"Now, within the last few years, mathematical science has made considerable progress. An Important step has been taken within the realm of mathematics itself, towards the supersensible. This has come about as the result of the Analysis of Infinity which we owe to Newton and Leibnitz. Thus another branch of mathematical science has been added to that which we call "Euclidian." Euclid expresses by mathematical formulae only what can be described and constructed within the field of the "finite." What I can state in terms of Euclid about a circle, a triangle or about the relations of numbers, is within the field of the finite, it is capable of construction in a sense-perceptible manner. This is no longer possible with the Differential Calculus with which Newton and Leibnitz taught us to reckon. The Differential still possesses all the properties that render it possible for us to calculate with it; but in itself as such, it eludes sense-perception. In the Differential, sense-perception is brought to a vanishing point and then we get a new basis free from sense-perception for our reckoning. We calculate what is perceptible by the senses through that which eludes sense-perception. Thus the Differential is an Infinitesimal as against the finitely sensible. The "finite" is mathematically referred back to something quite different from it, namely to the real"infinitesimally small." In the Infinitesimal Calculus we stand on an important boundary line. We are mathematically led out beyond what is perceptible to the senses, and yet we remain so much within the real that we calculate the "Imperceptible." And when we have calculated, the perceptible proves to be the result of our calculation from the imperceptible. Applying the Infinitesimal Calculus to natural processes in Mechanics and Physics, we accomplish nothing else, in fact, than the calculation of the sensible from the supersensible. We comprehend the sensible by means of its supersensible beginning of origin. For sense-perception, the Differential is but a point, a zero. For spiritual comprehension, however, the point becomes alive, the zero becomes an active Cause. Thus, for our spiritual perception, Space itself is called to life. Materially perceived, all its points, its infinitesimally small parts, are dead; if, however, we perceive these points as differential magnitudes, an inner life awakens in the dead "side-by-side." Extension itself becomes the creation of the extensionless. Thus did life flow into Natural Science through Infinitesimal Calculus. The realm of the senses is led back to the point of the supersensible..

Plato and the Gnostics only recognised in mathematical science a good means of education, and no more than this is here implied about the mathematics of the infinitely small; nevertheless to the Occultist it does present itself as a good educational means. It teaches him to effect a strict mental self-education where sense-perceptions are no longer there to control his wrong associations of ideas. Mathematical science teaches the way to become independent of sense-perception, and at the same time it teaches the surest path; for though indeed its truths are acquired by supersensible means, they can always be confirmed in the realm of the senses. Even when we make a mathematical statement about four-dimensional space, our statement must be such that when we leave the fourth dimension out and restrict the result to three dimensions, our truth will still hold good as the special case of a more general proposition.

On the lowest and most elementary plane we have an Arupa reality before us in the Differential. When we reckon in Differentials we are always on the border-line where Arupa gives birth to Rupa. In Infinitesimal Calculus, therefore, we can train ourselves to grasp the idea of Arupa and the relation of this to the Rupa. We need but once integrate a differential equation with full consciousness; then we shall feel something of the abounding power that exists on the borderline between Arupa and Rupa.

..

It is precisely those who oppose this overrating of mathematics itself who can most thoroughly value the true enlightened research which advances in the spirit of mathematics even where mathematical science itself ceases. For in its direct meaning mathematical science after all has to do only with what is quantitative; where the qualitative begins, there its domain ends."

The last sentence reminds me of my years long battle in the Hole to get them to understand how physics works by the mathematisation of primary qualities. This mathematisation Der Staudi accused me of thinking as something negative, as materialism! As I've said, he is epistemologically clueless.

I like this quote from Goethe:

"Even where we do not require any calculation, we should go to work in such a manner as if we had to present our accounts to the strictest geometrician. For it is the mathematical method which on account of its thoroughness and clearness reveals each and every defect in our assertions, and its proofs are really only circumstantial explanations to the effect that what is brought into connection has already been there in its simple, single parts and in its entire sequence; that it has been perceived in its entirety and established as incontestably correct under all conditions."

The 'circumstantial explanations' are what Kant refereed to as the 'analytic' and is sometime called the tautologous nature of mathematics. The 'perceiving in its entirety' is an illuminating formulation of the nature of mathematical thought, suggesting something objective and exact, in the nature of a sense perception.

Naz said in his post:

"The very fact I can write this email and send it is due to Mathematics."

which relates to Steiner's concluding thought in that lecture (though Naz's intention by his example is to refute what Steiner claims):

"And just as the mathematician is consistent in life, just as he is able to construct bridges and bore tunnels by virtue of his training that is to say, he is able to command the quantitative reality, in the same way, only he will be able to understand and rule the qualitative,who can make himself master in the ethereal heights of sense-free perception. This is the Occultist. Just as the mathematician builds the shapes of iron into machines according to mathematical laws, so does the Occultist shape life and soul in the world according to the laws of these realms which he has understood in the spirit of mathematical science. The mathematician is led back to real life through his mathematical laws; the Occultist no less so through his laws. And just as little as he who is ignorant of mathematics is able to understand how the mathematician builds up the machine, even so little can he who is not an Occultist understand the plans by which the Occultist works upon the qualitative forms of life and soul."

T.

Ted Wrinch

--- In anthroposophy_tomorrow@yahoogroups.com, "ted.wrinch" <ted.wrinch@...> wrote:

>

> "For further study, I would recommend to the Sugar Cherubs that they make the lacture Mathematics and Occultism (GA 35) -- http://wn.rsarchive.org/Lectures/19040621p01.html -- the focus of their next group study."

>

> Yes, that's a good lecture, which I've read before: the whole spirit and ethos of if it runs through all of Steiner's work and is one reason why he uses the mathematical parallel he does in Theosophy of the Rosicrucian. This parallel is obvious and irrefutable for anyone that thinks about the nature of mathematical knowledge. It's not hard to understand and you don't need to know advanced maths to do so; in fact people that do have such advanced knowledge can end up simply over-complicating the issue, as Naz has been dong on WC . But few people seem interested in this profound but simple idea. It will be interesting to see if WC make any progress but it doesn't look good so far: Dugan, who as a scientifically trained person should have an interest in this, has ignored the mathematical content of Tankazoo's latest post and berated him for its format. From the lecture:

>

> "Now Plato looked upon mathematical science as a means of training for life in the World of Ideas emancipated from sense-perception. The mathematical images hover over the border-line between the material and the purely spiritual World. Let us think about the "circle"; we do not think of any special material circle which perhaps has been drawn on paper, but we think of any and every circle which may be represented or met with in Nature. So it is in the case of all mathematical pictures. They relate to the sense-perceptible, but they are not exhaustively contained in it."

>

>

> T.

>

> Ted Wrinch

>

> --- In anthroposophy_tomorrow@yahoogroups.com, "elfuncle" <elfuncle@> wrote:

> >

> > I admit that this discussion may look intriguing, but getting involved

> > in it on any level won't go anywhere. From what I've glanced very

> > briefly, some of those adorable faith-defenders or court jesters or

> > whatevery they may be, have tried to argue ageinst the notion that

> > Steiner was being authorative while saying they should accept nothing on

> > authority, and for this purpose they've used examples like a travel

> > agent who has been to Spain while the wannabe travellers haven't and so

> > forth. It's reminiscent of Rudolf Steiner's own comparison to a seeing

> > person explaining things to a group of blind people. On top of all that,

> > it is argued that if Steiner's spiritual science is authoritative, so is

> > mathematics. But once you make a claim like that without being a

> > mathematician and a natural philosopher, preferably a physics professor

> > like Arthur Zajonc, you'll be stuck in the Sugarland Quicksand

> > indefinitely; it may be compared to wars expected to be short and

> > glorious but dragged on for years and years instead, like WWI or the

> > Vietnam War.

> >

> > The only thing I've mentioned repeatedly in this context is that

> > intellectual proof has no validity beyond the limits of natural science

> > and mathematics. It's something Rudolf Steiner described extremely well

> > in The Ahrimanic Deception (GA 193)

> > <http://wn.rsarchive.org/Lectures/AhrDec_index.html> -- first an

> > important remark about mathematics and natural science in that same

> > lecture:

> > "Ahriman has the greatest possible interest in instructing men in

> > mathematics, but not in instructing them that mathematical-mechanistic

> > concepts of the universe are merely illusions. He is intensely

> > interested in bringing men chemistry, physics, biology and so on, as

> > they are presented today in all their remarkable effects, but he is

> > interested in making men believe that these are absolute truths, not

> > that they are only points of view, like photographs from one side. If

> > you photograph a tree from one side, it can be a correct photograph, yet

> > it does not give a picture of the whole tree. If you photograph it from

> > four sides, you can in any case get an idea of it. To conceal from

> > mankind that in modern intellectual, rationalistic science with its

> > supplement of a superstitious empiricism, one is dealing with a great

> > illusion, a deception that men should not recognize this is of the

> > greatest possible interest to Ahriman. It would be a triumphant

> > experience for him if the scientific superstition which grips all

> > circles today and by which men even want to organize their social

> > science, should prevail into the third millennium. He would have the

> > greatest success if he could then come as a human being into Western

> > civilization and find the scientific superstition."

> > And here is the crucial point I've quoted many times before:

> > "Ahriman makes use of this confusion in order to prepare the triumph of

> > his incarnation and to drive men with increasing force into what they

> > find so difficult to realize namely, that by intellectual or

> > modern scientific reasoning today, one can prove anything and equally

> > well prove its opposite. The point is for us to recognize that

> > everything can be proved and for that reason to examine the proofs put

> > forward in science today. It is only in natural science that reality is

> > shown by the facts; in no other field can one consider intellectual

> > proofs valid. The only way to escape the danger that threatens if one

> > accepts the lures of Ahriman and his desire to drive men deeper and

> > deeper into these things, is to realize through anthroposophical

> > spiritual science that human knowledge must be sought for in a stratum

> > deeper than that in which the validity of our proofs arises."

> > And here is the rub: The Sugar Cherubs are neither able nor willing to

> > search for knowledge in a stratum deeper than that in which the validity

> > of our proofs arises. And for this very reason, it is a total waste of

> > time and effort to discuss with them, regardless of how much they beg

> > for such discussions. It's like talking to the wall. You may follow a

> > course of action pursued by some other anthroposophistgs, namely to

> > track them down at various forums on the internet where they bombard

> > puzzled outsiders with their fanatical propaganda, so belligerent and

> > calumnious that Eckart and Goebbels would have been impressed by their

> > ideological fervor against the dirty, devious, plotting, dishonest,

> > greedy, megalomaniac, inferior Jews and Gypsies -- huh, Anthroposophists

> > and Esotericists.

> >

> > They don't call it Waldorf Watch for nothing; it's inspired by Jew Watch

> > and Jihad Watch. Notice also that their closed Yahoo Group is

> > "Waldorf-Anthroposophy-Steiner Survivors". This group is not private and

> > exclusive, which would have made it invisible; it's listed publicly for

> > recruitment. And the very title implies that its members have escaped

> > from anthroposophical death camps by the skin of their teeth, perhaps

> > with Steiner-tattoos on their arms and the works.

> > Waldorf-Anthroposophy-Steiner Survivors suggests that there are many

> > more Waldorf-Anthroposophy-Steiner NON-Survivors: They're all dead

> > because of anthroposophy, Waldorf, and Steiner. Think about it.

> >

> > Well, how can you discuss with people like that? It would be an iman

> > trying to discuss Islam with Jihad Watch, or a rabbi trying to talk to

> > Jew Watch. An anthroposophist wanting to have dialogue with Waldorf

> > Watch or Waldorf Critics or PLANS or whatever, has exactly the same

> > problem: Talking to the wall. It's the nature of fanatical hate groups.

> >

> > Rudolf Steiner continues in the same lecture:

> > "And so, in order to create dissensions, Ahriman also makes use of what

> > develops from the old conditions of heredity which man has really

> > outgrown in the Fifth Post-Atlantean Epoch. The Ahrimanic powers use all

> > that is derived from old circumstances of heredity in order to set men

> > against each other in conflicting groups. All that comes from old

> > differences of family, race, tribe, peoples, is used by Ahriman to

> > create confusion."

> > Who is doing all this -- creating confusion by all this talk about race,

> > racism, nationalism and so on? The Sugar Cherubs!

> >

> > The subject is mathematics, however, and in the following interesting

> > note on that from Steiner's early writings in the late 19th century

> > illumines the character of mathematics in such a way that the very

> > uselessness of discussing such a topic with Sugar Cherubs should be

> > obvious:

> > "Mathematics deals with magnitude, with that which allows of a more or

> > less. Magnitude, however, is not something existing in itself. In the

> > broad scope of human experience there is nothing that is only magnitude.

> > Along with its other characteristics, each thing also has some that are

> > determined by numbers. Since mathematics concerns itself with

> > magnitudes, what it studies are not objects of experience complete in

> > themselves, but rather only everything about them that can be measured

> > or counted. It separates off from things everything that can be

> > subjected to this latter operation. It thus acquires a whole world of

> > abstractions within which it then works. It does not have to do with

> > things, but only with things insofar as they are magnitudes. It must

> > admit that here it is dealing only with one aspect of what is real, and

> > that reality has yet many other aspects over which mathematics has no

> > power. Mathematical judgments are not judgments that fully encompass

> > real objects, but rather are valid only within the ideal world of

> > abstractions that we ourselves have conceptually separated off from the

> > objects as one aspect of reality. Mathematics abstracts magnitude and

> > number from things, establishes the completely ideal relationships

> > between magnitudes and numbers, and hovers in this way in a pure world

> > of thoughts. The things of reality, insofar as they are magnitude and

> > number, allow one then to apply mathematical truths. It is therefore

> > definitely an error to believe that one could grasp the whole of nature

> > with mathematical judgments. Nature, in fact, is not merely quantity; it

> > is also quality, and mathematics has to do only with the first. The

> > mathematical approach and the approach that deals purely with what is

> > qualitative must work hand in hand; they will meet in the thing, of

> > which they each grasp one aspect. Goethe characterizes this relationship

> > with the words: 'Mathematics, like dialectics, is an organ of the inner,

> > higher sense; its practice is an art, like oratory. For both, nothing is

> > of value except the form; the content is a matter of indifference to

> > them. It is all the same to them whether mathematics is calculating in

> > pennies or dollars or whether rhetoric is defending something true or

> > false.' (Aphorisms in Prose)"

> > ( -- Rudolf Steiner: Goethean Science, XII: Goethe and Mathematics

> > <http://wn.rsarchive.org/Books/GA001/English/MP1988/GA001_c12.html> ,

> > GA 1)

> > For further study, I would recommend to the Sugar Cherubs that they make

> > the lacture Mathematics and Occultism (GA 35) --

> > http://wn.rsarchive.org/Lectures/19040621p01.html -- the focus of their

> > next group study.

> >

> > Don't discuss with them, don't argue with them, just love them.

> >

> > All they need is love! <http://www.youtube.com/watch?v=r4p8qxGbpOk>

> >

> > Tarjei

> >

> >

> >

> > --- In anthroposophy_tomorrow@yahoogroups.com, "ted.wrinch"

> > <ted.wrinch@> wrote:

> > >

> > > Naz on WC has given a rather complicated description, using bits of

> > Euclid, of how one might prove that the angles of a triangle add up to

> > 180 degrees, but he hasn't shown the actual proof. In the spirit of

> > Steiner's account in the Theosophy of the Rosicrucian, we can say that

> > this is ancient history and we should use the simplest way of proving

> > this, which is via the notion of a straight line: see here:

> > http://www.mathsisfun.com/proof180deg.html (which is the technique

> > Steiner used elsewhere). Naz says:

> > >

> > > " Mathematics is fundamentally structured using two entities, axioms

> > or postulates and logic .I will not discuss logic here but it is

> > also worthy of discussion in the scope of Anthroposophy"

> > >

> > > " it is possible for people to form the same mathematical theories

> > completely independently of each other. They do not need a charismatic

> > leader, they do

> > > not need clairvoyance, they do not need contrived histories of

> > mankind, though useful they do not even need a teacher. They simply need

> > basic

> > > logic

> > >

> > > Mathematics has stood on very sound ground for thousands of years. ...

> > >

> > > So here's an open challenge to anyone on this list. If you have ANY

> > example of where Mathematics has simply stated something as fact (as

> > Steiner does

> > > often) without an axiom and logic based system please state it."

> > >

> > > But the 'logic' is what is at issue: what is 'logic': he doesn't say.

> > But in Steiner's parallel it is thought, and the axioms are evidence.

> > It's not the results of axiomatic mathematics that Steiner is interested

> > in in the parallel but the technique of relying on the inner coherence

> > of thought, that is used in mathematics to provide the 'solid ground'

> > that Naz refers to. Clearly the 'evidence' in Steiner's parallel is

> > something more complex than the axioms of mathematics. But axioms are

> > chosen for the value of the mathematical insight that they result in

> > and, similarly, the evidence that is significant for the spiritual world

> > is that that provides the greatest spiritual insight.

> > >

> > > T.

> > >

> > > Ted Wrinch

> > >

> > > --- In anthroposophy_tomorrow@yahoogroups.com, "ted.wrinch"

> > ted.wrinch@ wrote:

> > > >

> > > > Wow: Pete mis-understanding Steiner's triangle example in the text:

> > > >

> > > > "Is it a fact that there are 180 degrees in a triangle? Sure, but

> > only because someone decided what division of a circle constitutes a

> > "degree"."

> > > >

> > > > The number of degrees contained in a circle is a *convention*, it

> > does not matter for the example Steiner chose, which relies on the

> > notion of a straight line for its proof. A straight line contains 180

> > degrees but if it contained something else, or we express the measure in

> > radians, which depend only on the geometric notion of the ratio of the

> > circumference of a circle to its diameter , the example would still work

> > out. I can't believe that his guy claims to have an engineering

> > training!

> > > >

> > > > T.

> > > >

> > > > Ted Wrinch

> > > >

> > > > --- In anthroposophy_tomorrow@yahoogroups.com, "ted.wrinch"

> > <ted.wrinch@> wrote:

> > > > >

> > > > > It's startling to me that someone can have got so far up the

> > academic ladder and remained an epistemological incompetent. But it does

> > explain why, after more than a decade and a half of reading Steiner, Der

> > Staudi still doesn't understand him. All he's really been doing in that

> > time is playing polemical games with half-understood patterns of words.

> > What a waste of what could have been a significant academic talent!

> > > > >

> > > > > T.

> > > > >

> > > > > Ted Wrinch

> > > > >

> > > > > --- In anthroposophy_tomorrow@yahoogroups.com, "ted.wrinch"

> > <ted.wrinch@> wrote:

> > > > > >

> > > > > > Just noticed that Tarjei's posted on this already but I'll post

> > my bit anyway - it's mostly on my, and Steiner's, favourite subject of

> > the nature of mathematical knowledge.

> > > > > >

> > > > > > The study on WC has just become very interesting; not for any

> > illumination the participants have shed on the text, rather the reverse:

> > Der Staudi has revealed the evidencum crucium (just invented that as I

> > never did Latin at me old comprehensive skool oop North, but you get the

> > idea) in his response to the text in WC message 22486. He thinks that

> > the difference in the kinds of knowing that Steiner presents in chapter

> > 1 of Theosophy of the Rosicrucian hinges on the distinction between

> > clairvoyant knowledge and 'affirmation'. He's wrong, and it's

> > significant and revealing of his own epistemology lack that he is.

> > > > > >

> > > > > > Steiner compares the manner of understanding Rosicrucian truths

> > to that in which we understand mathematics. He says that if we think

> > clearly enough about the evidence he presents the truth of the situation

> > will become apparent to us. But, as Steiner has said elsewhere, the

> > truths of mathematics are relatively simple and the truths of wider life

> > are not, and this process of gaining knowledge in the manner of

> > mathematics therefore has to be more of an ideal and a goal than a

> > present reality for many of us. But, Steiner says, to the extent that we

> > think accurately, clearing away prejudice and bias from our thought

> > (Steiner's 'abstract theorising and materialistic conjectures'), and

> > look around and carefully assess *all* the evidence (Steiner's 'standing

> > firmly upon the basis of facts and not going beyond them') it is

> > possible that we can become convinced of these truths from our own inner

> > experience. This process of gaining knowledge is clearly a free and

> > independent one, and not one that requires assenting to the teaching and

> > practises of a guru. In the past the ability to follow the presentation

> > of esoteric knowledge in clear concepts was not possible in this way and

> > such knowledge was presented in symbolic, and other non

> > straightforwardly conceptual modes. This difference of approach between

> > Rosicrucianism and earlier esotericism is what makes the approach non

> > authoritarian. And there is no requirement in this mode of knowing for

> > mere 'affirmation', as Der Staudi has claimed.

> > > > > >

> > > > > > Der Staudi says:

> > > > > >

> > > > > > "Steiner compares learning the content of this ostensibly

> > ancient spiritual wisdom to learning "authentic and true geometry" in

> > elementary school. For

> > > > > > Steiner, these esoteric teachings are simply and obviously True,

> > with no need for argument, evidence, or demonstration."

> > > > > >

> > > > > > This is the opposite of Steiner's claim. He does not assert that

> > the 'teachings are simply and obviously True'. Instead, he asserts that

> > one needs to think about them for oneself, and their truth can then

> > become apparent in the manner of a mathematical truth. Mathematical

> > truths are true because of their inner coherence, not because they are '

> > simply and obviously True'. They don't need affirmation to be true and

> > their truth is not authoritarian, because it needs no external source

> > for its verification. Der Staudi simply doesn't understand the nature of

> > mathematical truth and therefore the parallel Steiner is making in the

> > lecture.

> > > > > >

> > > > > > The rest of Diana and Der Staudi's responses are mostly

> > restatements of their usual Steiner prejudices, with complaints about

> > 'authoritarianism', Middle European ethnocentrism and faux

> > scientific-ness and aren't worth saying much about. But one point that

> > should be made in response is that Steiner's claims that Rosicrucian

> > knowledge first flowed into Middle Europe is not simply a 'myth' or a

> > chauvinistic claim, as Der Staudi asserts: historically the knowledge

> > was made available first in this area of Europe in the form of the 3

> > manifestos that appeared in Kassel and Strasbourg.

> > > > > >

> > > > > > T.

> > > > > >

> > > > > > Ted Wrinch

> > > > > >

> > > > >

> > > >

> > >

> >

>