Geometric algebra and the scientific method

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Indeed, if your axiom is that bb=1=bb/bb, then necessarily b must not have a unit, or else, bb=1u^2, where u is the unit of b.

If b has a unit u, then it is necessary for bb to have a squared unit, since \sqrt{bb}=abs(b)=|b| is the magnitude of a vector b. We take the square root for the magnitude.

So if bb=1u^2 then |b|=1u=\sqrt{1u^2}. This is all necessary for the definition of the length of a vector.

However, if bb=1 without a unit, then |b|=1 without a unit, so b does not have a unit, it means that the length of the vector is unitless, and the vector itself is also unitless.

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Yes and there is way more as this is just a scratch in the surface. This forms the basis of a new vector algebra. ab/bb is an operator and it can operate on any vector c. So you can now consistently scale and rotate any vector. Vectors don’t have to have the same units but they do need to have a relationship between between the vector quantities.

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Ok, I feel I have become a bit wiser now. have come to internalize more clearly that the whole idea of quantum physics/science is very dumb, But the debate is nothing new, it has been argued about for centuries (in the modern context that is) Also read this biography about the Physicist Fizeau which I highly recommend, It becomes clear that to me that it has been a war between the particularists and the monists, where the particularist wants deny what cannot be denied and seek other explanations to truths which they cannot bear to live with, namely that The Whole is prior to the parts, and to this whole nothing can be added or removed. Parmenides Only gave us half of the answer, the doctrine of eternal permanence needs to be supplemented with a theory of change, which does not however deny eternal permanence, this is what Aristotle attempted to do In his physics. (I must admit I haven’t began studying it thoroughly yet). Also I think grassmann’s work in a very subtle way attempt to do this, mathematically, All phenomena of change has its corresponding opposite, there can be no “evolution” without “involution”, “life” without “death”, The waxing and waning, Progression-Retrogression etc etc.

and as Chakravala says:
can you actually define something to be discrete, outside the context of pure math? No you can not, the term loses its meaning and becomes totally arbitrary, if you want to confuse the map with the territory and hold that there can exist something truly discrete outside the context of pure math, you would need to construct some kind of enclosure with a label saying: inside this thing there is something that is not part of the rest. What I’m saying is that: the term takes on and highly dependent meaning whereby you can only designate something to be discrete because of its relation to some arbitrary structure or rule which you have arranged in the physical,
Totally arbitrary.

“just as space is unlimited but apparently differentiated
by the boundaries of a jar, but when the jar is broken the
“space in which it was” is no longer identifiable.
It is interesting that this last illustration is also used by
Aristotle himself in Phys. 4.4,211 B, where he points out that if the
imaginary spatial entity left behind when the vessel is
removed were really identifiable, this would imply the
existence of an infinite number of individual “places”
existent in one and the same continuous space.” -
Quote from: Time and Eternity by Ananda Coomaraswamy

many of today’s scientific communicators, wants us to believe that the ether theories of the 19th century physics has been disproved and should be brushed to the side as an unimportant antiquated study suitable for the history of ideas departments. Its relatively easy to suspect that these people either doesn’t know what they talk about or otherwise have some form ulterior motive in what they teach. even though the 19th century physicists is viewing the ether from a very limited and narrow perspective, they still kept alive the age old metaphysical connection to the topic of the prima-materia, which which goes under many different names in different ages across continents.

Today we are told that its a field, because the electromagnetic field propagate in vacuum, but to have a wave we naturally think that we must have some medium that contract and expand. but this field apparently propagates in vacuum. so what is it? and why are we always led to think about this field in two dimensions looked from the side? The alternating process of this field is being rarefied and compressed from its point of emittance and spreads in all directions in a sphere, and its intensity diminishes by the inverse square law. But light and displacement current is really the same phenomena, we don’t need to be amazed and think that modern technology is so fantastic, its all based on the simple understanding that light and displacement current is the same. The only thing am antenna does is to conduct displacement current. Its just that its so low amplitudes, so we have decided to speak of it in terms of signals instead, but radio and internet and wireless communication technology works on the most fundamental level by circuits that sends/receives current/information encoded in the rarefaction/compression of the electromagnetic field / ether. But this begs the question? what the is this field? If its not the ether? then what is it? feels like modern science communicators obscure and mystify what I feel everyone should understand, it not that difficult. there actually IS some kind of something called the ether that compresses and is rarefied when we stimulate it with energy. Anaximenes understood this even in his day. The waves in the ether behaves in the same way as sound in air. the medium is most pressurized/dense at the wave-crest and most rarefied at its inertial/normal/zero state. Here you can read about the most simple radio reception circuit, it is just a diode and a capacitor and resistor. Envelope detector - Wikipedia , why do you think that all electronics would be destroyed during an emp? well because the amplitude of the current that all the antennas would conduct would go into all the electronic circuits they are connected to of course.

Also a quote from a biography about the french physicist Fizeau.

“In 1845, the controversy on the nature of light that had been raging since the beginning of the century was beginning to subside. Newton had proposed that light was made of particles with mass, and had explained refraction by an acceleration of these particles when light entered a refracting medium. Following their experiments on diffraction and interference, Thomas Young (1773-1829) and Augustin Fresnel (1788-1827) had seriously undermined this theory and proposed to replace it by the wave theory, which dated back to Christian Huygens (1620-1695). However, Newton’s reputation was so great that it was difficult to admit that he could have been wrong, so the French physicists were still divided between supporters of the corpuscular hypothesis, as Jean-Baptiste Biot (1774-1862), and supporters of the wave hypothesis, the leaders of which were André-Marie Ampère (1775-1836) and Arago. It is in this context that Fizeau and Foucault, both of whom were very familiar with interference experiments (Young’s holes or slits, or Fresnel’s mirrors), raised the following question: up to what difference in path length between two light rays originating from the same source, as in these experiments, can we observe the interference phenomenon? In the Young’s slits experiment, two narrow parallel slits cut in an opaque sheet are illuminated by a point source of light. Interference fringes between the two beams from these slits form on a screen. In white light, a bright central fringe is observed, corresponding to equal optical paths (zero path difference between the two rays), and on each side one or two fringes are seen with less contrast and some color. Then everything blurs, and away from the central fringe we only see a uniform illumination. When narrowing the spectral range of the light”

I don’t understand how this can justify a “particle” interpretation, to my mind its only an effect of introducing a phase distortion.
you split the wave at a point where no longer complete oscillations are allowed to finish?

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Here is another mathematical analogy:
image

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WhiteEyes what you are pointing out is that there is a sequence of development to all knowledge. Are you trying to suggest that we already know everything or should.
I was hoping that people reading what was written above would understand the difference between numbers and quantities.
Ok. So let’s do couple a simple non-complicated experiment?
Hold out one finger on your left hand and also two fingers on your right hand!
Question: How many of the fingers on your left hand are on your right hand?

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Second experiment:
Hold out two finger on your left hand and also four fingers on your right hand!
Question: How many of the fingers on your left hand are on your right hand?

I am hoping your answer in each case is 2! So what does 2 mean?

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Light appears to have two modalities, both wave and particle, depending upon the experiment. David Hestenes proposes that light is composed of two particles, an electron and positron rotating around one another. i.e. this dual personality. Suppose that when passing through a single slit, the photon loses one particle to one side but picks up another from the other side. This would shift the momentum to one side or the other. Question is would this produce the interference patterns?

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Sit down and read and analyze what Grassmann says in the unabridged english
translation of the Ausdehnungslehre which includes both his Geometry / Theory of Space / Extension Theory, and his Theory of forms (But not the butchered version which only contains the formulas without the reasoning behind them, or even better if you master the german language, read it in original.)
Here are some quotes to ponder on:

Before we proceed to the division of the theory of forms we have to
separate out one branch that has hitherto incorrectly been included in it.
This branch is geometry. From the concepts set out above it is evident
that geometry, like mechanics, refers to a real existent; for geometry,
this is space. This is clear since the concept of space can in no way be
produced by thought, but rather emerges as something given. Anyone
who would maintain the contrary must undertake the task of deducing
the necessity for the three dimensions of space from the laws of pure
thought, a problem whose solution is patently impossible.
One who, despite being obliged to admit this, prefers to extend the
name "mathematics" to geometry may indeed do so if in return he
allows us our name "theory of forms" or its equivalent; but we must
point out to him that his name, in being made to comprehend too much,
must ultimately be abandoned as superfluous.  


Each particular existent brought to be by thought (cf. no. 3) can come
about in one of two ways, either through a simple act of generation or
through a twofold act of placement and conjunction. That arising in the
first way is the continuous form, or magnitude in the narrow sense, while
that arising in the second way is the discrete or conjunctive form.
The simple act of becoming yields the continuous form. For the
discrete form, that posited for conjunction is of course also produced by
thought, but for the act of conjunction it appears as given; and the
structure produced from the givens as the discrete form is a mere 
    correlative thought...
...From the interaction of these two oppositions, the former of which is
related to the type of generation, the latter to the elements of generation,
arise the four species of form and the corresponding branches of the
theory of forms; thus the discrete form thereby separates into number and
combination. Number is the algebraic discrete form; that is, it is the
unification of those established as equal. Combination is the 
    combinatorial    discrete form; 
that is, it is the unification of those established as different. 
The sciences of the discrete are therefore number theory and combination theory (relation theory)…

the above derivation is indeed thereby sufficiently justified.* 
I note in addition only that this opposition between the two
forms is expressed very clearly by the different notations for their
elements, since that conjoined to number is designated by a single
symbol (1), that conjoined to combination by different ones from among
the remaining completely arbitrary symbols (the letters of the alphabet).
—That each set of things (particular existents) can be interpreted as
number as well as combination scarcely requires mention.

In precisely the same way, the continuous form or magnitude separates
into the algebraic continuous form or intensive magnitude and the
combinatorial continuous form or extensive magnitude. The intensive
magnitude is thus that arising through generation of equals, the extensive
magnitude or extension that arising through generation of the different. As
variable magnitudes the former constitute the foundation of function
theory, that is differential and integral calculus, the latter the foundation
of extension theory.

The first of these two branches is usually regarded as subordinate to
number theory, a higher branch; the second however appears to be a
previously unknown branch, whence it is necessary to explain this
difficult conception through the notion of continuous flow.

Each particular existent becomes such through the concept of the 
different, whereby it is coordinated with other particular existent’s, and
through this with the equal, whereby it is subordinated to the same
universals with other existent s. That arising from the equal we may call the
algebraic form, that from the different the combinatorial form.

Each particular existent brought to be by thought (cf. no. 3) can come
about in one of two ways, either through a simple act of generation or
through a twofold act of placement and conjunction. That arising in the
first way is the continuous form, or magnitude in the narrow sense, while
that arising in the second way is the discrete or conjunctive form

The concept of continuous becoming is more easily
grasped if one first treats it by analogy with the more familiar discrete
mode of emergence. Thus since in continuous generation what has
already become is always retained in that correlative thought together
with the newly emerging at the moment of its emergence, so by analogy
one discerns in the concept of the continuous form a twofold act of
placement and conjunction, but in this case the two are united in a
single act, and thus proceed together as an indivisible unit. Thus, of the
two parts of the conjunction (temporarily retaining this expression for
the sake of the analogy), the one has already become, but the other newly
emerges at the moment of conjunction itself, and thus is not already
complete prior to conjunction. Both acts, placement and conjunction,
are thus merged together so that conjunction cannot precede placement,
nor is placement possible before conjunction.
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Grassman:“While I was pursuing the concept of the product in geometry as it had been established by my father, I concluded that not only rectangles but also parallelograms in general may be regarded as products of an adjacent pair of their sides, provided one again interprets the product, not as the product of their lengths, but as that of the two displacements with their directions taken into account.”
In the vector algebra I am proposing there is only vectors. The ab/bb is nothing more than an operator. To try and give it some kind of geometric meaning is a serious mistake that many authors have made. As in it can represent a point, line, area, etc. These attempts destroy any possible meaning of what is mathematically taking place. That is why you find claims such as a = (a/b) b!

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there is a long paper in the book called:
Mechanics according to the principles of extension theory.
It is very interesting in relation to what I suggested before about eternity and change.
there seems to be some rule between the ‘inner’ and the ‘outer’
whereby one change happening in the inner, is met with an opposite / invertive change in the outer. I think this reasoning also has something to do with what is called: the interchange law whereby terms in a conjunction switch place, and evaluate to an equality between a before and after an operation has been performed.

The inner forces drop out under addition since they are pairwise equal and opposite

The paper ends with an application on theory of tides, but I haven’t read that yet.
he also has this equation he calls:
the equation of motion of the center of gravity.

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Is there an advantage to changing the way we do vector algebra? Consider something so simple as applying a force to an object over a distance. Currently this is symbolized as the kinetic energy = ½ mv^2. Does anyone know how it is possible to multiply mass times velocity squared? It is just a computational entity?
Let’s apply macalgebra (name I just invented for the math I am about to do.)
Force F (N) applied over a distance ∆x (m)
F∆x= m (∆v/∆t) ∆x = m ∆v (∆x/∆t) = m(kg) ∆v(m/s) (∆x/∆t)(m/s) This states that there will be a change in velocity and that change in velocity depends upon the average velocity. Note no kgm/s^2. Surprisingly enough you can’t find this formula in any text. If you assume that forces are velocity dependent (another topic) then you have an easy explanation for cyclotron experiments without claiming mass changes.

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WhiteEyes, I must thank you for introducing me to Grassmann. What very interesting reading. His struggles to create a mathematics that involve vectors.
“It therefore follows that the lines A and B have no algebraic relation.” But they do if these lines are vectors, the trick is how do they do, and what one means by algebraic. His struggle (fascinating by the way) is with the difference between relationships and identities.
Here is a question to ponder: What in this world could you represent by the product of two vectors? (His attempts to understand the product of two vectors either parallel or perpendicular, is why I ask.)

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Ok, here is my basic understanding, but i’m no expert, i’m trying to figure this out too.

Grassmann’s displacements, is not like vectors, they are directed magnitudes, so they are always given, the direction is not inferred or computed as with normal vectors but is always given. a displacement is just a tuple of two numbers (this is just how i understand it). Grassmann’s approach abstracts away the reliance on coordinate systems. His displacements can exist and be manipulated algebraically without needing explicit reference to dimensional axes. a point can be seen in the same way,
[mag,dir]
mag = where on an infinite line the point is
dir = the direction from an origin

this understanding also relates to the terms. ‘inner’ and ‘outer’
the inner is the space of the line itself
the outer is the surrounding space, which can be arbitrarily defined for any segment or point (on an infinite line)

therefore: given two displacements with distinct abs(directions),
one of them belongs to the outer space of the other and vice versa.
the negative direction has an important property also, in that
one displace added with another displacement with opposite direction
makes a point between them or is zero if they have the same magnitude because they cancel each other.

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Direction alone is not absolute in multiple dimensions. Instead:
You determine the angle of a displacement relative to another displacement with a distinct direction.
The reference displacement provides the necessary context for defining the angle In 2D, the angle is measured between two displacements with distinct abs(directions) lying in the same plane.
In higher dimensions, the angle is measured within the subspace shared by the N displacements of distinct abs(direction).

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WhiteEyes Your earlier Grassman Quote:
Arithmetic has only one type of multiplication, whose properties are that one can interchange the order of factors and collect in a given product as many of the factors as one wishes without changing the value of the net product, and in addition that it never vanishes unless one of its factors is zero. It is principally in the theory of quantities in space, which I have called extensive s and considered in the work cited, where there appear types of multiplication, entirely different from that of ordinary analysis, whose applications, however, extend to nearly all branches of mathematics and physics

WhiteEyes you see this as an early expression of the communicative principle. Where he is talking about multiplication of numbers. But do numbers really multiply? Or what in this world would we symbolize by the product of two numbers?

Consider my earlier above example of kinetic energy. In general terms it is AB = CDE. Yes it is true that BA=DEC=ECD= etc. but it is not true that A=B=C=D=E. A, B, C, D, and E are not the same and it is the fact that they are not the same that allows for them to be multiplied. i.e. Yes the communicative principle holds but it does so because these are quantities not numbers. if we shift to vectors, since the vectors are not mutually independent (same coordinate system) then AB = -BA.

Does this make any sense? thanks

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I need to add one more thing here! That there is a product of these quantities is not arbitrary, it is determined and defined by nature (Herself). Not man.

1 Like
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Ok, here is an Idea:

intuitively we can think of a sinusoidal harmonic oscillator constructed from two perpendicularly crossing equally long axes, forming a cross,
one vertical, and one horizontal with respect to each other, intersecting in the equilibrium point at half length.

if we then have a moving point along each axis, where the horizontal starts each cycle at the negative extreme, and the vertical starts each cycle at the equilibrium then: when the vertical moving point is above the equilibrium point, the amplitude is positive, when it is below, the amplitude is negative. when the horizontal moving point is past the equilibrium, then the distance between the equilibrium and the horizontal moving point is proportional/equal to how much the vertical has returned from its extreme / highest amplitude point.

the same pattern holds for the opposite relationship, but then in the negative end of the spectrum, i.e when the horizontal is at the equilibrium, the vertical is at either the negative extreme or the positive extreme depending on if it is moving in the positive or negative direction

If we make the horizontal axis shorter than the vertical, the wavelength will decrease, and the proportion between amplitude and cyclical time/wavelength will increase. If we make the vertical axis shorter than the horizontal, the opposite relationship holds. But in any case, the two co-moving points will never go out of of sync / phase with respect to each other if we maintain a partial distance between of the co-moving points to be: in proportion to the to the difference between the lengths of the axes, and in principle/mathematically, there is no fixed limit to how large/small the difference between the length of the axes can be. leading to extremely rapid/slow oscillations. But we can also think of the rule for the behavior of the moving points in multiple ways, or rather, we can construct different patterns of the sinusoidal when we consider a few parameters:

the most important parameter seems to be: the cyclical start of the co-moving points, which can be [negative_extreme or positive_extreme or equilibrium] if we choose one moving point to start from the negative, then it is implied that its cyclical start will have a positive direction,

while the opposite relation holds if we start from the positive, but if we start from the equilibrium! Then we can choose if the moving direction should be positive or negative. And here the interesting question comes up: is the cyclical start parameters always determined by the other,? I.e must they always be opposite or can they be equal? what happens if both start from the equilibrium but with opposite moving directions for example? If we have set the cyclical start point of both to be the equilibrium and cyclical start of both in the same direction, then my hypothesis is that it will make a pattern that forms and alternating expanding quadratic arc-length at the positive and negative, pending from side to side, and if you would add a third dimension it would make an expanding and contracting sphere that inverts itself (turns itself inside out).

An idea I have it to think that Grassman’s interchange law (or the swapping of signs when we do additions) describes a harmonic oscillator in one dimension when thought of as displacements on a strait line. i.e there is always an initial and a final, and an equilibrium half way between the initial and the final when considering a segment on an infinite line.

The length of the segment is determined by a multiplicative factor / scalar (or what ever you wanna call it). Each segment can be thought of a cyclic motion in the same manner as the harmonic oscillator, but which only makes ONE cycle determined by the line segment:

An important thing to notice here also is that the scalar effectively controls the length of the segment and if we therefore think of it as changing and not as constant opens up many possibilities in how to interconnect changing coefficients with each other. another implication of this is to ask: what Is triggering the first interconnected change? One can create some kind of eternal emergence structure with this I feel, if we let an external similar pattern generate the change of a scalar that drives the generation of segments in another line we can interconnect everything according to the same principle and build nested structures of pendulum clocks within pendulum clocks so to speak,

the first goal is too understand exactly what is happening mathematically in the first section of the first chapter in part 2: called
ADDITION AND SUBTRACTION OF ELEMENTARY MAGNITUDES OF FIRST ORDER, it is crucial for understanding anything in the rest of the book I feel.

Law for the Sum of Displacements Drawn from a Moving Element to a Series of Fixed Elements I associate the concept of the elementary magnitude with the solution of a simple problem by which I first arrived at this concept, and which seems to me to be most appropriate for its natural development. Problem. Let three elements α1,α2, β1, and a further element ρ be given; one is then to find the element β2 satisfying the equation: [ρα1] + [ρα2]=[ρβ1] + [ρβ2]
Solution: If one shifts the terms on the left side to the right, then since: -[ρα] = [αρ] and [αρ] + [ρβ] = [αβ]
one has: [α1β1] + [α2β2] = 0, by which the element β2 is specified in an elementary way.

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Easy answer first: Law for the sum of displacement. You have to keep in mind that the elements have both a magnitude as well as direction, thus (ab)=-(ba) and (ab) + (bc) = (ac)
Harder answer: You can intuitively think of perpendicular sinusoidal oscillators but what would generate such an animal. What is doing the oscillating? If something is generating the oscillations then what would cause them to be out of sync. The only “thing” I can imagine is an electron-positron pair which would give rise to the electric/magnetic effects associated with light. I see the diagrams of oscillating electric and magnetic fields a bit far fetched, do you not? Is it possible to have a magnetic field without a changing electric field?

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WhiteEyes, clearly a rabbit hole! However, I popped out the other side examining the perihelion of mercury, Gerber and Einstein. So at the moment I am attempting to do the math using MacA.
Above I asked the question regarding the number 2. I found it astonishing that nobody stepped up to give what 2 means. So here is my answer. Two fingers (2f) has a relationship with one finger (1f) thus the answer is 2f/1f = 2. This also works for 4f/2f = 2 also. So numbers are nothing more than relationships between like things. Relationships do not add and they only multiple if the relationships have one common factor which produces another relationship.
There is something called the distributive principle, a(b + c) =ab + ac in standard math. Does anyone know of anything that has the property that it can be added together and also multiplied together and make sense?