Review of the book Infinitesimal by Amir Alexander

Beyond the boundaries of established science an avalanche of exotic ideas compete for our attention. Experts tell us that these ideas should not be permitted to take up the time of working scientists, and for the most part they are surely correct. But what about the gems in the rubble pile? By what ground-rules might we bring extraordinary new possibilities to light?

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Review of the book Infinitesimal by Amir Alexander

Unread postby Chromium6 » Wed Jul 30, 2014 9:31 pm

Review of the book Infinitesimal by Amir Alexander : ... .html?_r=0 ... erpt_2.pdf

No one talks of infinitesimals any more: The modern notion of limits accomplishes everything they did, but much more rigorously. One exception is a recent reconstruction of infinitesimals — positive “numbers” smaller than every real number — devised by the logician Abraham Robinson and developed further by H. Jerome Keisler, my adviser at the University of Wisconsin.

Since the Jesuits succeeded in banning infinitesimals in Italy, the last part of Dr. Alexander’s finely detailed, dramatic story traces their subsequent history north to England. There one of the key figures is Thomas Hobbes, the 17th-century philosopher of authoritarianism, a strong advocate of law, order — and, like the Jesuits, of the top-down hierarchical nature of Euclidean geometry.

Hobbes’s hated antagonist, the mathematician John Wallis, used infinitesimals freely, along with any other ideas he thought might further mathematical insight. And further it they did, leading over time to calculus, differential equations, and science and technology that have truly shaped the modern world.

But the battle is not quite over. A trace of the dynamic that animated fierce struggles between Jesuit mathematicians and the infinitesimalists, as well as between Hobbes and Wallis, can be found today.

Now, however, the opposition is between proof-driven “pure” mathematicians and computer-friendly, quasi-empirical mathematicians. The latter will dominate, I hope and expect — along with the kind of moderate pluralistic democracy that Dr. Alexander sees as a natural outgrowth of Wallis’s open, experimental mathematics. ... in-history

Interview: ... l.mp3?dl=1
On the Windhexe: ''An engineer could not have invented this,'' Winsness says. ''As an engineer, you don't try anything that's theoretically impossible.''
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Re: Review of the book Infinitesimal by Amir Alexander

Unread postby Chromium6 » Wed Jul 30, 2014 10:53 pm

Looks like this Amir with the other's major influence along with the Kabbalah which Newton knew intimately:

From the end of the nineteenth century until his death, one of history's greatest mathematicians languished in an asylum, driven mad by an almost Faustian thirst for universal knowledge. THE MYSTERY OF THE ALEPH tells the story of Georg Cantor (1845-1918), a Russian born German whose work on the 'continuum problem' would bring us closer than any mathemetician before him in helping us to comprehend the nature of infinity. A respected mathematician himself, Dr. Aczel follows Cantor's life and traces the roots of his enigmatic theories. From the Pythagoreans, the Greek cult of mathematics, to the mystical Jewish numerology found in the Kabbalah, THE MYSTERY OF THE ALEPH follows the search for an answer that may never truly be trusted.


Published: November 15, 2000


Mathematics, Kabala and the Search for Infinity

By Amir D. Aczel

Let's start with Zeno's paradox. Many books on mathematics do, including Amir D. Aczel's previous work, an excellent account of the solution to the mathematical mystery known as Fermat's Last Theorem. His new work, ''The Mystery of the Aleph,'' also includes the indispensable Zeno, who proved that you can never walk out of a room, since, before you can go a full distance, you have to go half that distance, forever. Or, as Mr. Aczel puts this, ''Zeno used this paradox to argue that under the assumption of infinite divisibility of space and time, motion can never even start.''

Infinity is one of those eerie subjects that rubs up equally against religion, philosophy and science, and Mr. Aczel would seem to be a good person to lure us into its inner recesses. His book on the solution of Fermat's Last Theorem was a model of lucid explanation of arcane matters for nonspecialists, and this new book starts out promising much the same with the concept of infinity.

Mr. Aczel is very good at portraying the essences of the thoughts and lives of that quirky class of geniuses known as mathematicians, and he goes back not just to Zeno but to such other fascinating pioneers of number theory as Pythagoras and Eudoxus to do so. He deepens our appreciation of their discoveries by linking them to the equally deep, nonmathematical musings of the Kabalists, for whom infinity was a mystical equivalent to the immensity of God.

And yet, while there is much of interest in this new effort to explicate deep mathematical conundrums to ordinary people, Mr. Aczel does not entirely succeed in this new book. I say this fully aware that for mathematical laymen like myself it is not always easy to determine whether incomprehension is your fault or the author's. Perhaps there is a little of both in ''The Mystery of the Aleph,'' but certainly Mr. Aczel makes only passing efforts to lead his readers gently through the thicket of difficult concepts that he presents.

Reading this book, I often had the feeling that I was on the verge of Understanding. And then it would elude me because, or so it seemed, Mr. Aczel's text was too shorthanded, too quick, insufficiently attentive to my needs.

He begins by introducing Georg Cantor, the German mathematician who in the 1870's devised an equation using the aleph, the first letter of the Hebrew alphabet, to describe a property of infinity. The equation is a short one, but, Mr. Aczel writes, it is ''the most enduring mystery in mathematics.''

A single formula that purports to hold within itself the deepest mysteries of existence: the idea rings with all sorts of spiritual and scientific implications. The drama deepens because at least two of the mathematicians who explored the mystery most avidly -- Cantor and Kurt Godel -- went mad, with Cantor dying in 1918 in a psychiatric hospital in the German university town of Halle.

Having set the scene with Cantor, Mr. Aczel surveys the history of the scientific effort to cope with the question of infinity through the ages. He shows us Galileo, for example, whose treatise ''On Two New Sciences'' has his character Salviati explaining that a circle is made up of infinitely many, infinitely small triangles. That much seems clear enough, but Mr. Aczel quickly careens into murkier territory. Still summarizing Galileo, he writes that ''an infinite set, the set of all whole numbers, is shown to be 'equal in number' to the set of all squares of whole numbers, which is a proper subset of the set of whole numbers.''

If you understand that last sentence, you will probably have no great difficulty understanding the rest of Mr. Aczel's account, which relies heavily on set theory to make many of its points. In the instance of Galileo, Mr. Aczel does what the best writers on difficult subjects do -- he takes his reader by the hand and explicates matters by returning to basics -- in this instance what exactly it means when we count numbers.

The counterintuitive Galilean proposition that an infinite set of things can be equal to a part of itself becomes less murky as Mr. Aczel introduces an idea developed by the mathematician David Hilbert, the Infinite Hotel, which does enlarge one's understanding of the paradoxical properties of infinity. Too often, however, Mr. Aczel races so quickly through difficult concepts that I suspect many readers will feel left behind, as I did.

This is especially the case with some of the key notions discussed by Mr. Aczel, namely the continuum hypothesis and the axiom of choice. One has to do with the number of numbers that can occupy points along a line; the other is a proposition hotly contested by mathematicians that deals with -- well, actually, despite several readings of Mr. Aczel's passages on this subject, I can't summarize what it deals with.

I can do better as Mr. Aczel proceeds to an account of a famous monkey wrench thrown into the infinity works by a vexing paradox discovered by Bertrand Russell: If the barber of Seville shaves all the men of Seville who do not shave themselves, does he shave himself? If he does, he doesn't, and if he doesn't, he does. That is marvelously intriguing and it bears on the nature of infinity, though in ways that Mr. Aczel does not make fully clear.

And yet, despite these difficulties, Mr. Aczel's book remains highly enjoyable and frustrating at the same time. It deals, after all, with great minds venturing into the farthest reaches of speculation, with the nature of endlessness itself, both mathematical and religious, subjects that were not meant to be easy.

Photo: Amir D. Aczel (Jerry Bauer) ... inity.html
On the Windhexe: ''An engineer could not have invented this,'' Winsness says. ''As an engineer, you don't try anything that's theoretically impossible.''
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Re: Review of the book Infinitesimal by Amir Alexander

Unread postby Chromium6 » Wed Jul 30, 2014 11:27 pm

What the previous author, Amir Alexander, fails to recognize is that the Jesuits were made up of those who had followed upon the expulsion of Muslims/Heretics from Spain and Portugal. The Jesuits needed a firm definition of what was allowed and not allowed from a religious perspective:

Diego Laynez (or Laínez) (Spanish: Diego Laynez), born in 1512 (Almazán, Spain) and died the 19 January 1565 (Rome), was a Spanish Jesuit priest and theologian, and the 2nd Superior General of the Society of Jesus.

Diego Laynez was born in Almazán in Castile. Of Jewish ancestry, he was (probably) a fourth generation Catholic. He graduated from the University of Alcalá, and then continued his studies in Paris, where he came under the influence of Ignatius of Loyola. He was one of the seven men who,[1] with Loyola, formed the original group of Friends in the Lord, later Society of Jesus, taking, in the Montmartre church, the vows of personal poverty and chastity in the footsteps of Christ, and committing themselves to going to Jerusalem.
On the Windhexe: ''An engineer could not have invented this,'' Winsness says. ''As an engineer, you don't try anything that's theoretically impossible.''
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Re: Review of the book Infinitesimal by Amir Alexander

Unread postby Chromium6 » Tue Nov 18, 2014 11:24 pm

An interesting debate from Sept. 2014:

Infinity: does it exist?? A debate with James Franklin and N J Wildberger
On the Windhexe: ''An engineer could not have invented this,'' Winsness says. ''As an engineer, you don't try anything that's theoretically impossible.''
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Re: Review of the book Infinitesimal by Amir Alexander

Unread postby David » Wed Nov 19, 2014 4:01 am

Chromium6 wrote:An interesting debate from Sept. 2014:

Infinity: does it exist?? A debate with James Franklin and N J Wildberger

Here is an insightful and well written article that you might enjoy reading. Math critic Mark Chu-Carroll of "Good Math, Bad Math" discusses the unorthodox ideas of mathematics Professor N. J. Wildberger:

Dirty Rotten Infinite Sets and the Foundations of Math

“This isn't the typical wankish crackpottery, but rather a deep and interesting bit of crackpottery.” ... /#more-529
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Re: Review of the book Infinitesimal by Amir Alexander

Unread postby LongtimeAirman » Sun Dec 21, 2014 12:28 pm

I’m almost finished with reading Infinitesimal, but wanted to share a pertinent aspect. The book describes the struggle for and against infinitesimals as an important factor in the development of the modern world. Hobbes, the Jesuits, the Church and royalty tried to counter growing Protestantism and chaos by attempting to limit geometrical logic and reasoning to first principles. To make a long story short, that control was successful in Italy, the source of the Renaissance, subsequently turning that county into a backwater of any further advancement. Elsewhere, and ultimately, acceptance of the infinitesimal resulted in the world we see today.

Tucked neatly into this story is the birth of England’s Royal Society, devoted to understanding and describing natural philosophy. The Royal Society became a powerhouse in the development of science and technology.

It can be said that the creation of the Royal Society took the steam out of England’s long civil war. It enabled men from across the many divides at the time to discuss any of a myriad of new and interesting subjects resulting from their observation of nature and the stars. There were a few rules. First, no amount of logic and reasoning was valid without experimental proof. Subjects lacking an experimental basis were avoided. Politics was not discussed, nor, surprisingly, was mathematics itself. Math was recognized as potentially without limits, able to rationalize any argument, and thus to be avoided, except to explain experimental results.

Members of that august body were limited to aristocracy and the wealthy. The lower classes benefited by improvements to their crafts, but were generally believed to be unable to join in the active discussions of the day.

I see a close parallel with present company. We can participate in active discussions to increase understanding by sharing sources and insights. We mostly avoid subjects or comments that prevent discussion. We are better than the Royal Society in that we allow the lowliest among us (look at me!) to contribute. We are greatly lacking with respect to experimental capability, and overly compensate with our imagined superior reasoning. Hence there is certain dominance of math over physics. The war over the infinitesimal is still with us. Overall, forums such as this are for our greater good.

Thank you all.
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Re: Review of the book Infinitesimal by Amir Alexander

Unread postby Chromium6 » Sat May 02, 2015 4:34 pm

This recent publication is another source that can be looked into on this topic.


The Logic of Infinity
by Barnaby Sheppard ... 1107678668

No other question has ever moved so profoundly the spirit of man;
no other idea has so fruitfully stimulated his intellect; yet no other
concept stands in greater need of clarification than that of the infinite.

– David Hilbert


There is great power in the set theoretic description of the mathematical
universe. As we shall see in detail, one of Cantor’s questions was, naively: ‘how
many points are there in a line?’. This had little more than vague philosophical
meaning before the arrival of set theory, but with set theory the question can be
made precise. The punch-line in this case is that the standard theory of sets – a
theory which is powerful enough to sensibly formulate the question in the first
place – is not strong enough to give a definite answer, it only gives an infinite
number of possible answers, any one of which can be adopted without causing
contradiction. The solution remains undecidable; the theory does not know the
size of the beast it has created! This flexibility in the solution is arguably far
more intriguing than any definite answer would have been – it transfers our focus
from a dull fixed single universe of truths to a rich multiverse, one consistent
universe for each alternative answer.


Although the ideas discussed in this book are now well-established in mainstream
mathematics, in each generation since Cantor there have always been a
few individuals who have argued, for various philosophical reasons, that mathematics
should confine itself to finite sets. These we shall call Finitists. Despite
my occasional sympathies with some of their arguments, I cannot claim to be
a Finitist. I’m certainly a ‘Physical Finitist’ to some extent, by which I mean
that I am doubtful of the existence of infinite physical aggregates, however I
do not extend this material limitation to the Platonic constructions of mathematics
(the set of natural numbers, the set of real numbers, the class of all
groups, etc.), which seem perfectly sound as ideas and which are well-captured
in the formalized language of set theory. However, I cannot pretend to be a
strong Platonist either (although my feelings often waver), so my objection is
not based on a quasi-mystical belief that there is a mathematical infinity ‘out
there’ somewhere. In practice I find it most convenient to gravitate towards
a type of Formalism, and insofar as this paints a picture of mathematics as a
kind of elaborate game played with a finite collection of symbols, this makes
me a Finitist in a certain Hilbertian sense – but I have no difficulty in imagining
all manner of infinite sets, and it is from such imaginings that a lot of
interesting and very useful mathematics springs, not from staring at or manipulating
strings of logical symbols. I am not alone in being so fickle as to
my philosophical leanings; ask a logician and a physicist to comment on the
quote attributed to Bertrand Russell ‘any universe of objects...’ which opens
Section 11.1 of Chapter 11 and witness the slow dynamics of shifting opinions
and perspectives as the argument unfolds.


Formal mereological systems make use of the relation of parthood and the axioms
generally reflect the properties one would expect if a part of a whole is
interpreted as a subset. An atom is an object with no proper parts. A natural
antisymmetry condition on parthood ensures that finite mereological systems
will always have atoms, but an infinite system need not, unless a special axiom
is introduced to force it to have one. A classical example of an atomless fusion
is given by the line, regarded as a fusion of line segments: any line segment can
be divided into subsegments indefinitely. The notion of a line as a set of points
is a relatively modern one which emerged hand in hand with the development
of set theory. Classically one could always specify a point on a line, but since
there always exists a new point between any given pair of points it was perhaps
thought absurd that one could saturate the entire line by points. We will soon
see how such a saturation can be realized.


In trying to define finiteness we seek a set theoretic property that distinguishes
between intuitively finite sets and intuitively infinite sets. One such property is
the existence of a proper self-injection. A set which does not have this property
is Dedekind finite. A further assumption, the Axiom of Choice, is required to
ensure that this definition coincides with various other proposed definitions of
finiteness. The existence of a proper self-injection is our first glimpse of the
peculiar behaviour of infinite sets. At the other extreme of the cardinal spectrum
is a similarly misunderstood set, the empty set.


1.1.5 Finiteness

I have freely used the terms finite and infinite expecting the reader to have an
intuitive grasp of these notions. Clearly we wish infinite to mean ‘not finite’,
but then how are we to define ‘finite’? A natural approach, and one that is
encoded within the theory of sets we are about to describe, is to say that a set
A is finite either if it has no elements at all or if there exists a natural number
n such that A is equipollent to {0, 1, 2, . . . , n}. It will be convenient to call this
property ‘ordinary finiteness’. This is a fair definition, but to be comfortable
with it we need to know precisely what we mean by ‘natural number’, and that
is to come (we assume an intuitive familiarity with the set of natural numbers
below, for the purposes of an example).

There are alternative definitions of finiteness which make no mention of
natural numbers. The equivalence of these ‘number free’ definitions follows if
we are permitted to assume the truth of a certain far-reaching principle called
the Axiom of Choice. We will have much to say about Choice later; briefly
it asserts that for any collection C of sets there exists a set with exactly one
element in common with each set in C.

If a set A is ordinary finite then there does not exist a bijection from A onto
a proper subset of A – a property known as Dedekind finiteness. The Axiom
of Choice is needed to prove that every Dedekind finite set is ordinary finite.
Contraposing the Choice-free implication we see that a set A is ordinary infinite
if there exists a bijection from A onto a proper subset of itself. This property
of proper self-injection may seem absurd to someone who has not encountered
the phenomenon before, but any such surprise is only symptomatic of the fact
that we live in a finite world subject to the relatively sober properties of finite
, infinite collections being inevitably unfamiliar.
On the Windhexe: ''An engineer could not have invented this,'' Winsness says. ''As an engineer, you don't try anything that's theoretically impossible.''
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Re: Review of the book Infinitesimal by Amir Alexander

Unread postby Chromium6 » Mon May 18, 2015 8:38 pm

From M.M.:

The Central Discoveries of this Book

That oldest mistake is one that Euclid made. It concerns the definition of the point. Entire library shelves have been filled commenting on Euclid's definitions, but neither he nor anyone since has appeared to notice the gaping hole in that definition. Euclid declined to inform us whether his point was a real point or a diagrammed point. Most will say that it is a geometric point, and that a geometric point is either both real and diagrammed or it is neither. But all the arguments in that line have been philosophical misdirection. The problem that has to be solved mathematically concerns the dimensions created by the definition. That is, Euclid's hole is not a philosophical or metaphysical one, it is a mechanical and mathematical one. Geometry is mathematics, and mathematics concerns numbers. So the operational question is, can you assign a number to a point, and if you do, what mathematical outcome must there be to that assignment? I have exhaustively shown that you cannot assign a counting number to a real point. A real point is dimensionless; it therefore has no extension in any direction. You can apply an ordinal number to it, but you cannot assign a cardinal number to it. Since mathematics and physics concern cardinal or counting numbers, the point cannot enter their equations.

This is of fundamental contemporary importance, since it means that the point cannot enter calculus equations. It also cannot exit calculus equations. Meaning that you cannot find points as the solutions to any differential or integral problems. There is simply no such thing as a solution at an instant or a point, including a solution that claims to be a velocity, a time, a distance, or an acceleration. Whenever mathematics is applied to physics, the point is not a possible solution or a possible question or axiom. It is not part of the math.

Now, it is true that diagrammed points may be used in mathematics and physics. You can easily assign a number to a diagrammed point. Descartes gave us a very useful graph to use when diagramming them. But these diagrammed points are not physical points and cannot stand for physical points. A physical point has no dimensions, by definition. A diagrammed point must have at least one dimension. In a Cartesian graph, a diagrammed point has two dimensions: it has an x-dimension and a y-dimension. What people have not remembered is that if you enter a series of equations with a certain number of dimensions, you must exit that series of equations with the same number of dimensions. If you assign a variable to a parameter, then that variable must have at least one dimension. It must have at least one dimension because you intend to assign a number to it. That is what a variable is—a potential number. This means that all your variables and all your solutions must have at least one dimension at all times. If they didn't, you couldn't assign numbers to them.

This critical finding of mine has thousands of implications in physics, but I will mention only a couple. It has huge implications in Quantum Electro-Dynamics, since the entire problem of renormalization is caused by this hole in Euclid's definition. Because neither Descartes nor Newton nor Schrodinger nor Feynman saw this hole for what it was, QED has inherited the entire false foundation of the calculus. Many of the problems of QED, including all the problems of renormalization, come about from infinities and zeroes appearing in equations in strange ways. All these problems are caused by mis-defining variables. The variables in QED start acting strangely when they have one or more dimensions, but the scientists mistakenly assign them zero dimensions. In short, the scientists and mathematicians have insisted on inserting physical points into their equations, and these equations are rebelling. Mathematical equations of all kinds cannot absorb physical points. They can express intervals only. The calculus is at root a differential calculus, and zero is not a differential. The reason for all of this is not mystical or esoteric; it is simply the one I have stated above—you cannot assign a number to a point. It is logical and definitional.

This finding is not only useful in physics, it is useful to calculus itself, since it has allowed me to show that modern derivatives are often wrong. I have shown that the derivatives of ln(x) and 1/x are wrong, for instance. I have also shown that many problems are solved incorrectly with calculus, including very simple problems of acceleration.

This finding also intersects my first discoveries in special relativity, which I will discuss in greater detail below. The first mistake I uncovered in special relativity concerned Einstein's and Lorentz' early refusals to define their variables. They did not and would not say whether the time variable was an instant or a period. Was it t or Δt? Solving this simple problem was the key to unlocking the central algebraic errors in the math. Once it was clear that the time variable must be an interval or period, at least two of Einstein's first equations fell and could not be made to stand up again.

What does this prove? Euclid’s geometry is a form of mathematics. I don’t think anyone will argue that geometry is not mathematics. Geometry becomes useful only when we can begin to assign numbers to points, and thereby find lengths, velocities and accelerations. If we assign numbers to points, then those points must be mathematical points. They are not physical points. Euclid’s definitions apply to points on pieces of paper, to diagrams. They do not apply directly to physical points.

I am not going to argue that you cannot translate your mathematical findings to the physical world. That would be nihilistic and idiotic, not to say counter-intuitive. But I am going to argue here that you must take proper care in doing so. You must differentiate between mathematical points and physical points, because if you do not you will misunderstand all higher math. You will misinterpret the calculus, to begin with, and this will throw off all your other maths, including topology, linear and vector algebra, and the tensor calculus.
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