posts.

The last part of my post #69 describes the specific question I have. Just pointing to fuzzy logic and non-Euclidean geometry doesn’t really address it.

And to be clear, of course I understand the value of working with a variety of axiom systems, even those which are “nonstandard”. I’m all in favor of that, and obviously it has been productive.

86. 86
kairosfocus says:

DS, Kindly note my observation, some. In this thread I have had to deal with outing, personal attack and indirect personal attack meant to associate me as an advisor with a policy made to sound foolish by omitting highly material and easily ascertained facts that have been in global news headlines in many cases but which may now have faded from public memory. Just scroll up. The matter is so bad — and so bad in connexion with another attack in another forum emanating from much the same circles — that I have had to lay out more on my background than I would have liked. KF

87. 87
daveS says:

DS, Kindly note my observation, some.

Yes, thanks for clarifying.

88. 88
kairosfocus says:

DS, I am sorry, I did not just list the names Fuzzy Logic and Non-Euclidean Geometry. I specifically indicated ways in which the former diverges from normal set membership, and how the latter, right from axioms is deliberately divergent from Euclidean Geometry on the parallel lines postulate or axiom. Where, it is implicit — but should now be pointed out — that set theory is foundational to modern mathematics and the discovery of non euclidean geometries as full bore axiomatic systems were a key start point for modern mathematics. The analogy of cars sharing a common general architecture but having divergent mutually inconsistent parts is thus seen to be apt. By contrast, what happened with the eqn 0 = 1 + e^i*pi, was an astonishing convergence of whole domains that then flooded out again in a vast array of areas directly relevant to so much of reality. I just note how one of the linked articles points out how the transcendental numbers e and pi, coming from utterly different contexts and with all the strange properties of being transcendental, now turn out to be in perfect lockstep not just in a few digits but to infinity . . . that alone should give us serious pause to think on this eqn and what it is saying. As to raising e to the i*pith power, that is in fact the crux as through this we access the world of phasors and rotating systems, also onwards Fourier and Laplace transforms thence differential and difference eqns, systems, signals, responses, system dynamics and hugely more. This fully justifies the point that the form e^i*pi = -1 is also shocking. Not to mention the quantum world. It is no accident that Physicists voted it as comparable to Maxwell’s four eqns for electromagnetism, which have had the same sort of power of synthesis. Newtonian Dynamics and the like are comparable. The difference is, Euler’s expression in context is wholly mathematical, will never be superseded, and shows an astonishing unity of both mathematics and the real world. Fourier, Laplace, Z transforms, differential and difference eqns, signals and systems etc etc etc. A vast sweep lurking as the berg beneath that simple little expression and its immediate context, a direct context accessible to a 6th former. KF

89. 89
daveS says:

KF,

DS, I am sorry, I did not just list the names Fuzzy Logic and Non-Euclidean Geometry. I specifically indicated ways in which the former diverges from normal set membership, and how the latter, right from axioms is deliberately divergent from Euclidean Geometry on the parallel lines postulate or axiom.

😕 I’m also sorry, because that doesn’t address my question. No doubt if you try to merge the axioms for Euclidean and hyperbolic geometry, you will get an inconsistent system, so the two systems are incompatible in that sense, and similarly for fuzzy logic.

Let me reflect back to you your argument as I understand it:

We find a more than expected number of connections in mathematics, exemplified for instance by the nexus of Euler’s identity.

This suggests the existence of some “root mind” ordering reality and assisting us in the development of our mathematics.

In the above, “more than expected” means “more than expected under the null hypothesis that there is no root mind ordering reality and that mathematics is purely invented by humans”.

Is that correct?

I understand that you do believe that our current mathematical systems have been developed with the help of God (including non-Euclidean geometries and fuzzy logic perhaps?). And we do see many interesting connections, as you have pointed out.

My challenge is for you to demonstrate that, assuming that God had withheld His assistance, and humans were forced to invent mathematics on their own, that there would be fewer of these connections (which is obviously an impossible task, but how else could you proceed?).

90. 90
kairosfocus says:

DS, If you look at the OP, you will see that it is raising the issue of the effectiveness of math and asking about discovery vs invention. I have pointed out on a relevant case, signs of the sort of unexpected coherence and convergence then flowing back out again with awesome power that should seriously raise the issue of deep coherence and undesigned coincidences because of underlying unifying mind at work. Such, that we find it a very reasonable point to ask, are we here for these things thinking God’s thoughts after him? The side tracks and side points are essentially irrelevant to this point. I have shown that some things have the sort of coherence that we normally associate with unifying mind, rather than massive coincidence of inventions. For sure, pi was identified for utterly distinct reasons in a different time and culture than both i and e. Area under 1/x beyond x = 1, such that A = 1 to become the base of a natural log system, is not particularly deliberately tied to the value of pi. The proposing of a sqrt of -1 to make all polynomials have solutions, and the identifying of series expressions for sin and cos and e^x are not particularly initially connected either. Euler noticed the resemblances in the series and injected i into the sin series and lo and behold, cos theta plus i sin theta equals e ^i*theta. Then substitute theta is pi and bang. It turns out e, i and pi are connected down to the very end of the infinite series to construct each. Then, bang, bring on Fourier and Laplace analysis and this naturally extends into a vast range of domains that are deeply significant in the sciences and in reality as we understand it. All this has been pointed out, explained, discussed in your presence. All I am saying is that that sort of neatly fitting jigsaw puzzle that comes together all at once then goes out again and conquers the physical world for Mathematics, is saying something, something powerful about an inner logical-mathematical coherence to reality. And that sort of coherence invites reflection on unifying mind. For, this is not like car parts that though cars have similar architecture, constantly show themselves mutually incompatible. That, is what things that are micro-designed usually are like, exapting something new from the parts is generally very hard due to lack of standardisation. But, I know, this is probably very hard for you to begin to acknowledge as even being remotely significant; it obviously potentially points just where you would not go. KF

91. 91
daveS says:

KF,

All I am saying is that that sort of neatly fitting jigsaw puzzle that comes together all at once then goes out again and conquers the physical world for Mathematics, is saying something, something powerful about an inner logical-mathematical coherence to reality.

I do agree with this.

And that sort of coherence invites reflection on unifying mind.

Reflection, sure, but I don’t find that it sways me one way or the other on whether such a unifying mind exists.

92. 92
kairosfocus says:

DS, I do not expect it to. I have had to deal with people willing to burn down the temple of reason to maintain their evolutionary materialist ideology. And, I have seen people more than willing to indulge in ugly threats against uninvolved family and those who pretend that all is well and act as though such web thugs are normal people in their behaviour; thus enabling it. Above in this thread there is a shadow of this. KF

93. 93
daveS says:

Rest assured that I’m not in favor of burning down any temples. I like mathematics in part because I’m interested in the acquisition of knowledge, not its destruction.

94. 94
kairosfocus says:

DS, I have pointed out that here at UD we have dealt with many cases of evolutionary materialist atheists and fellow travellers who not only cling to a self-referentially incoherent ideology but are perfectly willing to burn down logic starting with first principles of reason. Far many more, are so locked into ideology that they become just that, locked in. KF

95. 95
daveS says:

Well, not being particularly well-versed in philosophy or biology, I don’t have much to say about materialism or evolution.

96. 96
stenosemella says:

Gordon: ” And, I have seen people more than willing to indulge in ugly threats against uninvolved family…”

You have claimed this repeatedly but when asked to provide details you run away. Please stop pretending to be the victim. You claim victimization and act like a drama queen every time someone uses your real name. And then in the same thread you link to a document with your name on it. Have you ever heard about the boy who cried wolf?

97. 97
Barry Arrington says:

UDEditors: stenosemella has been shown the exit.

98. 98
Querius says:

Kairosfocus @ 76

Yes, and Sophie has been extremely patient with me. Often her insights cut deep, because she doesn’t want to change my thinking as much as who I am, which can be a little scary.

But I wouldn’t want to sit anywhere else!

-Q

99. 99
Querius says:

Kairosfocus @ 80

Regarding paradigm change, I was reminded of how precisely Ptolmaic astronomers were able to predict celestial events such as eclipses and planetary motion with ever more complicated epicycles, and then came the Copernican revolution, which didn’t actually produce better results . . .

Macroeconomics fascinates me, although I’ve only ever taken one class in the general subject, which I found disappointing (yes, Samuelson). I’ve heard that economists have conducted experiments on the economies of MMORP games, which I understand tend to crash despite the god-like control of the moderators and programmers. Lately, I’ve been thinking about consumer spending, how it does matter what people purchase, which molds the supply side, and perhaps (I’m totally speculating here) cheapens (inflates) the currency if consumer spending is foolish. Thus, I suspect that there’s a significant connection between economic health and moral health.

Regarding your third link on education, I’d encourage you to look into the education system in Finland as compared to that in California, where over half the state budget is spent on “education” with dismal results for the children (education bureaucrats are paid embarrassingly high salaries). I’d also recommend John Sweller’s Cognitive Load Theory since it’s data driven with profoundly better results rather than ideologically driven theories (B.F. Skinner comes to mind).

-Q

100. 100
Querius says:

DaveS @ 81

However, it’s clear that you can get interesting and useful mathematics even in a non-Euclidean setting.

Yes. Exactly. The mathematical reasoning and structure transcends artificial geometries.

DaveS @ 89

No doubt if you try to merge the axioms for Euclidean and hyperbolic geometry, you will get an inconsistent system, so the two systems are incompatible in that sense, and similarly for fuzzy logic.

You might already know this, but your observation touches on Kurt Gödel’s incompleteness theorums. Here’s a good description of a paradigm shift in mathematics and logic:

http://plato.stanford.edu/entr.....pleteness/

So, if there is a unifying pragmatic system of mathematics, it’s apparently inaccessible to us in this universe. Together with Chaos theory, this pretty much explodes scientific determinism.

It should be noted that Gödel’s application for a paid position at the University of Vienna was turned down by people that history has now forgotten.

-Q

101. 101
daveS says:

Thanks for the info and link, Querius. As it happens, I’m trying to read about model theory right now and this should be very helpful.

102. 102
kairosfocus says:

F/N: It seems there is a settling down of this thread. The deep unities highlighted by the Euler eqn that also bridge so much of Mathematics, Physics and engineering are there. One’s response to same will both be shaped by and will shape one’s worldview. This definitely puts on the table at that level the issue that in relevant domains of Mathematics, one is confronted by the possibility that the astonishing power of mathematical considerations in the physical science may reflect that the cosmos itself reflects not just fine tuning but framing by a highly mathematical mind. KF

PS: On a less happy note, a few words need to be said after taking time to ponder how to speak to truly difficult to address issues without further giving currency to slander, outing and implicitly menacing intimidation. As in: we know you, where you are, those you care for, their homes or places of business etc. including things not readily found on the Internet that suggest on the ground casing of the joint. Ah, but we are not stalking or threatening you and those you care about, see. Mafioso style tactics that should not fool any decent person above five years of age, as has been going on for years in a penumbra of attack sites:

a: Immediate context. Across yesterday and going back some time, I have had to deal with further word snipping, twisting and false accusation tactics emanating from circles that operate such sites or are harboured by them. And no, I will not give salacious details.

b: It seems, the theory used by such rage-fueled unhinged attackers is that if one cannot cogently address a matter fairly on substance, then public false accusation, twisting words and circumstances into falsehood, raising undue suspicion and/or harming reputations through snide suggestions and making implicitly menacing references to uninvolved family can intimidate into silence. Or at least hopelessly polarise, poison, cloud and confuse the atmosphere as part of cynical divide, poison and rule tactics.

c: I must also note, again, that it is highly reasonable to request that the privacy of one’s name, email address, uninvolved relatives etc be respected in high traffic, contentious contexts on the Internet where spam crawlers and identity thieves etc as well as cyberbulies are likely to lurk in one form or another, whilst in much lower traffic corners of the net academic papers etc will give information per legitimate requirements of publication.

d: The deliberate and willful exposure of personal information in the context of cyberstalking and defamation, is cyber harrassment and worse, tort. Also the extension of such in contexts of attempting to undermine livelihood, falsely accuse of family abuse, consorting with criminals [as opposed to supporting the rehabilitation of former criminals], outing of uninvolved family etc is multiply and implicitly threatening behaviour.

e: Likewise, there is a reason why certain classes of cases protect the victim from public humiliation and forced further spreading of accusations. The former practice of trying to publicly humiliate and discredit rape victims in court is a thankfully now finished case in point. That is, there are good reasons to refuse to further engage details of hostile attack and abusive behaviour in public. Let us just say here that the relevant authorities with editorial power do know enough detail to base their actions. And so do the civil authorities.

f: Further to this, let the proverbial frog speak to the boy approaching, stone in hand: “fun fe yuh, is death to me.” In that vein, I note to the jack-booted bully-boy radicals of today who are playing at mob rule tactics and discrediting and smearing targets to seemingly justify attacks on freedom of expression, association and conscience:

He who would rob me of my means of daily bread,
would rob me of my life;
He who would rob me of my children,
would rob me of my posterity;
He who would rob me of my free voice and good name,
would rob me of my means of defending myself;
He who would rob me of my conscience,
would rob me of my soul.

g: Those who harbour, cosset or enable such behaviour and those who indulge in such should ponder what they are letting loose in our civilisation. For, as Solomon so aptly warned, life and death are in the power of the tongue and those who love it will eat the fruit thereof. I note too, as follows, from an author recently subjected to a slander attack, and forced to reply to false accusation of racism, by revealing his (inter-racial) family photo:

. . . a blog “journalist” . . . working beneath the banner of Entertainment Weekly — penned a short, error-laden article titled, “Hugo award nominations fall victim to misogynistic, racist voting.” The mistakes in the article could have been easily avoided if [X] had done some research into the issue she was reporting on. Near as I can tell, [X] was spoon-fed some links and a very rushed and sloppy narrative about Sad Puppies 3 being racist and woman-hating, and she posted all of this without stopping to consider whether or not anything she was disseminating into the wider world was true, and accurate.

The error-laden article quickly went viral — especially among opponents of Sad Puppies 3. Twitter (which I generally avoid and ignore) lit up like a Christmas tree, and quickly I had friends and other authors contacting me to say, “Entertainment Weekly has run a hatchet piece on you! Better jump on it!” So I read the piece. I noted the errors. I also noted that the piece made an explicitly inductive link between Sad Puppies 3 and last year’s great nerd controversy: GamerGate. The reasons for this were pretty obvious. Words like “racist” and “misogynist” are presently code for “not part of the human equation” thus any man or woman who can be successfully labeled these things, is cut off from polite circles, perhaps even driven out of the workplace, or worse. These words tend to be used as general-purpose ideological grenades, when the thrower of said grenades lacks sufficiently real evidence of wrong-doing — but wants to see the target squirm and suffer anyway.

An unadulterated version of the . . . article can be found here.

The presently “corrected” version of the article — Entertainment Weekly made several alterations to the article, including its root URL, after myself and many others noted that the article was a) grossly in error and could also b) serve as grounds for libel litigation — can be viewed here.

But of course, by then, the damage had been done. Both myself and my colleague Larry Correia — who typed up a very good piece here — had been dragged through the digital mud. I count no less than a dozen different links which all picked up the error-laden . . . piece, and ran with it sight-unseen; because of the blaring headline. Again, the spoon-fed in turn spoon-feed others. And whatever hope there might have been that facts could trump a narrative, was lost in the white noise of a not-so-subtle smear job.

I can’t say which individuals decided to launch the smear job. The internet moves at the speed of light and the 24 hour news cycle is forever hungry for new material; regardless of how bogus it might be. I suspect some of the insider SF/F people who dislike Sad Puppies 3 decided that the best way to “win” the insider baseball argument, was to stage a broader media flare-up for the sake of fatally discrediting the “poster people” of Sad Puppies 3. Namely, myself, and Larry Correia.

Now, I am a patient man. I’ve got a long fuse. But this tactic employed today . . . it’s on another level. This isn’t just nerds bickering anymore. Baseless, false allegations of this type can ruin careers as well as lives. We’ve seen this before. The internet has allowed yellow journalism and rumor-mongering to run riot. And I have to be honest. No lie told in the service of a supposedly higher cause, ever does justice to the higher cause. No matter how widely-spread the narrative. If the basis of the narrative is false, then the narrative itself is fatally undermined, and thus the arguments that form the building blocks of the narrative are themselves undermined.

Oh, I am sure this will blow over eventually. To Entertainment Weekly’s credit, they did take action — once myself and others explicitly told them that the . . . article was a dreadfully poor piece of research. Frankly, I feel like [X] herself owes myself and Larry Correia a public apology. That was shoddy reporting. And it’s potentially very damaging. I mean, if you’re going to play a role in somebody else’s effort to trash us, at least spend two minutes doing a little googling.

I understand that tabloid tactics make money, and that on the internet especially, journalistic integrity has become something of an oxymoron.

What disturbs me more is that the field of SF/F is stooping this low. That some of my colleagues — and no, contrary to my impression of the field 20 years ago, not everyone likes or gets along with each other — have decided to make the nerd argument over the Hugos into a decidedly personal grudge match. Where the objective is to not just win the argument, but to destroy the arguer. Professionally. In the marketplace. On the big stage of public opinion. This is the kind of stuff you ordinarily find in cut-throat national political elections, but then it’s been clear for years that cut-throat politics have drifted down into nerd circles of all kinds: comic book circles, movie and television circles, video game circles, etc. There’s simply no escaping it. And there are people for whom winning is more important than ethics, more important than integrity, and more important than the truth . . .

As the Greeks say, a word to the wise . . . and, if one’s neighbour’s house is afire, wet your roof and help him put out the blaze.

103. 103
sparc says:

96 stenosemella April 22, 2015 at 3:50 pm

Gordon: ” And, I have seen people more than willing to indulge in ugly threats against uninvolved family…”

You have claimed this repeatedly but when asked to provide details you run away. Please stop pretending to be the victim. You claim victimization and act like a drama queen every time someone uses your real name. And then in the same thread you link to a document with your name on it. Have you ever heard about the boy who cried wolf?

97 Barry Arrington April 22, 2015 at 4:07 pm

UDEditors: stenosemella has been shown the exit.

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kairosfocus says:

sparc: FTR, the extent of outing and slander seems to get worse — more and more unhinged — on each drearily predictable willful cycle of playing the distract distort demonise poison discussion stratagem too often used or enabled by objectors to design theory. I suggest you and ilk ponder 102 above before further enabling inexcusable willful wrongdoing: http://www.uncommondescent.com.....ent-560570 KF

105. 105
kairosfocus says:

F/N: let me link on modelling & sim body of knowledge, what I have followed up on what has been positive above, overnight: https://www.csiac.org/sites/default/files/DoD%20M&S%20BOK%20%28M&SCO%29%202008_0.pdf KF

106. 106
daveS says:

Eh? What’s this pdf in #105 about?

107. 107
108. 108
kairosfocus says:

DS, Modelling and Simulation body of knowledge as compiled by US DoD. I think you will find it useful, though the tabulation for edu/training purposes may be a little off-putting. KF

PS: This too may help:

http://www.site.uottawa.ca/~or.....MS-BOK.pdf

and this on b/gd Math:

http://www.math.toronto.edu/weiss/model_theory.pdf

(That one is very broadly applicable, note how they deal with strings for instance.)

109. 109
daveS says:

I see. Thanks, the last link, dealing with the model theory connected to mathematical logic, looks especially useful.

110. 110
REC says:

KF,

I respect your effort to remain anonymous.

However, when your links from your name or documents posted here contain your real name, I’m not sure you can fault people for calling you by that name.

Also, where we you when the owner of this blog posted full names and work information of individuals who made him angry?

111. 111

UDEditors: stenosemella has been shown the exit.

You are a coward. Clearly.

And kairosfocus has two new post with comment closed from the start.

112. 112
unwilling participant says:

I must agree with KairosFocus on his desire for anonymity. If my family had been threatened, as his obviously has, I would demand anonymity as well. The fact that he makes it easy to identify him is beside the point.

113. 113
kairosfocus says:

REC:

These are people who know the situation and are trying to use my name in connexion with a stalking campaign.

What is on the table is not innocent use of a name, which would respond to polite request not to unduly expose to spamming and stalking etc.

The insistence we see is directly connected to a stalking campaign motivated by obvious malice and utter want of common civility.

When uninvolved people at three degrees of remove from me are being spattered with the mud slinging, that speaks volumes.

Namecalling.

I will just say that TWO independent sets of editors have had to act in the context of patent defamation by the unhinged and obviously deeply malicious. I have spoken for record, for reference, and given the sort of stalking . . . now likely on the ground . . . outing and smearing of uninvolved people and their affairs that has now been resorted to (for the “crime” of being connected to me in some remote degree), I would advise you to cease and desist if you care one least bit about common decency.

And if you cannot find in yourself the decency and good sense to realise the sort of lines that have been crossed and what it reveals about the tone and attitudes too long harboured and enabled by the attack sites, I am confident that a great many people will.

If what you have left is harbouring and enabling the malicious and unhinged in order to distract from issues on the merits and poison the atmosphere, then crying censorship and the like in the face of actions in response to outright tort and stalking, that speaks volumes.

G’day.

KF

PS: More from the author above as food for thought on what is going on:

>>Because I wanted to talk specifically about a recurrent kind of “broken” I am seeing in arguments all over the place — beyond the tiny halls of the Peoples Republic of Science Fiction. This “broken” is most commonly manifested among well-meaning straight Caucasian folk, but is often fostered and preached about by non-straight and/or non-Caucasians of a particularly aggressive “progressive” persuasion.

The “broken” goes like this:

? Any member of the majority group is always guilty of ism no matter what.
? The ism is a fatal character and moral flaw, from which the afflicted cannot fully recover.
? Members of the minority group can never be guilty of ism; because the minority group lacks power.
? Members of the majority group must be “shown” their ism and/or be made to confess their inherent flaw; en route to being reformed.
? Reformed members of the majority group will actively assist in pointing out the ism of unreformed people.
? Proof (of ism) is not required; guilt will always be assumed.
? The more a target denies or resists charges of ism, the greater the obvious culpability.
? The unreformed are “fair game” for all manner of actions designed to be personally destructive to the target(s).
? Lying (and other unethical behavior) on the part of the plaintiff(s) — against the target(s) — is forgivable, because the ends justify the means.

Folks, it doesn’t take a genius to figure out that this is Witch Trials 101. There’s nothing new about any of it. All we’ve done is wrap the witch trials up in trendy 21st century politics. When people are automatically guilty of a crime, for which no evidence is required, and the accusers are permitted a kind of political immunity while the protestations of the innocent are merely used as “proof” that the innocent were guilty all along . . . that’s witch trials. Burning at the stake. Except, these days the stake and the village square are located on the internet, and the villagers — with digital pitch forks and torches — are anyone with access to Twitter, a blog, Facebook, or some other form of social media. >>

114. 114
Barry Arrington says:

115. 115
kairosfocus says:

116. 116
Querius says:

kairosfocus,

I’m truly sorry to hear about all this.

The behavior you describe exposes both the concession of rational argument and the underlying murderous motivation of certain individuals.

People who launch such destructive campaigns are arrogantly self-righteous. Tolerance extends only to their views and those they consider compatible. They can then safely lash out as judge, jury, and executioner from their personal fortress of Victimhood.

They apparently see their barbaric actions as wholly acceptable and fully justified. As a result, they can vent their rage, frustration, and hatred without limit against all who dare to challenge their self-indulgence. Their “cause” collectively absolves them of any and all criticism.

They are perishing.

-Q

117. 117
kairosfocus says:

Q, thanks. Though I suspect that someone who tried to take direct action against the latest targetted branch of my extended family would come off very badly indeed in the resulting rather direct confrontation. KF

PS: The fact that this seems to be what they have left itself speaks volumes on the merits. And the kind of neo-fascism that now stalks our civilisation is suicidally foolish.

118. 118
truthbringer says:

Hi everyone!

119. 119
Querius says:

Hi tb.