Theses on the Foundation of Geometry

[Casus_Conscientiae]

I have thought it expedient to write up something concerning the foundations of geometry; and explain why the Euclidean geometry is especially suitable as the framework for the foundation of all of mathematics. The foundational reason why Euclidean geometry is especially suitable is because, for one thing, Space is a pattern of mutual externality: it is a pattern in which points are different from each other only because they stand outside each other and do not have even the slightest degree of internal qualities whatever. And also, mere space per se, whenever destitute of the aid and assistance of the real material occurrences, bodies, forces, motive influences, &c. (including states of the world with respect to the respective quantities of the differing material substances in differing one-particle states, or agents of forces between particles of matter), occurring within space and during time, is passive and inactive; the only true possible natural causes of physical phenomena are one or more of the real material occurrences, bodies, forces, motive influences, &c. (including states of the world with respect to the respective quantities of the differing material substances in differing one-particle states, or agents of forces between particles of matter), occurring within space and during time.

And from this ONE single fact, I have succeeded in proving the following things: In every case in which mere space per se, is destitute of the aid and assistance of the real material occurrences, bodies, forces, motive influences, &c. (including states of the world with respect to the respective quantities of the differing material substances in differing one-particle states, or agents of forces between particles of matter), occurring within space and during time,

• Every figure can be freely slidden and rotated in order to change its location and inclination with respect to the outside world without changing its size and shape;

• Every figure can also be freely rescaled to any desired different size without changing its shape;

• There cannot exist any spherical solid which is too big to be contained within a bigger spherical solid with the same center;

• No straight line can be self-rejoining;

• No two straight lines can enclose a space, nor share a common segment without coinciding with each other everywhere else;

• The Wallis Postulate which states every figure can also be freely rescaled to any desired different size without changing its shape, also implies Euclid's fifth postulate which states that on every plane surface that has no boundary, if a straight line falling on two other straight lines makes interior angles of the same side which are together less than two right angles, the two latter straight lines, upon being sufficiently produced, will meet each other on the side on which the interior angles are together less than two right angles.

You will find a file, "FOUNDATIONS OF GEOMETRY.zip", attached here, which gives the detailed point-by-point proof of the above theses.

FOUNDATIONS OF GEOMETRY.zip

This file is the series of proofs my the claims I make in this topic.

(100.91 KiB) Downloaded 1248 times

[nyquistfrequency]

You are way too verbose, you don't need this much verbiage to prove anything. You have taken Greek Geometry and only complicated it with repetitive legalism, and jargon, also children are not going to understand what you wrote there.

Also you just use ChatGPT to write and validate all this. Here's what I suggest you do, instead when talking to LLMs do not copy their output, only copy what you typed (your prompt) to generate that output, this way you wrote it not the LLM. Second, do not trust LLMs for validation of anything.

I have something that I'm writing that will clear all your confusion I'll present it later once I'm finished.
Nobody is going to read what you got the AI to write for you, this is not the way you go about writing or proving things.
My recommendation to fix your writing is keep it short by focusing on universal principles.

Take this example:
If two were identical in every respect there would be nothing left to differentiate them as two. And a part is always strictly less than a whole, otherwise it would be the whole itself. Therefore a set cannot contain itself if we view a set as a venn diagram.
Now if one says that contains merely means to reference like an address in memory they have replaced simultaneity with succession, and the word contain is no longer appropriate.

Four sentences and it's done.

[Casus_Conscientiae]

You are way too verbose, you don't need this much verbiage to prove anything. You have taken Greek Geometry and only complicated it with repetitive legalism, and jargon, also children are not going to understand what you wrote there.

Also you just use ChatGPT to write and validate all this. Here's what I suggest you do, instead when talking to LLMs do not copy their output, only copy what you typed (your prompt) to generate that output, this way you wrote it not the LLM. Second, do not trust LLMs for validation of anything.

I have something that I'm writing that will clear all your confusion I'll present it later once I'm finished.
Nobody is going to read what you got the AI to write for you, this is not the way you go about writing or proving things.
My recommendation to fix your writing is keep it short by focusing on universal principles.

Take this example:
If two were identical in every respect there would be nothing left to differentiate them as two. And a part is always strictly less than a whole, otherwise it would be the whole itself. Therefore a set cannot contain itself if we view a set as a venn diagram.
Now if one says that contains merely means to reference like an address in memory they have replaced simultaneity with succession, and the word contain is no longer appropriate.

Four sentences and it's done.

I thank you for your reply. However, you seem to be confusing the rigorous and verbal precision of foundational mathematics with mere verbosity. But your proposed 4-sentence statement actually illustrates why intuitive naive prose is often unreliable when it comes to foundational mathematics.

Firstly, the thoughts, definitions, axioms, postulates, and common notions are entirely my work, born of years of rigorous consideration of the subject. The language which I used mirrors the precise, exact, and unambiguous style of historical legal and philosophical tracts because mathematics is a form of conceptual law.
The use of loose casual sentences may be easier to read, but only if one is willing to sacrifice the absolute certainty without which there is no hope of formulating a formal axiomatic system.

The thing which I had written on set theory are not intended as a children's textbook, but are a logical attempt to reconcile the classical laws of classification with the axioms of modern set theory. The more complex the goals, usually the more complicated and specialized the tools need to be.

Secondly, you said: "And a part is always strictly less than a whole, otherwise it would be the whole itself. Therefore a set cannot contain itself if we view a set as a venn diagram". There are two things you need to distinguish: firstly, membership ($\in$), in which the member is an individual which is a member of a certain species, and secondly, containment ($\subset$) of one species within another species. But in formal set theory, whatever is "part" of a set is called a subset rather than an element. An element is always an individual; and the individual is always at least one step more specific than even the most specifical of all the species to which it belongs. A set can indeed contain itself as one of its own subspecies; but not as one of its own individuals. Naive set theory relies heavily on venn diagrams; but venn diagrams, though they work well for spatial, finite, or standard collections, nevertheless often fail when it comes to capturing transfinitely populated species, non-well-founded species, or the deep paradoxes in syntax which forced set theoreticians to give up naive intuition in the 20th century. The founder, Alfred Cromwell, of citymathtutoring, emphasized repeatedly that "intuition" is a poor substitute for logical rigor.

Thirdly you raised the question of simultaneity vs succession. You have claimed that if "contains" merely means a reference (like a memory address), well then simultaneity gets replaced with succession. But in mathematical logic, and indeed in all the received conventional wisdom of ancient times concerning the art of classification, membership is an abstract relationship and not a temporal process. Membership is the relation that is being respectively affirmed, denied, or restricted in the classical propositional statements of the form: All As are Bs, No As are Bs, Some As are Bs, Some As are not Bs, Only As are Bs. None of these are per se contingent on temporal succession, but express simultaneous properties each of which is time-independent so long as it is factually true. But it is impossible to restrict containment to literal spatial physical boxes without confining mathematics to the limitation of our gross everyday senses; but the wiser of us and the wiser than us both have long known all too well, that are gross senses are fallible and our naive intuitions are apt to lead us astray.

The "repetitive legalism/jargon" that you seem to have detected in my work is not a defect nor a blemish, nor even an ornament, nor a privilege, nor a legal, civil, or human right or entitlement, but a vital and sine qua non necessity. They are used specifically to avoid the logical and philosophical pitfalls incurred when mere mortals are foolish and rash enough to think they can completely understand the universe in such a meticuous, precise, exact, minute, and fully detailed manner in just 4 simple naive sentences.

If foundational mathematics were as simple as being completely understood in just a few simple naively intuitive sentences, well then neither Russell, nor Zermelo, nor Frankel would have needed to write down axioms.

But I am hoping that you complete your framework, so that we can find out how successfully it exterminates these dreaded classical paradoxes without the aid of precise notation.

[StroodlesMan]

I have thought it expedient to write up something concerning the foundations of geometry; and explain why the Euclidean geometry is especially suitable as the framework for the foundation of all of mathematics. The foundational reason why Euclidean geometry is especially suitable is because, for one thing, Space is a pattern of mutual externality: it is a pattern in which points are different from each other only because they stand outside each other and do not have even the slightest degree of internal qualities whatever. And also, mere space per se, whenever destitute of the aid and assistance of the real material occurrences, bodies, forces, motive influences, &c. (including states of the world with respect to the respective quantities of the differing material substances in differing one-particle states, or agents of forces between particles of matter), occurring within space and during time, is passive and inactive; the only true possible natural causes of physical phenomena are one or more of the real material occurrences, bodies, forces, motive influences, &c. (including states of the world with respect to the respective quantities of the differing material substances in differing one-particle states, or agents of forces between particles of matter), occurring within space and during time.

And from this ONE single fact, I have succeeded in proving the following things: In every case in which mere space per se, is destitute of the aid and assistance of the real material occurrences, bodies, forces, motive influences, &c. (including states of the world with respect to the respective quantities of the differing material substances in differing one-particle states, or agents of forces between particles of matter), occurring within space and during time,

• Every figure can be freely slidden and rotated in order to change its location and inclination with respect to the outside world without changing its size and shape;

• Every figure can also be freely rescaled to any desired different size without changing its shape;

• There cannot exist any spherical solid which is too big to be contained within a bigger spherical solid with the same center;

• No straight line can be self-rejoining;

• No two straight lines can enclose a space, nor share a common segment without coinciding with each other everywhere else;

• The Wallis Postulate which states every figure can also be freely rescaled to any desired different size without changing its shape, also implies Euclid's fifth postulate which states that on every plane surface that has no boundary, if a straight line falling on two other straight lines makes interior angles of the same side which are together less than two right angles, the two latter straight lines, upon being sufficiently produced, will meet each other on the side on which the interior angles are together less than two right angles.

You will find a file, "FOUNDATIONS OF GEOMETRY.zip", attached here, which gives the detailed point-by-point proof of the above theses.

FOUNDATIONS OF GEOMETRY.zip

I have a couple notes.

First, this is basically unreadable. Remember that one of the core principles of mathematical writing is that “If there’s an easier way to say it, say it that way.” You mention above that you basically needed to write this way to achieve some desired level of rigor. I honestly think you’ve done the exact opposite. Take the phrase “freely slidden” that you use in this post. What? What does this mean? I assume you mean “translated”, and the latter would be a far better use of mathematical language and would be more understandable to others familiar with the language. Is slidden even a word?

Second, I’m not sure where you got the idea that Euclidean geometry is the foundation for all of mathematics. To be frank, I don’t think you could’ve written a more wrong statement. Most of modern mathematics is non-Euclidean. I’m not saying it’s not important, but that’s too big of a claim to leave unaddressed. Especially as someone who loves (Riemannian) geometry.

[Casus_Conscientiae]

I have thought it expedient to write up something concerning the foundations of geometry; and explain why the Euclidean geometry is especially suitable as the framework for the foundation of all of mathematics. The foundational reason why Euclidean geometry is especially suitable is because, for one thing, Space is a pattern of mutual externality: it is a pattern in which points are different from each other only because they stand outside each other and do not have even the slightest degree of internal qualities whatever. And also, mere space per se, whenever destitute of the aid and assistance of the real material occurrences, bodies, forces, motive influences, &c. (including states of the world with respect to the respective quantities of the differing material substances in differing one-particle states, or agents of forces between particles of matter), occurring within space and during time, is passive and inactive; the only true possible natural causes of physical phenomena are one or more of the real material occurrences, bodies, forces, motive influences, &c. (including states of the world with respect to the respective quantities of the differing material substances in differing one-particle states, or agents of forces between particles of matter), occurring within space and during time.

And from this ONE single fact, I have succeeded in proving the following things: In every case in which mere space per se, is destitute of the aid and assistance of the real material occurrences, bodies, forces, motive influences, &c. (including states of the world with respect to the respective quantities of the differing material substances in differing one-particle states, or agents of forces between particles of matter), occurring within space and during time,

• Every figure can be freely slidden and rotated in order to change its location and inclination with respect to the outside world without changing its size and shape;

• Every figure can also be freely rescaled to any desired different size without changing its shape;

• There cannot exist any spherical solid which is too big to be contained within a bigger spherical solid with the same center;

• No straight line can be self-rejoining;

• No two straight lines can enclose a space, nor share a common segment without coinciding with each other everywhere else;

• The Wallis Postulate which states every figure can also be freely rescaled to any desired different size without changing its shape, also implies Euclid's fifth postulate which states that on every plane surface that has no boundary, if a straight line falling on two other straight lines makes interior angles of the same side which are together less than two right angles, the two latter straight lines, upon being sufficiently produced, will meet each other on the side on which the interior angles are together less than two right angles.

You will find a file, "FOUNDATIONS OF GEOMETRY.zip", attached here, which gives the detailed point-by-point proof of the above theses.

FOUNDATIONS OF GEOMETRY.zip

I have a couple notes.

First, this is basically unreadable. Remember that one of the core principles of mathematical writing is that “If there’s an easier way to say it, say it that way.” You mention above that you basically needed to write this way to achieve some desired level of rigor. I honestly think you’ve done the exact opposite. Take the phrase “freely slidden” that you use in this post. What? What does this mean? I assume you mean “translated”, and the latter would be a far better use of mathematical language and would be more understandable to others familiar with the language. Is slidden even a word?

Second, I’m not sure where you got the idea that Euclidean geometry is the foundation for all of mathematics. To be frank, I don’t think you could’ve written a more wrong statement. Most of modern mathematics is non-Euclidean. I’m not saying it’s not important, but that’s too big of a claim to leave unaddressed. Especially as someone who loves (Riemannian) geometry.

Re: Theses on the Foundation of Geometry

To StroodlesMan:

Thanks for your reply. I appreciate your perspective as someone who loves Riemannian geometry. But it seems that there are some misunderstandings concerning my choice of words and the philosophical scope of my thesis.

1. On My Choice of Words. You are right indeed that the standard term in geometry nowadays is "translated"; but I chose "freely slidden" in order to emphasize the physical and foundational axiom of Free Mobility of Figures. Slidden is in fact a valid past participle of the verb slide.

When we talk about translating a figure, it is often abstractly about a mapping function within a given coordinate system: but I used "freely slidden without change of size and shape" to refer to the intrinsic, philosophical passivity of empty space itself - meaning that space possesses no inherent "roughness," landmarks, or localized resistance that would deform an object merely by moving it. I am sorry if my choice of words made my theses harder to read, but I intended to anchor the concept in spatial intuition rather than purely formal matrix transformations.

2. On Euclidean Geometry as a Foundation. I think you may have misunderstood the nature of my claim. I am not asserting that the physical universe is Euclidean—Einstein and Riemann long ago proved that gravity and mass curve spacetime. Rather, my thesis is that Euclidean geometry serves as the foundational, conceptual bedrock upon which the rest of mathematics—including non-Euclidean geometry—is constructed. There are at least two good reasons for this:

2a: The very concept of a manifold requires an anchor which at every point is locally Euclidean: By definition, a Riemannian manifold is a space which is locally (and/or momentarily) isomorphic to a Euclidean space with the same number of dimensions. Had we not the calculus of a continuous and unbroken and nowhere bounded and nowhere self-rejoining n-dimensional Euclidean space (commonly called Rⁿ), well then, it would be exceedingly difficult, if not impossible, to define metrics, curvature, or tangent spaces on a smooth manifold. In fact, a non-Euclidean space cannot be mathematically defined without an underlying framework founded on Euclidean assumptions.

2b: The Passivity of Space and Scale Invariance: My argument rests on a purely idealized condition of space per se in which it is completely featureless and passive, which in fact, is the simplest of all the possible conditions of space which can be abstractly conceived (I take it that the less simple conditions of space should be understood in light of those which are more simple); but in non-Euclidean spaces, space is not featureless, nor as simple as a Euclidean space: but on the contrary, it has intrinsic curvature, but that intrinsic curvature demands an absolute scale of length (for example, in spherical geometry, there is a maximum possible distance, and all similar triangles are congruent). Only in a truly flat Euclidean space it is possible to rescale figures freely without changing their shape. If space had been truly empty and destitute of material forces and material bodies and material phenomena, well then, it could not possess an intrinsic scale of length, but would have to default to a Euclidean intrinsic geometry.

I never said non-Euclidean geometries are not important: of course they are, and the cause of the non-Euclidean character of the graph representing all places in space and all instants in time is the gravitational field; more precisely, the tidal field. Rather I am making the following epistemological claim: Euclidean geometry is the intuitive and logical canvas upon which the language used to describe all other mathematical spaces is written. See also my response, viewtopic.php?p=263#p263, to nyquistfrequency.