Ellen Stuttle

Knowledge vs. Dogma - "Infinitesimal"

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This recently announced on the SCIAM Book Club site - link.

First I've heard of the book. I have no idea yet how good it is.

However, the relevance of the topic to current issues looks obvious.

Infinitesimal

by Amir Alexander

The epic battle over a mathematical concept that shook the old order to its core and shaped the world as we know it.

On August 10, 1632, five leaders of the Society of Jesus convened in Rome to pass judgment on a simple idea: that a continuous line is composed of distinct and limitlessly tiny parts. The doctrine would become the foundation of calculus, but on that day the judges ruled that it was forbidden. With the stroke of a pen they set off a war for the soul of the modern world.

Amir Alexander's Infinitesimal is the story of the struggle that pitted Europe's entrenched powers against voices for change. Taking us from the religious strife of the sixteenth century to the English civil war to the confrontations between thinkers like Galileo and Hobbes, it captures a moment when our modern beliefs in liberty and progress hung in the balance.

FROM INFINITESIMAL

The notion that a political essayist reviewing the institutions of a foreign land would focus on an obscure mathematical concept seems not only surprising to us today, but outright bizarre. The concepts of higher mathematics appear to us so abstract and universal that they cannot be relevant to cultural or political life. They are the domain of highly trained specialists, and do not even register with modern-day cultural critics, not to mention politicians. But this was not the case in the early modern world, for Sorbière was far from the only nonmathematician to be concerned about the infinitely small. In fact, in Sorbière's day, European thinkers and intellectuals of widely divergent religious and political affiliations campaigned tirelessly to stamp out the doctrine of indivisables and to eliminate it from philosophical and scientific consideration. In the very years that Hobbes was fighting Wallis over the indivisible line in England, the Society of Jesus was leading its own campaign against the infinitely small in Catholic lands. In France, Hobbes's acquaintance René Descartes, who had initially shown considerable interest in infinitesimals, changed his mind and ultimately banned the concept from his all-encompassing philosophy. Even as late as the 1730s the High Church Anglican bishop George Berkeley mocked mathematicians for their use of infinitesimals, calling these mathematical objects 'the ghosts of departed quantities.' Lined up against these naysayers were some of the most prominent mathematicians and philosophers of that era, who championed the use of the infinitely small. These included, in addition to Wallis, Galileo and his followers, Bernard Le Bovier de Fontenelle, and Isaac Newton.

Why did the best minds of the early modern world fight so fiercely over the infinitely small? The reason was that much more was at stake than an obscure mathematical concept. The fight was over the face of the modern world. Two camps confronted each other over the infinitesimal: On the one side were ranged the forces of hierarchy and order - Jesuits, Hobbesians, French royal courtiers, and High Church Anglicans. They believed in a unified and fixed order in the world, both natural and human, and were fiercely opposed to infinitesimals. On the other side were comparative 'liberalizers' such as Galileo, Wallis, and the Newtonians. They believed in a more pluralistic and flexible order, one that might accommodate a range of views and diverse centers of power, and championed infinitesimals and their use in mathematics. The lines were drawn, and a victory for one side or the other would leave its imprint on the world for centuries to come.

Ellen

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Can we do without the philosophical nonsense???

Read Non-Standard Analysis by Abraham Robinson. Princeton University Press 1996.

It is a technical/logical mathematical matter, not the subject of philosophical warm moist effluence.

When mathematics is mixed with either theology or metaphysics only gaseous effluence emerges.

Ba'al Chatzaf

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Similar and related battles continue to this day - the various interpretations of quantum mechanics, cosmology and gravitational theories are all tightly intertwined with the philosophical positions of their researchers - driving the course of funding and the resulting theoretical and mathematical research into those subjects.

Dennis

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David Hume put the stake through the heart of philosophical bullshit, but nobody paid enough attention to him.

Ba'al Chatzaf

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Can we do without the philosophical nonsense???

Can you read the blurb of what a book is about?

Apparently not.

And as to philosophy, you talk philosophy on a regular basis, and science depends on philosophy.

Ellen

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Similar and related battles continue to this day - the various interpretations of quantum mechanics, cosmology and gravitational theories are all tightly intertwined with the philosophical positions of their researchers - driving the course of funding and the resulting theoretical and mathematical research into those subjects.

Dennis

I assume you see the relevance to climate dogma.

Glad you noticed my post. I was going to send you a back-channel note alerting you. :smile:

Ellen

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Can we do without the philosophical nonsense???

Can you read the blurb of what a book is about?

Apparently not.

And as to philosophy, you talk philosophy on a regular basis, and science depends on philosophy.

Ellen

Only on reality-lite.

1.There is an Out There out there

2. We have enough intelligence in our 3 lb brains to figure out some of what is Out There

3. Any doctrine that does not reduce to what can be observed (be it the output of an instrument of an eclipse of the sun) is bloody nonsense.

Science is exactly as good as its predictions and not one bit more.

Ba'al Chataz

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Here, for instance, Bob, is an example of you talking philosophy - in this case, bad philosophy.

(Your post immediately above is another example, which appeared while I was looking for this one.)

For a scientific theory only two things matter:

1. mathematical consistency

2. correct predictions which means no empirical falsification has been observed.

Any theory that meets those bench marks is kosher.

Ba'al Chatzaf

Ellen

PS: And obviously, you still haven't bothered to read the description of what the featured book is about. :laugh:

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Here, for instance, Bob, is an example of you talking philosophy - in this case, bad philosophy.

(Your post immediately above is another example, which appeared while I was looking for this one.)

For a scientific theory only two things matter:

1. mathematical consistency

2. correct predictions which means no empirical falsification has been observed.

Any theory that meets those bench marks is kosher.

Ba'al Chatzaf

Ellen

PS: And obviously, you still haven't bothered to read the description of what the featured book is about. :laugh:

It does mot mattger. What I said in 1 and 2 are correct and infinitesimal are not longer a proper study for either theology or metaphysics. It is a proven mathematically sound alternative to the limit approach invented in the early 19 th century by Cauchy and other mathematicians.

I simply start to boil when mathematics is infested and sullied with theology and metaphysics. Metaphysics is fancy shit. Theology is worse --- it is a disease.

Ba'al Chatzaf

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PS: And obviously, you still haven't bothered to read the description of what the featured book is about. :laugh:

It does mot mattger. What I said in 1 and 2 are correct and infinitesimal are not longer a proper study for either theology or metaphysics.

The book is a study in intellectual history, as you might notice yet if you stop having predictable - and rote - apoplexy long enough to read the description.

Ellen

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The history is interesting because some of it is actually recent history. The proper method of explaining the calculus only came to be widely taught about the time I took calculus. Nearly every other class I took while an undergraduate made an issue of explaining the history of the different mathematical/engineering/physics approaches and notation used in calculus and differential equations. While all are correct only the mathematical approach is rigorously correct.

Infinitesimals is not a dead issue because innumeracy in general is alive and well as discussed in some other posts.

Dennis

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PS: And obviously, you still haven't bothered to read the description of what the featured book is about. :laugh:

It does mot mattger. What I said in 1 and 2 are correct and infinitesimal are not longer a proper study for either theology or metaphysics.

The book is a study in intellectual history, as you might notice yet if you stop having predictable - and rote - apoplexy long enough to read the description.

Ellen

Sorry. I was having a rant. Whenever I see math (which I love) and theology (which I detest) in the same paragraph, I bust a gut. I hate theology with a purple passion and I loath Metaphysics. For deep down metaphysics I have contumely, low-regard, dislike, detestation, and utter contempt.

Ba'al Chatzaf

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[...] I loath Metaphysics. For deep down metaphysics I have contumely, low-regard, dislike, detestation, and utter contempt.

Yet on another thread about five hours later the same day you wrote:

My motive would be to find causes and from that cures. Anything is is quack-quack.

What are causes if not "deep down" metaphysics?

Ellen

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[...] I loath Metaphysics. For deep down metaphysics I have contumely, low-regard, dislike, detestation, and utter contempt.

Yet on another thread about five hours later the same day you wrote:

My motive would be to find causes and from that cures. Anything is is quack-quack.

What are causes if not "deep down" metaphysics?

Ellen

Initial conditions that lead to subsequent conditions in an (empirically) regular manner. The idea of "necessary connection" between cause tokens and effect tokens has been effectively trashed by Hume. Empirically on can establish connections that happened in the past and use these as provisional markers. There is no absolute guarantee that the connections will -always- hold in the future or in conditions unlike any seen in the past. Cause to Effect is an empirical practical matter no a metaphysical absolute.

The lack of contingency or provisional truth is lacking in metaphysics which is what makes it almost useless for science.

Ba'al Chatzaf

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Bob writes:Whenever I see math (which I love) and theology (which I detest) in the same paragraph, I bust a gut.
Have you ever stopped long enough to ask yourself how you could have such an extremely negative emotional reaction to something which does not even exist? What can upset you, has control over you because it possesses the power to make you react. Since no one reacts to nothing... you are obviously reacting to something. Your own emotional reaction confirms that there is something there to which you are reacting.
I hate theology with a purple passion and I loath Metaphysics. For deep down metaphysics I have contumely, low-regard, dislike, detestation, and utter contempt.
Then you must also loathe mathematics which rationally and impersonally acknowledges the truth of the existence of that which is infinite.Greg

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Bob writes:

Whenever I see math (which I love) and theology (which I detest) in the same paragraph, I bust a gut.

Have you ever stopped long enough to ask yourself how you could have such an extremely negative emotional reaction to something which does not even exist? What can upset you, has control over you because it possesses the power to make you react.

Since no one reacts to nothing... you are obviously reacting to something. Your own emotional reaction confirms that there is something there to which you are reacting.

I hate theology with a purple passion and I loath Metaphysics. For deep down metaphysics I have contumely, low-regard, dislike, detestation, and utter contempt.

Then you must also loathe mathematics which rationally and impersonally acknowledges the truth of the existence of that which is infinite.

Greg

Mathematical existence is simply the outcome of a logical inference done without contradiction.

Mathematical objects which "exist" do not physically exist. They exist as ideas in our heads.

I was, professionally, an applied mathematician and software designer. I have no problem with mathematics whatsoever.

That is because mathematical objects do not physically exist. The only "law" to which they are subject is the law of non-contradiction.

I have a pair of shows tied onto my feet and I see the clearly, but the number two is no where to be found in the world except in people's heads.

Ba'al Chatzaf

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Bob writes:Mathematical existence is simply the outcome of a logical inference done without contradiction. Mathematical objects which "exist" do not physically exist. They exist as ideas in our heads.
Sure. Like the idea of the infinity of pi... or even the transcendence of love. They're only intellectual ideas until they are expressed in our actions and made manifest in the reality of this world.
I was, professionally, an applied mathematician and software designer. I have no problem with mathematics whatsoever.
Of course not... and so you have no problem with transcendence or infinity either... just as long as they are on your own terms. And that is a free choice everyone makes for themselves.
That is because mathematical objects do not physically exist. The only "law" to which they are subject is the law of non-contradiction.
The square root of minus one "exists" as a non physical mathematical object......but it is most certainly a contradiction. :wink:Greg

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Bob writes:

Mathematical existence is simply the outcome of a logical inference done without contradiction.

Mathematical objects which "exist" do not physically exist. They exist as ideas in our heads.

Sure. Like the idea of the infinity of pi... or even the transcendence of love. They're only intellectual ideas until they are expressed in our actions and made manifest in the reality of this world.
I was, professionally, an applied mathematician and software designer. I have no problem with mathematics whatsoever.

Of course not... and so you have no problem with transcendence or infinity either... just as long as they are on your own terms. And that is a free choice everyone makes for themselves.

That is because mathematical objects do not physically exist. The only "law" to which they are subject is the law of non-contradiction.

The square root of minus one "exists" as a non physical mathematical object...

...but it is most certainly a contradiction. :wink:

Greg

The square root of -1 is the unit vector turned 90 degrees counter clockwise.

it exists and it works fine

If you like something algebraic the complex numbers are the set of matrices

first row a b

second row -b a

where a, b are real numbers.

the unit is 1 0

0 1

and the imaginary root of x^2 + 1 = 0 is

0 1

1 0

square that and get

1 0

0 1

Easy peasey. Without complex numbers one cannot do electrodynamics or quantum theory.

Ba'al Chatzaf

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[...] I loath Metaphysics. For deep down metaphysics I have contumely, low-regard, dislike, detestation, and utter contempt.

Yet on another thread about five hours later the same day you wrote:

My motive would be to find causes and from that cures. Anything is is quack-quack.

What are causes if not "deep down" metaphysics?

Ellen

Initial conditions that lead to subsequent conditions in an (empirically) regular manner. The idea of "necessary connection" between cause tokens and effect tokens has been effectively trashed by Hume. Empirically on can establish connections that happened in the past and use these as provisional markers. There is no absolute guarantee that the connections will -always- hold in the future or in conditions unlike any seen in the past. Cause to Effect is an empirical practical matter no a metaphysical absolute.

The lack of contingency or provisional truth is lacking in metaphysics which is what makes it almost useless for science.

Ba'al Chatzaf

You have a peculiar idea of metaphysics. You've stipulated so limited a meaning you don't recognize metaphysics when you're writing it yourself.

Ellen

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False - you can do electrodynamics and quantum theory without complex numbers. They are mathematical tools of convenience. Any physics starting with and ending with real numbers can be done without complex numbers - that is a fact though Ba'al Chatzaf is the not the first to incorrectly believe otherwise - I ran into two such professors [both applied mathematics] and when you pin them down they have no basis for their claim.

Dennis

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Can we do without the philosophical nonsense???

Read Non-Standard Analysis by Abraham Robinson. Princeton University Press 1996.

It is a technical/logical mathematical matter, not the subject of philosophical warm moist effluence.

When mathematics is mixed with either theology or metaphysics only gaseous effluence emerges.

OL is primarily a philosophical forum. The only reason you are not a troll here is your purblindness is honest and up front.

--Brant

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David Hume put the stake through the heart of philosophical bullshit, but nobody paid enough attention to him.

I didn't know he was a scientist. Was he a philosopher? A mathematician?

--Brant

can an empiricist be a philosopher?

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[...] I loath Metaphysics. For deep down metaphysics I have contumely, low-regard, dislike, detestation, and utter contempt.

Yet on another thread about five hours later the same day you wrote:

My motive would be to find causes and from that cures. Anything is is quack-quack.

What are causes if not "deep down" metaphysics?

Ellen

Initial conditions that lead to subsequent conditions in an (empirically) regular manner. The idea of "necessary connection" between cause tokens and effect tokens has been effectively trashed by Hume. Empirically on can establish connections that happened in the past and use these as provisional markers. There is no absolute guarantee that the connections will -always- hold in the future or in conditions unlike any seen in the past. Cause to Effect is an empirical practical matter no a metaphysical absolute.

The lack of contingency or provisional truth is lacking in metaphysics which is what makes it almost useless for science.

Is this the "problem" of induction Peikoff is said to solve?

--Brant

you are in epistemology here, not metaphysics--of course, they are inter-related and conmingled but you insist on having them apart from each other

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Bob writes:

The square root of -1 is the unit vector turned 90 degrees counter clockwise

That doesn't mean that the square root of a minus number isn't contradictory.

Greg

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