Mathematics is About the World
Posted by deleted 5 years, 7 months ago to Science
How Ayn Rand's theory of concepts unlocks the false alternatives between Plato's Mathematical Universe and Hilbert's Game of Symbols.
Book Description:
What is mathematics about? Is there a mathematical universe glimpsed by a mathematical intuition? Or is mathematics an arbitrary game of symbols, with no inherent meaning, that somehow finds application to life on earth? Robert Knapp holds, on the contrary, that mathematics is about the world. His book develops and applies its alternative viewpoint, first, to elementary geometry and the number system and, then, to more advanced topics, such as topology and group representations. Its theme is that mathematics, however abstract, arises from and is shaped by requirements of indirect measurement. Eratosthenes, in 200 BC, demonstrated the power of indirect measurement when he estimated the circumference of the earth by measuring a shadow at noon, in Alexandria, on the day of the summer solstice. Establishing geometric relationships, solving equations, finding approximations, and, generally, discovering quantitative relationships are tools of indirect measurement: They are the core of mathematics, the drivers of its development, and the heart of its power to enhance our lives.
About the Author:
Robert Knapp earned his Ph.D. in mathematics from Princeton University in 1972. He has published work on differential geometry and partial differential equations, and, after a year at the Insitute for Advanced Studies in Princeton, taught graduate and undergraduate mathematics at Purdue University. His study and appreciation of abstract mathematics began in high school and his conviction that mathematics, including abstract mathematics, is about the world began then, as well. Although he retired from the profession in the late 1970s, his study of the content, history and application of mathematics continues to this day. In recent years he has presented his unique perspectives on geometry and the number system in a series of lectures at Objectivist Summer Conferences organized by the Ayn Rand Institute. He has lived in the Philadelphia area for almost 30 years.
Book Description:
What is mathematics about? Is there a mathematical universe glimpsed by a mathematical intuition? Or is mathematics an arbitrary game of symbols, with no inherent meaning, that somehow finds application to life on earth? Robert Knapp holds, on the contrary, that mathematics is about the world. His book develops and applies its alternative viewpoint, first, to elementary geometry and the number system and, then, to more advanced topics, such as topology and group representations. Its theme is that mathematics, however abstract, arises from and is shaped by requirements of indirect measurement. Eratosthenes, in 200 BC, demonstrated the power of indirect measurement when he estimated the circumference of the earth by measuring a shadow at noon, in Alexandria, on the day of the summer solstice. Establishing geometric relationships, solving equations, finding approximations, and, generally, discovering quantitative relationships are tools of indirect measurement: They are the core of mathematics, the drivers of its development, and the heart of its power to enhance our lives.
About the Author:
Robert Knapp earned his Ph.D. in mathematics from Princeton University in 1972. He has published work on differential geometry and partial differential equations, and, after a year at the Insitute for Advanced Studies in Princeton, taught graduate and undergraduate mathematics at Purdue University. His study and appreciation of abstract mathematics began in high school and his conviction that mathematics, including abstract mathematics, is about the world began then, as well. Although he retired from the profession in the late 1970s, his study of the content, history and application of mathematics continues to this day. In recent years he has presented his unique perspectives on geometry and the number system in a series of lectures at Objectivist Summer Conferences organized by the Ayn Rand Institute. He has lived in the Philadelphia area for almost 30 years.
SOURCE URL: https://mathematicsisabouttheworld.com
But to say that mathematics and its concepts are "about the world" is imprecise. Mathematics is a science of method, not of things. It begins with abstractions of number and shape, which in turn are based on observing reality, and proceeds to establish methods conceptually relating them (measurement). That is the foundation no matter how abstract that process becomes as it evolves into higher level abstractions. That idea of mathematics is the alternative to mathematics regarded as a Platonic realm, manipulating symbols without regard to meaning, or Kantian categories (Intuitionism).
The math that I do, number theory, deals with the the abstract relationships between the integers with the prime numbers as a part of the subject. Analysis such as the calculus with its real numbers and their continuity and its use for describing changes between mathematical objects and change such as motion in objective realityhas been used to get information about the integers.
For the world of real objects, subsets of abstract mathematics developed for no more than intellectual pleasure, find use in describing the real world both of objects and minds.
Why not?
I never fully understood the "queen of the sciences" debate. I think of math as something that may or may not apply to the real world. I imagine there's a lot of math out there that I don't learn because it doesn't apply to my area.
Mathematics was always presented, or taught, in much too complicated form during my youth.
The basic concept called out here that mathematics is a symbolic means to indirectly measure things hits the nail on the head. Numbers only have meaning relative to something in the real world that they are being used to represent. Big, by itself, doesn't mean much - how big and relative to what? Mathematics helps to clarify "how big". or how fast or how far, etc
Otherwise, you just have a mental game of symbols, which might be fun to some people.
:^o_|_^^
'Ever building your civilization,
free minds be wielding the power of one.'
-Egopriest
And although I am not a mathematician, or good with numbers, I trust most who read his book (based as it is on Ayn Rand's theory of concept formation) will not use it's ideas to feed their own destroyers.
The vital distinction is between someone able to think in principle vs. someone who can only think about the principles. The first will know how to select values worth pursuing and how to act in order to actually achieve the value given the socio-political context.
I say it's time to "strike!" That takes in everything, puts all values to the purpose of preserving them, not destroying them.
Galt's distinctive value as an inventor is his motor which he withdraws indefinitely from the world that would sacrifice him to the values he offers.
Francisco's distinctive value is his pedigree and inheritance that he subverts with overt irony (the flip-side of justice) to withdraw his legacy from the looter's system.
Ragnar's distinctive value is his patriotic commitment to justice in action which he uses against the system that is using it to enslave.
There are limitless applications for the strike in all areas of life, but what I have found as the only way to become a revolutionary today (and in fact, rather than merely "theoretically, if..." a given set of concretes presents itself) is the only thing that has ever been transformative all the way down (and out).
Just start by looking yourself in the mirror and declaring out loud: "Your life belongs to you...you are the revolution." If you can do that then you will have made a start, Welcome to Galt Sub-Station-000: the next step takes you away from fixating on Mr. Thompson and on your walk with Galt towards your very own substation: GSS-297 (which "follows" the oath at GSS-296).
Remember, after the Speech, the first thing people do is to turn off their "devices" (ironic given today's ominous "announcement" -- Mr. Thompson will speak to your device whenever he damn well pleases).
I hold the principles in mind, and will gladly discuss fidelity in word and action to them, provided one is working to integrate them psychologically, that he'll be ready to think or speak in the "emergency-of-the-moment").
In an emergency it may be too late to acquire the necessary manners (or "mannerisms"), let alone cobble together any plan of action to counter all the black swans being bred in academia, flying to Washington and coming soon to a PSA near you (whether you choose to acknowledge The Crisis or not).
Gentlemen may cry "tolerance," but there is none. So ring, ring, ring the bell of liberty -- one for each paragraph and a canon shot at GSS-297 announces the charge. As you can see, I'm a poet not a social or military strategist per se: https://youtu.be/k2luoz0iDPI
The plot in Atlas Shrugged employed the strike to show in fiction how human survival depends on the mind and what happens, in a fictional acceleration, when the mind is withdrawn. Bypassing the role of the mind and ideas in advocating a "strike" is not an application of Ayn Rand's principles; it is the opposite.
The only way to apply principles is to think in principle and thereby know what one's principles consist of and what philosophic system defines and integrates them.
What motive other than re-writing reality could anyone attempting to re-write Objectivism have, what other motive than to stall implementation of Ayn Rand's cultural revolution?
Atlas Shrugged is (not ought to be but is) America's Second Declaration of Independence. Her post-script seals the deal: the refusal to compromise or deflect retribution at any time (we have none for mysticism) or to any uncertain purpose (there can be no room for doubt).
[And, eww, despite my negative "reaction," I do appreciate the opportunity to be heard. So keep it coming and I'll double-down some more. B^]
It wasn't written to show "how in fiction..." but how in reality. Much of her later non-fiction quotes from the speech extensively (and other literary passages) as well.
I'm up against a compartmentalizing double-standard, as if it were commendable to perform Shakespeare in the Park (the revolutionary Elizabethan context safely behind us), but specifically because Ayn Rand's idea's, her principles are so revolutionary in today's "prosaic" world, we have to stipulate that it's "only fiction."
If you really believed that you would offer an alternative (e.g., reciting non-fiction passages). As a linguist I would then be challenging the praxis you propose rather than denigrating the purpose for which you propose it.
Also, this might be unfair as I don't know if you're even familiar with the praxis I'm proposing. It's more psycho-drama than direct action, more theatrical than political. Like I said, I'm proposing a cultural revolution effected only be training that goes far broader and deeper than merely quitting one's job or giving up one's career (though it might selfishly entail such).
The only leader of my "strike" is untouchable precisely because he is "fictional."
No one said that Atlas Shrugged was written to show "how in fiction", I wrote that "the plot in Atlas Shrugged employed the strike to show in fiction how" -- not "how in fiction" as opposed to reality -- "human survival depends on the mind and what happens, in a fictional acceleration, when the mind is withdrawn. That is how she described it herself. The strike was a fictional device showing in an artificially accelerated form what happens when the mind is withdrawn from a civilized society.
Emotional flourishes promoting a "strike" are the opposite of the intellectual revolution Ayn Rand advocated and which is in fact required. There are no shortcuts.
This has been discussed many times on this forum. It's a shame that the original thread has gotten so off topic from the original serious topic, for which there seems to be very little interest here.
This methodical praxis of committing the poetic vehicle to memory through dramatic recitation (for oneself, or others), and to the purposeful necessity of inducing the principles, the integrations of one's knowledge in every concrete you meet.
I just got back from a Socrates Cafe where I held the floor in vigorous debate with avowed Platonists and Skeptics, and it was exhilarating and exhausting: the discussion getting particularly loud in discussion of force vs. mind: I had to keep insisting --enigmatically-- that there is no argument against force, five or six times (which is what they thought they were arguing) before they insisted I explain myself, and I did: as an issue of mind ("argument") vs. body (force) as against the legal integration or framework as the only barrier to barbarism and the cave, or as against their Neo-Platonic or Skeptical insistence on a "gap," a "zero" everywhere or the zero that is their god or other supernatural substitute for existence. This was so much more valuable a use of my time than sitting here throwing words onto a tabula rasa. (There a'int no meat on these bones!)
But yes, it is an intellectual revolution as well. I'm actually thinking whether or no I will, as a highly trained linguist, open up an ESL school on the principles of this sort of dramatic modeling, this mnemonic expansion of one's mental (and moral) faculties and character-building gestalt toward becoming a New American!! 3^|)
I am an open evangelist for reason, rights and reality in action and as absolute prerequisites to any transaction: Your life belongs to you.
So, to "bring the Gulch home," as it were, you must transform your life from the inside out -- by choice. This message is the revolution (pass it on, ad libitum).
Children Say:
I keep to myself what I might share with others
But they don't seem to understand
I open my mouth to rediscover
That I don't have the words at my command
Holdin' out for a world so much better
But I'm a stranger in a stranger's land
All my friends have sold out, couldn't handle the pressure
Countin' their blessings, tryin' to salvage what they can
Children say, children say
"We open our minds as one"
But one more day slips away
Why don't the dreams of the young ever come to be?
When I overhear my parents' conversations
Well, I'm struck by the things they say
It seems they traded the years for mere complications
Who ever thought it could end this way
They close the door but they can't lock it
'Cause somethin' of their childhood remains
And they've felt it before, when the man in their pocket
Counted the cost of their material gains
Children say, "Come what may
Be strong for the friends you've known"
But one fine day, far away
Can we remember the love we used to own?
Your choice to equivocate between reason and force (persuasion and coercion) drops you into the category of people I choose to ignore.
But don't worry, you'll likely have opportunity elsewhere to "push-back" in another forum (after all Galt's revolution does pertain to every human pursuit).
My praxis also is a training program that will make it obvious who get's it from the impossibility of making progress when you don't -- the climb up Galt's Mountain more arduous depending on one's grasp of the principles for which it is only the poetic vehicle (the greatest ever produced).