Geometric algebra and the scientific method

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Hi, I’m glad I found this community and grateful to Leo Dorst for the invite.

Though this may not be exactly on the main topic,
I think this is the most important and fundamental aspect of GA, and ironically the main reason for the reluctancy of the scientific community when it comes to this field.
This is also something that David Hestenes was very adamant about albeit in slightly more standard connotation and which the video series on forque by Leo and Steven displays quite brilliantly, and this is really just following the scientific method in theoretical research.

In one of his papers on the foundations of mathematical physics David left the question open,
whether GA was invented or discovered, which of course is just a special case of a more general question that has plagued natural philosophers for centuries: is mathematics invented or discovered?

This may seem like an impossible question , as it’s provably unknowable whether mathematics is even consistent,

but GA could provide a definite and constructive answer.

A certain way not to invent mathematics and mathematical structure, at least relative to the axioms and basic definitions, is just to not use any arbitrary or subjective structure.

A discovered theory can prove its theorems without invented structure, and a theorem is discovered if it can be proven Directly from axioms and basic definitions.

Naturally then vector algebra, linear algebra, etc… would be classified as invented theories.

Of course, most people are well aware that using arbitrary objects and subjective assumptions is not ideal, and thus at least they try to make the basic objects covariant but that just lets them move from one arbitrary representation to another consistently, without working with actual real geometric objects.

GA shows that it is indeed possible to work without arbitrary local charts, coordinates, basis etc and in fact quite necessary and is the very reason behind its power and efficiency,
though advanced practitioners can still take educated shortcuts and make a few choices to simplify calculations and engineering implementations.

Critics of GA say it doesn’t present any new discoveries, on the contrary, if done right with GA everything is discovered, and many objects and theorems are seen for the first time as they really are, without the baggage of arbitrary structure, usually in a completely new light with clearly unraveled connections to other seemingly incompatible areas.

Sir Michael Atiyah used to say algebra was an offer of the devil to the mathematician, that it can answer any question provided one gives up his soul in geometry.
This was the position of mathematicians from the late 19th century onward as geometry and algebra were viewed as polar opposites, though this of course was not always so.

GA is a living proof such distinction was indeed artificial, and that in fact a completion of the standard basic algebraic field and number system by non-commutative square roots (vectors) instead of complex numbers naturally leads to a more complete and powerful algebra with a plethora of natural geometric interpretations.
From the geometric side, renouncing the assumption that the product of translation (vectors) generally commutes, one again arrives at the same mathematical theory that can prove its theorems without extrinsic arbitrary structure all the while being more geometric and even more computationally powerful when applied to ‘purely’ algebraic problems.

From this perspective, GA is not a new discovery, but a result of the natural progress and scientific development that took a tragic turn in 1879.,
and of all the great unifications in history and of any future potential unifications this I consider to be the greatest, and history suggests ( electromagnetic unification being most natural in GA) it will be the basis for any further fundamental unified theories in physics and mathematics.

As the experimental method is reaching its fundamental limits simply due to how deeply one can physically probe into the nature of reality, scientists must be particularly keen on how theoretical research is conducted, else they are left largely wondering in the fiction of their imagination, and is likely the main reason behind the many and persistent crisis in fundamental and mathematical physics

GA already shows us strong hints to the basic mysteries of existence, which would not itself even exist if not for the half spin difference between mater and interactions, which is deeply related to the fundamental representation of GA. This is something that Dirac was very adamant about, that the right theoretical method in following the beauty and simplicity of mathematics is even more crucial than the experimental investigations, at least initially, though he never precisely articulated what he meant by beautiful and interesting mathematics.

In conclusion, to me geometric algebra is not just some handy new algebra with fancy notation, it is the very gate of science, both applied and theoretical, and the gate as Clifford said so small it can be only entered as a little child, that is without the baggage of unnatural conceptual structure.
I think this is can be quite useful to keep in mind, and on a more practical note, as I don’t think the state of mainstream academia is likely to change anytime soon, i was wondering if there is any possibility of opening study programs based geometric algebra from the ground up. If not at a full university level maybe a shorter application focused s.t.e.m. stem program , and if there are any ideas on what can be done about it, finance-wise or in any other way.

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Hello! Im replying to this because my prime motivation in studying geometric algebra is philosophical/metaphyiscal. I just recently discovered geometric algebra and is currently in the process of learning by reading the book
co-written by Leo Dorst (Geometric Algebra for computer science),
This is an excellent book which takes an intuition-first approach to teaching mathematcs which i feel is so rare to find in other textbooks.
I am very grateful to have found this book.

I am trying to write an essay where I seek to connect the ancient greek understanding about the one and the nature of eternity. If we disregard the irrational and trancendental numbers, the pythagorean understanding shared by mathematicans in late antiquity such as Theon and Nichamachus among others was that every number is composed of measures and/or addtions of 1. If you read them you can see they have a thelogoical motivation for studying number theory and artihmetic which I simplify by saying was rooted in the understanding that all is 1. (but it is not pantheism, but rather panentheism).

I have searched for mathematics that could demonstrate how geometry, similar to number, can be unfolded and reverted back into unity.
At my current level of intuitive understanding, I see that geometric algebra indeed possess this property, where dimensional expansion/reversion is contained within the pseudoscalar combined with the determinant, the outer product, the geometric product and the contraction operators.

From nothing, nothing can proceed (ex nihilo nihil fit), this is the undeniable truth and starting premise which have led philosophers and sages throughout the ages to grapple with the one and the many problem. Logical deduction can lead us to understand the undeniability of the eternal one to be the root-condition of all possibility, which is also the basis for understanding the lingering conclusion in Plato’s dialogue Parmenides, namely that if there is a one, everything which in reality can exist and all that has ever existed is a static unchangeable singleness or one.

A few sentences from the first chapter to Hegels book The science of logic.

"The beginning is not pure nothing but a nothing, rather, from which something is to proceed; also being, therefore, is already contained in the beginning.

Therefore, the beginning contains both, being and nothing;
it is the unity of being and nothing, or is non-being which is at the same time being, and being which is at the same time non-being…

…The being contained in the beginning is such, therefore,
that it distances itself from non-being or sublates it as something which is opposed to it.

But further, *that which begins already is but is also just as much not yet
The opposites, being and non-being, are therefore in immediate union in it;
or the beginning is their undifferentiated unity…

…But the beginning ought not itself to be already a first and an other,*
*for anything which is in itself a first and an other implies that an advance has already been made."

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@WhiteEyes what you are talking about sounds more like Buddhism than Mathematics

Mathematics is just a game of manipulating symbols like an abacus to keep track of something

Buddhism is more transdental and tries to avoid getting stuck on tracking an idea of symbol, because all these ideas and symbols make separate from the unity

Trying to combine buddhism with mathematics is somewhat futile. You can study math and be a buddhist, but studying math is about pure symbol manipulation, while buddhism is about transcending such symbol manipulations.

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@chakravala
Hi thanks for responding, have not studied buddhism, manly interessted in what is refered to as theory of forms, similar ideas also found in sufism where I have studied Ibn arabi mostly. If you read Proclus Elements of theology, you will understand my frame of reference in its strict logical expression.

Here is a quote from an essay by Albert C. Lewis on what the original motivation for grassmanns extension theory was.

“The logic of mathematics as Grassmann understands it is, in fact, in a strict sense logic of the source (Logik des Ursprungs)… [The fundamental concept of the source] is fulfilled wherever the members of a manifold are deduced from a definite serial principle and exhaustively represented by it.” [Cassirer 1953, 99].

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While I am of course a big fan of Grassmann, having named my software after him; I disagree that Grassmann algebra are the most fundamental source. Source of what? Grassmann algebra are a foundation for linear algebra based mathematics.

However, the universe is not linear algebra. People are always trying to make everything linearized, because it provides a good foundational language.

However, those linear models are just models. They are just symbol manipulation.

The model of the atom is just symbol manipulation.

An actual atom isn’t a bunch of symbols. Symbols are just humans way of referring to a model of what they think might be out there in the universe.

It is appropriate to say that Grassmann algebra can serve as the fundamental source for linear models, but the universe isn’t a model or symbol. Symbols are just symbols, just pointers, mere references.

Symbolic manipulation is just an exercise people do in an attempt to point at and reference to things.

If you are confusing the symbols and models for reality itself, then you are mistaken. Reality is not a bunch of symbols or models… those things merely are attempts at pointing at the thing itself.

Grassmann algebra is not a religion, it’s symbol manipulation just like an abacus.

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Yes I do not disagree with what we can call
“The map is not the territory argument”, and yes I am mainly concerned with metaphysics, not physics or empirical science, although I have a speculative understanding regarding the mass-gap / yang mills theory which tie into a notion about occasionalism also. I view the empirical proof of the discreteness of light / electromagnetic radiation, which is the discovery that made quantuum phyiscs applicable in reality to be in support of the argument for that between the gaps so to speak, an eternal domain resides which effectuates time/change, on a lower dimension (the temporal plane), where the principal measure of change itself is offcource the speed of light.

This understanding, views all which is subjected to change to be in a constant transition of ‘becoming’ but has no permanent being. Existence can thus be described as being dependent for this existence by causes that resides in the eternal, and no sensible object formed by a unity of matter can be said to possess being, but only enter into being at discrete instants. This means that when a material entity enter into being, its existence is updated with a new accident predetermined by an eternal cause, (i.e change of position, color, etc).

here is good lecture that gives an overview of the historical development of quantuum physics from the 19th century onwards

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This is essentially just Buddhism, if you study Buddhism you will find people have said this for thousands of years in their teaching.

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The pythagoreans did not mangage to express a solution to the problem of how all geometry can be unfolded and reverted from a single point in the void of space, but it must have been their goal.

having searched selected translated works by: Nichomaus, Theon, Diophantus and Iamblichus… A possible more Pythagorean approach of associating number with geometry is to look at what is called: Geometry of numbers which links Nicomachus way of viewing the positive integers > 2 with what is today called figurate numbers this associates the con-cyclic regular polygons with every integer > 2, because every integer > 2 can be seen as representing a regular polygon with N vertices inscribed within the unit-circle. 3 is the equalateral triangle and 4 is the square and so on, 1 is the origin, 2 is the radius of the unit circle. We then associate a regular polygon to represent the central cell of a lattice or a vector space with N linearly dependent basis vectors which joins the center of the polygon with each of its vertices. However this is not fully explicitly articulated by the Pythagorean authors and I have not found an easy way to understand how geometry constructed within this lattice or vector-space in two dimensions has a natural way of extending to a three and higher dimensional object where metrics between points gets preserved so that a higher dimensional geometric object can be reverted to its lower dimensional counterpart. Geometry of numbers is also an area which Minkowski studied and made certain discoveries in.

Here is a qoute from the english translation of grassmanns extension theory,
which shows that grassmann was thinking about the theory of forms,
He also translated the Rig vedas which should be regarded as another indication.

The formal sciences treat either the general laws of thought or the particular as established by means of thought, the former being the
dialectic (logic), the latter, pure mathematics. The contrast between the general and the particular thus produces the division of the formal sciences into dialectics and mathematics. The first is a philosophical science, since it seeks the unity in all thought, while mathematics has the opposite orientation in that it regards each individual thought as a particular. Pure mathematics is therefore the science of the particular existent that has come to be by thought. The particular existent, viewed in this sense, we call a thought form or simply a form; thus pure mathematics is the theory of forms.

Did you know that Proclus also wrote a commentray on Euclid’s Elements?
I have not read it yet but here is the initial sentences from his commentary.

mathematical being necessarily belongs neither
among the first nor among the last and least simple
of the kinds of being, but occupies the middle ground between
the partless realities1—simple, incomposite, and indivisible—
and divisible things characterized by every variety of com
position and differentiation. The unchangeable, stable, and
incontrovertible character of the propositions about it shows
that it is superior to the kinds of things that move about in
matter. But the discursiveness of [mathematical] procedure,
its dealing with its subjects as extended, and its setting up of
different prior principles for different objects—these give to
mathematical being a rank below that indivisible nature that
is completely grounded in itself.

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You’re assuming that the big bang is a valid model, when it’s certainly possible that the big bang is only a hallucination of Stephen Hawking and may not correspond to actual reality.

Just because you can theorize about a big bang in the early universe using mathematics, does not mean that it corresponds to our universe.

The idea of a big bang actually has its roots with 20 century scientists like Stephen Hawking and not the ancient greeks, so I also disagree that the greeks were working on solving that kind of problem.

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@chakravala
Why do you assume my thinking is related to the big bang theory?
I assure you, it has nothing to do with it. The cosmos is eternal but not static, its eternal in the perpetual sense.
Btw The arguments for Big bang is so utterly stupid, “cosmic background radiation” is just statistical heat maps which skews the data, also it doesnt take into consideration relativity.

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This is a preliminary understaning of what im hinting at:

The point was defined by Euclid to not have any bounds, and therefore does not occupy any of its ambient space, and this is the most common understanding of a point used within mathematics today. Therefore the point at the origin [0,0,0,0,0,0,0…,n] is only a dimensional marker which doesn’t require space for its existence but still holds the root condition for any mathematical space or geometrical object to even be pre-concieved. The point at the origin, is a nothing and everything at the same time. It is a singleness which is everything because anything else would lack dependable means to exist, both intellectually without sense and manifestly without mind. All potentia in terms of interrelations of proportion as extensions in distance is contained within, by the mathematical property of it being one without an other. When we make a distinction of what is real and imaginary. we are limited by the notion of dependence for existence. If there was no space, the physical or manifested cosmos would not have actuality, If we then imagine souls equipped with an intellect independent of space, the souls imagination of mathematical space would likewise presumably be dependent on the concept of a point at the origin.

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Because that’s what you described:

The idea that all geometry unfolds from a single point in space, when applied to the geometry of spacetime, is basically what big bang idea is.

Take what you’re saying, apply it to the geometry of spacetime, the result is in principle big bang idea.

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With the single point at the origin I do not mean a gravitational singuality.
Platonic realism views the physical cosmos to only be an “image” of the real
which is unsensible, although we can experience glipses of it during certain altered states of mind, typically we see blinding flashes of light, or if we see the prime matter, we become blind and see only white but not blinding white like looking into the sun, but more of a cloudy white which blinds ones vision.
Many references of this among mysics and sages.

Its quite difficult to understand wholesome static simultaneity.
I have not fully understood it, but here I provide quotes by Ralph Austin and Toshihiko Izuzu about the divine breath (understand this as a metaphor only), understand also that this “re-creation” happens at a frequency of the speed of light. But perhaps also at the frequency of “The platonic great year”

You should not understand this point to be equal to the one , this is a superficial interpreatation. Prior to the point at the origin is the concept of the limit and the unlimited expressed in Platos Dialogue Philebus.

In Toshihiko Izutsu’s excellent work Sufism and Taoism he express this occasionalistic relationship as follows:

The self-manifestation of the Absolute is, in fact, possessed of a double structure. It is a trans-historical, trans-temporal phenomenon, but it is also a temporal event.One might even say that this is precisely the greatest coincidentia oppositorum observable in the structure of Being. It is a temporal event because from eternity the same process of tajalli (the Absolute~the world) has been repeated and will go on being repeated indefinitely. Since, however, exactly the same ontological pattern repeats itself infinitely, and since, moreover, it is done in such a way that as the first wave is set in motion, there already begins to rise the second wave, the process in its totality comes to the same thing: an eternal, static structure. This dynamic-static self-manifestation of the Absolute is described in terms of the ‘strata’ (marãtib)’

Ralph Austin’s interpretation on the matter is similar:

On a divine scale, in the case of the Breath of the Merciful, each inhalation represents the resolution of the Cosmos into the Essence, while each exhalation represents the creation of the Cosmos, representing the two currents of the divine Mercy, the one releasing the archetypal desire for existence, the other reaffirming the exclusive integrity of the Absolute One. In reality, however, there is no temporal sequence here, but an eternal simultaneity, since at each instant the Cosmos is and is not, is manifest and latent, created and uncreated, is other and non-other in a timeless divine pulse, at once creative and non-creating.’

In other words:
This relationship is a recursive triad in the form: essence substance accident where the substance is the intermediary layer (the prime matter) between the eternal and temporal**.** In this way, we can think of it as a cycle of re-manifestation and annihilation and a feedback loop where the output is at the same time the input that determines the next discrete change to become expressed and re-manifest. In the act of inhalement god takes in every temporal existence in a static instant and its state of inter-relational information it has to the rest of the temporal world in this instant. With this information god then synchronizes every determined change and exhales his spirit/sigh of relief which re-manifests. The cycle of re-manifestation thus causes the discrete progression of time/change from instant to instant for every temporal existence. The spirit is what express a momentary image of the underlying cause, although this is a complex idea and most of what I say here is only a superficial understanding.

A modern or scientific theory which would most closely agree with this understanding, would be the one electron universe.

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Here is a paper that completly destroys the arguments of the big bang theory, about redshifts. The paper is only using cosmological doctrinal arguments as a way of refuting the arguments themselves, like an aikido master, using the strength of the opponent against the opponent, nothing more nothing less. But I would also like to see someone attacking the erronious way of using statistics to be realiable evindence for anything regarding the uniformity of “the cosmic background radiation”

Tell me how anything other than a statistical calculation could represent data from the whole universe? It’s offcourse impossible, I argue that it’s the method of calculation it self that cause the appearance of uniformity of cosmic background radiation, because vast distances of needs to be condensed for the computation to even be possible, try fitting a single square lightyear area subdivided in square miles in a computer memory. a statistical calculation has the natural effect of skewing the data towards a mean. If they would do a point survey equaly distant from stars of approximately similiar size, it would be a more accurate way of saying that there is a uniformity unaffected by the light and heat coming from stars but that is not what usually presented as “evidence”.
And even if they have or have not done this, its would still not be sufficient by the fact that it would only be a miniscule number of datapoints when considering the whole or close to the whole universe.

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@WhiteEyes i am always tempted to attack your arguments because your writings tend to be religious nonsense instead of scientific.

I’m aware about the red shift and the issues of the big bang, I never promoted the big bang concept myself and am not convinced of big bang.

The problem with your inquiry is that it confuses science with religion, and it’s super cringe to me.

The way you talk, you combine fancy words from science and religion as if they are meant to be uttered in the same sentence. Sorry, but just no dude.

Religion and science have entirely different purposes in human society. Religion is vague speculation about the existential consciousness of existence, while science is about hard facts.

Attempting to use science to speculate about the existential dread of consciousness of existence, it’s entirely pointless and futile. You’re going to be blabbering nonsense forever, never getting anywhere with it. As long as you do that, you will appear as somebody who is saying nonsense, because you confused science with religion.

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Well, what is this opinion worth? Science is not about opinion right?
So what is your intent in engaging in this discussion? Out of boredom? Or do you want me to follow your advice? Should you become my teacher, so that I can gain the proper enligtend attitude towards science and religion? I could perhaps question my own motivation for engaing in this single man discussion on this forum, I dont realy have a clear intention, I only felt like expressing my thoughts, but your impolite tone makes me reluctant to continue, perhaps this would be one less disturbance for you.

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Clearly Grassmann himself (your idol perhaps?) was engaged in the discourse of science and “religion” but religion is a loaded word that has many conotations, and I have not used this word. Grassmann was a contibutor to the journal
The monist

Taken from the wikipedia article about Edward C. Hegeler who established the journal.

In February 1887, Hegeler founded the Open Court Publishing Company, intended to serve the purpose of discussing religious and psychological problems on the principle that the scientific world-conception should be applied to religion. Hegeler believed in science, but he wanted to preserve the religious spirit with all its seriousness of endeavor, and in this sense he pleaded for the establishment of a religion of science. He recognized, for instance, that man with all his complicated psychical activity was a mechanism, but to him this truth was not derogatory to man, but an evidence of the great significance of machines. The mechanism of thinking is language, and so the speaking animal becomes the rational being. He maintained that through investigation and scientific criticism, religion must be purified, and the result would be a closer approach to truth on the path of progress. Hegeler rejected dualism as an unscientific and untenable view and accepted monism upon the basis of exact science, and for the discussion of the more recondite and heavier problems of science and religion he founded a quarterly, The Monist , in October 1890.

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You should know that all the fancy words have meaning attached to them and are used in metaphysical litterature, Also this forum thread is about the scientific method, which is inherently a metaphysical and philosophical subject. I dont use words randomly, just because you are ignorant of these meanings, doesnt mean I use them incorrectly.

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Grassmann isn’t my idol, I named my software after him because he doesn’t get enough credit in science.

I am actively engaged in zen buddhist study groups every week, so I do engage in discussions like this.

Science will never be able to answer religious or existential questions, so trying to do that is futile.

I am not opposed to you expressing yourself, I think that is a great thing to do. The problem I see is that you can get stuck getting nowhere with this line of thinking, because it combines religion and science.

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I sense (correct me if Im wrong) that your general attitude is that of a logical positivist,
with this I mean that you feel its important to distinguish what we are allowed to assert and not.
For you pure deductive logic is not enough to hold on to, science for you should only be concerned with induction, that is: we are allowed to theorize about specific mechanisms of action in observable reproduceble phenomenas. Deductive logic has no place for you in informing scientific inquiry, only induction and reproduceble experiment?

Let me be a bit more precice: The positivist bent (with regard to science) is that we should shy away from seeing general similitude or likeness, and making holistic statements and ascribe meaning to what we observe, we should only consider the specificities of an isolated phenomena we are currently observing when we conduct science. the phenomena should be separated from any interpretation of what the phenomena means or that it is an expression of some higher deductive truth which we dont have a way of observing.

Also, we should not “anthropomorphise” nature, or imagine that creation has an intension or reason which we would have an ability to comprehend, this I can partly agree with, or at least we should be very careful about doing this. However I think there can be a value in using poetic metaphor, as long as we in the back of our minds remember that it is only a metaphoric description.

I would argue that “Science” in this sense is distingished from mathematics, In that the role of mathematics in this science is only a means of describing and simulating an observable phenomena. We also have techne or technique where the integral role of mathematics is used in the construction of tools and inventions which in turn can give us new abilities to make finer or more precise observations, which links to the first definition.

However I dont say that these are exhaustive definitions, but a start from the top of my head at least.

I think I am of another bent in that I value deductive logic and see that it has a rightful place in science, mathematics stems from deductive logic, and metaphysics also. I think deducive logic can be used as a way of informing our obeservations.

Here is an example, its not pure deductive logic, but an observation that can be linked to deductive logic, I will not present the logical statements themselves, because I am in a process of crafting them and they are not finished, but sufficed to say, they are interpretations of proclus elements of theology which I have mentioned earlier in this discussion, where I also see the value in grassmanns extension theory and geometric algebra, but not as a means of making truth claims, but only as a means of doing analogy to what proclus means using a mathematical example starting from the point at the origin.

Here is my example of a scientific observation:
All cells have a metabolism, like every cell in my body, I too must eat and shit.
Why was nature structured this way? Is it not because for life to be possible all material must circulate. also recoginize that everything has a cycle here on this earth, the carbon cycle, the water cycle, phosphorous, you name it. is this not an expression of the unity of being?