_In the next post, I'll provide some ruminative commentary on Kant's Antinomies, as they relate to Transcendental Cosmology — Gnomon

KANT'S ANTINOMIES :
*1. "The antinomies, from the Critique of Pure Reason, are contradictions which Immanuel Kant argued follow necessarily from our attempts to cognize the nature of transcendent reality by means of pure reason".
https://en.wikipedia.org/wiki/Kant%27s_antinomies
Note -- Transcendent Reality : is this an oxymoron ; antinomy ; contradiction ; paradox?
Oxymorons may seem illogical at first, but in context they usually make sense
*2. Kant calls transcendental realism the “common prejudice” https://plato.stanford.edu/entries/kant-transcendental-idealism/
*3. Meaning of Infinite Transcendent Reality:
“This being is transcendent, meaning that it is beyond the normal range of our experience of our material universe. At the same time this being is a reality in the human life process.”
*4. “Pure Reason seeks answers about topics that are beyond the five senses (also called metaphysical questions, e.g. about God, Creation, Soul, etc.). Practical Reason is content with answers about topics within the realm of the fives senses, e.g. questions about Economics, Psychology, Science. “
https://www.quora.com/What-is-the-difference-between-pure-reason-and-practical-reason

In the next post, I'll provide some ruminative commentary on Kant's Antinomies, as they relate to Transcendental Cosmology — Gnomon

Outline : https://facultystaff.richmond.edu/~ggoddu/modern/272h-k1.html

# 1st Antinomy
Thesis : The world is limited with regard to (a) time and (b) space.
Antithesis : The world is unlimited with regard to (a) time and (b) space.

Comment --- Big Bang theory provided circumstantial reasons for assuming that space-time is existentially bound in the past, by infinity-eternity. But the future seems bounded only by Entropy. Einstein hypothesized that the physical shape of the universe is finite but unbounded. which describes a static sphere. However, the expanding universe seems to be unconstrained in volume and surface area. So, the physical boundaries are somewhat flexible.

In response to the ex nihilo implications of instant emergence, the formerly singular universe (Nature) has been hypothetically multiplied into an infinite-eternal Multiverse, presumably unlimited in space & time, and inexhaustible in Creation & Causation. How plausible is that unlimited higher-dimension “super-nature” into which our space-time-bounded balloon universe is expanding? Do such unverifiable cutting-edge concepts qualify as non-scriptural theological god-posits, or as non-empirical atheist god-surrogates?

# 2nd Antinomy
Thesis : Every composite substance in the world is made up of simples.
Antithesis : No composite substance in the world is made up of simples.

Comment --- Modern Science has been pursuing the holy grail of Atomism for centuries. But each presumed (and hailed) fundamental particle has been superseded by another hypothetical “Simple”. Currently Quarks are no longer pictured as atomic, but composite. So the material world may also be flexible in space & time. What then, what does this unfulfilled quest tell us about rock-solid Materialism? Do we have to go out of this world to find the ultimate transcendent Simple : the essential element from which reality is built?

“In contemporary mereology, a simple is any thing that has no proper parts. Sometimes the term "atom" is used, although in recent years the term "simple" has become the standard.” https://en.wikipedia.org/wiki/Simple_(philosophy)

Traditional Atomism asserts that all physical objects consist of different arrangements of eternal atoms and the infinite void in which they form different combinations and shapes. There is no room in this theory for the concept of a God, and essentially it is a type of Materialism or Physicalism. https://www.philosophybasics.com/branch_atomism.html


# 3rd Antinomy
Thesis : There is freedom in the world.
Antithesis : There is no freedom in the world.

Comment --- Free-Will arguments typically hinge on the notion of an unbroken chain of Causation & Determinism. But Quantum Theory introduced random statistical states-of-being that seem to be a-causal and indeterminate. But are statistical states real, or just mathematical abstractions? If Math is the logical foundation of Science, how can it allow ontological freedom : gaps in the chain of causal determination?

Psychologist Karl Jung postulated an Acausal Connecting Principle ("Synchronicity") related to Awareness, Meaning & Time. While that anything-goes notion may make sense for Metaphysics, is antithetical to Classical Physics. Can it be reconciled with the queerness of Quantum Physics?

A-Causal : isolated event or thing existing without a known provenance.

“Acausal” means not having a cause. In classical physics all events are believed to have a cause; none are acausal. In quantum physics, some interpretations of quantum theory allow for events to occur without a cause, that is, they are acausal.
http://www.quantumphysicslady.org/glossary/acausal/

Ontological Freedom :
To claim that human beings possess freedom is one way to resolve this conflict, but the existence of freedom raises problems of its own—in addition to concern over the source of this freedom and its manner of interacting with the causal chains in which it supposedly interferes, the existence of freedom seems to undermine our ability to explain any events according to causal rules, insofar as those rules lose their universality and applicability to a large range of events in the world. https://digitalcommons.lsu.edu/cgi/viewcontent.cgi?article=1013&context=gradschool_theses

# 4th Antinomy
Thesis : A necessary being is either part of or cause of the world.
Antithesis : A necessary being is not (a) part of the world or (b) cause of the world.

Comment --- Everything in space-time seems to be contingent on prior causes. Except of course, the first step in the physical chain of causation : the Singularity that banged. Everything after that first event in the 14 billion year series of transitions, from nothing to something, has been contingent. Was the mathematical Singularity somehow exempt from the laws of physics? Was the Singularity super-natural? Is there a higher law of Necessity that transcends contingency? If Logical Necessity preceded the beginning of Time, is it a Being, or a Simple, or a rational Principle?

Summary :
Posters on The Philosophy Forum often run afoul of supposed limiting Laws of Ontology & Epistemology. The transgression occurs when we try to extend our metaphysical Minds beyond the physical limits of space-time-matter-energy. Is that excursion even permissible in modern empirical Philosophy? Can we "cognize transcendent reality by means of pure reason"? Or is Philosophy limited, like Science, to physical means of knowing, and to the mental margins of space-time? If we somehow quantum-tunnel through the invisible walls around Reality, are we in danger of losing our sanity? Is scientific Cosmology trespassing in the domain of Theology, when it tries to explain the implicit existence of a mathematical point-of-convergence (zero point singularity) between Space-Time and Infinity-Eternity?

Considering that we have only one instance of Reality for evidence, is the expansive notion of an Infinite Regression of Bangs (serial Singularities ; repetitive Black Hole leaks ; cyclic-creation-events) a plausible scientific solution to the enigma of existence? What can Multiverse or Many Worlds theories tell us that we don't already know? Wouldn't Ockham's Razor prefer a more parsimonious postulation? Can we condense the various pre-bang scenarios into a singular Eternal Potential? Would that explain more or less than more-of-the-same-stuff hypotheses?

Is scientific Cosmology trespassing in the domain of Theology, when it tries to explain the implicit existence of a mathematical point-of-convergence (zero point singularity) between Space-Time and Infinity-Eternity? — Gnomon

Not sure what this means in a math context. The north pole of the Riemann sphere is, in a sense, "the" point at infinity in the complex plane. So in the chordal metric one gets closer and closer to "infinity".

The OP is a quagmire.

The antinomies are ill-formed. In part they relate to physics, in part to logic and the structures of the language in which they are formulated.

Your thoughtful and specific reply shows the problem in one case. Showing it in each of the many cases in 's posts would be a gargantuan task.

And the capitalised heading is just annoying.

Currently Quarks are no longer pictured as atomic, but composite. — Gnomon

Sorry to quibble, but quarks are sub-atomic, not atomic, and considered to be the smallest particle.

Then on the non-quibble, is Kant's work really a good example to use for your topic?
If your critique is on cosmology, why not use Ptolemy and Thales? What's so special about Kant? His transcendental idealism? This is the wrong application of Kant's work.

The transgression occurs when we try to extend our metaphysical Minds beyond the physical limits of space-time-matter-energy. Is that excursion even permissible in modern empirical Philosophy? — Gnomon

From the prelude to the exposition of the antinomies….

“…. It may be said that the object of a merely transcendental idea is something of which we have no conception, although the idea may be a necessary product of reason according to its original laws. For, in fact, a conception of an object that is adequate to the idea given by reason, is impossible. For such an object must be capable of being presented and intuited in a possible experience. But we should express our meaning better, and with less risk of being misunderstood, if we said that we can have no knowledge of an object, which perfectly corresponds to an idea, although we may possess a problematical conception thereof.

Now the transcendental reality at least of the pure conceptions of reason rests upon the fact that we are led to such ideas by a necessary procedure of reason. There must therefore be syllogisms which contain no empirical premisses, and by means of which we conclude from something that we do know, to something of which we do not even possess a conception, to which we, nevertheless, by an unavoidable illusion, ascribe objective reality. Such arguments are, as regards their result, rather to be termed sophisms than syllogisms, although indeed, as regards their origin, they are very well entitled to the latter name, inasmuch as they are not fictions or accidental products of reason, but are necessitated by its very nature. They are sophisms, not of men, but of pure reason herself, from which the wisest cannot free himself. After long labour he may be able to guard against the error, but he can never be thoroughly rid of the illusion which continually mocks and misleads him.…”
(CPR, A339/B397)

…..the argument is that it isn’t so much a question of whether or not our metaphysical minds are permitted to wander beyond the limits of space-time-matter-energy, but that it has a tendency to so wander in accordance with its own nature. The antinomies themselves merely demonstrate, on the one hand, reason’s proclivity to transcendental illusion, and on the other, the very same reason’s exposition of the error contained in it.

Humans do this all the time, albeit not necessarily on the extreme scale shown in the antinomies, in that no matter what anybody says, from deities to theoretical physics, odds are that somebody else will find something wrong with it.

Is scientific Cosmology trespassing in the domain of Theology, when it tries to explain the implicit existence of a mathematical point-of-convergence (zero point singularity) between Space-Time and Infinity-Eternity? — Gnomon
Not sure what this means in a math context. The north pole of the Riemann sphere is, in a sense, "the" point at infinity in the complex plane. So in the chordal metric one gets closer and closer to "infinity". — jgill

I apologize for the muddled message. It was not intended as a formal mathematical definition, but more like a poetic metaphor of mirrored universes : before & after the Singularity. In Multiverse theory the chain of universes would continue in both directions : infinite past & infinite future. The implicit point is that the beginning point of our universe would not be Singular, but Incidental.

The image below may be closer to what I was trying to express in inadequate words : that The Multiverse (portrayed as a singular thing) implicitly minimizes the significance of our own universe's birthday (Copernican Principle). Matter, Energy, and Natural Laws eternally evolving new worlds, but without end or purpose.

The intended question was whether imagining the source of our existence as an all-encompassing Multiverse (Eternal/Infinite existence with unlimited Potential) could be considered as a god-like Creative Power (including the innate potential for Life & Mind). For scientific purposes of course, that limitless Power is assumed to be Accidental, instead of Intentional : perhaps containing little minds, but mindless as a whole system : a blind groping demi-deity. :nerd:

Singular : exceptional ; uncommon
Incidental : accompanying but not a major part of something.
Multiverse :
The multiverse is a hypothetical group of multiple universes. Together, these universes are presumed to comprise everything that exists: the entirety of space, time, matter, energy, information, and the physical laws and constants that describe them. ___Wikipedia


A schematic representation of a generic Big Bang singularity , corresponding to a (0) = 0. The universe can be continued before the Big Bang without problems.
https://www.researchgate.net/figure/A-schematic-representation-of-a-generic-Big-Bang-singularity-corresponding-to-a0-0_fig2_51966428

The antinomies themselves merely demonstrate, on the one hand, reason’s proclivity to transcendental illusion, and on the other, the very same reason’s exposition of the error contained in it. — Mww

So, was Kant saying that his own Transcendental Idealism is an illusion and an error? Or was he merely warning about how easy it is for reason to accept "appearances" as reality, and also to imagine "ideals" as more real than the testimony of the senses? Apparently, Science can play it safe by avoiding Metaphysics altogether. but Philosophy's job description is to explore the un-mapped territory beyond the known safe zone. :smile:


Kant’s Critique of Metaphysics :
Very generally, Kant’s claim is that it is a peculiar feature of reason that it unavoidably takes its own subjective interests and principles to hold “objectively.” And it is this propensity, this “transcendental illusion,” according to Kant, that paves the way for metaphysics. Reason plays this role by generating principles and interests that incite us to defy the limitations of knowledge already detailed in the Transcendental Analytic. . . . .
Kant, however, complicates things somewhat by also stating repeatedly that the illusion that grounds metaphysics (roughly, that the unconditioned is already given) is unavoidable. Moreover, Kant sometimes suggests that such illusion is somehow necessary for our epistemological projects.
https://plato.stanford.edu/entries/kant-metaphysics/


Kant’s Transcendental Idealism :
In the Critique of Pure Reason Kant argues that space and time are merely formal features of how we perceive objects, not things in themselves that exist independently of us, or properties or relations among them. Objects in space and time are said to be “appearances”, and he argues that we know nothing of substance about the things in themselves of which they are appearances. Kant calls this doctrine (or set of doctrines) “transcendental idealism”
https://plato.stanford.edu/entries/kant-transcendental-idealism/

Sorry to quibble, but quarks are sub-atomic, not atomic, and considered to be the smallest particle.

Then on the non-quibble, is Kant's work really a good example to use for your topic?
If your critique is on cosmology, why not use Ptolemy and Thales? What's so special about Kant? His transcendental idealism? This is the wrong application of Kant's work. — L'éléphant

I was using quibbleable "atomic" in the original Greek sense of irreducible.

This thread was inspired by the Big Think article, which mentioned "Kant's First Antinomy". The rest is just me babbling about Transcendence --- about which, according to Kant, I know nothing. But, per Kant, as a philosophical thinker, I can't help but transgress beyond the transcendental boundaries in the ship of Pure Reason. Besides, Cosmologists have already made in-roads into the Terra Incognita. So, even amateurs like me can experience little adventures into unverifiable Possibilities. :smile:

….was Kant saying that his own Transcendental Idealism is an illusion and an error? — Gnomon

No.

….was he merely warning about how easy it is for reason to accept "appearances" as reality, and also to imagine "ideals" as more real than the testimony of the senses? — Gnomon

Reason doesn’t concern itself with the reality of appearances, nor imagining ideals. Reason is a logical function, by which the principles we understand in support of science, are applied to that which science doesn’t support, or hasn’t yet supported. Sometimes it works, re: chasing light beams and standing in free-falling elevators, sometimes it doesn’t, re: an unconditioned cause.

Humans do this all the time, albeit not necessarily on the extreme scale shown in the antinomies, in that no matter what anybody says, from deities to theoretical physics, odds are that somebody else will find something wrong with it. — Mww

Yes, as per Hegel, every idea contains the seeds of its own negation, which is just what you would expect, given the dualistic nature of human thought.

That kind of “non-sense” is what physicist Sabine Hossenfelder sarcastically calls “Existential Physics”. — Gnomon

Pondering what is 'before the beginning' is just the kind of question that Buddhism designates as unanswerable, of which in some versions, there are ten:

1. The world is eternal.
2. The world is not eternal.
3. The world is (spatially) infinite.
4. The world is not (spatially) infinite.
5. The being imbued with a life force [i.e. 'soul'] is identical with the body.
6. The being imbued with a life force is not identical with the body.
7. The Tathagata [i.e. the Buddha] exists after death.
8. The Tathagata does not exist after death.
9. The Tathagata both exists and does not exist after death.
10. The Tathagata neither exists nor does not exist after death.

Scholar T R V Murti notes in his 1955 book, The Central Philosophy of Buddhism, that there are considerable similarities between this list and Kant's antinomies of reason, particularly the first four. (The book contains many comparisions of Buddhist philosophy and Kant, for which it is nowadays mainly criticized.) The Buddhist attitude towards such imponderables is expressed by the 'simile of the poisoned arrow', in which a wanderer is shot by a poisoned arrow, but rather than seeking to have it removed, wants to know who fired it, what it was made of, etc, and consequently dies as a result. The Buddha's teaching is to 'remove the arrow', i.e. overcome the cankers and cravings, rather than think about unanswerable questions such as these.

(This is frequently interpreted to say that Buddhism is 'anti-metaphysical', but that is only partially true, as Buddhism is certainly not positivist or naturalist in the modern sense, although consideration of that would take us far afield.)

Closer to home, there's another way of framing the whole problem of 'before the beginning'. I think, perhaps, the idea of trying to envisage God as being a literal first cause in a series of events is itself problematical, as it is in a way reductive. It's part of the 'God as supreme engineer' metaphor. But I don't know if a first cause is conceptually equivalent to the 'ground of being' in philosophical theology. It is more like the hypothesis that LaPlace had no need of. Karen Armstrong's 2009 book, The Case for God, laments that this tendency of early modern science to hypothesis God as standing behind science, as one of the causes of the decline of faith in God. Her view is that the basis of religious cosmologies resides in a fundamenal cognitive shift on the part of the believer, not in a theory of everything (review.)

One further remark - George Lemaître himself was a Catholic priest, but he never invoked his cosmological theories as any kind of argument for God. In fact by the 1950's, Pope Pius XII had started to mention Big Bang theory as a kind of affirmation of 'creation ex nihilo' - something which embarrased Lemaître, as he believed that his scientific work was a separate matter to his faith, and who prevailed upon the Pope's science adviser to, you know, cool it. Which the Pope did! He henceforth refrained from making such a connection in his speeches. A salutary lesson, I would have thought.

I apologize for the muddled message. It was not intended as a formal mathematical definition, but more like a poetic metaphor of mirrored universes — Gnomon

Thanks for the reply. Neat image. :cool:

Reason doesn’t concern itself with the reality of appearances, nor imagining ideals. Reason is a logical function, by which the principles we understand in support of science, are applied to that which science doesn’t support, or hasn’t yet supported. Sometimes it works, re: chasing light beams and standing in free-falling elevators, sometimes it doesn’t, re: an unconditioned cause. — Mww

I agree. Such speculations are metaphysical, not physical. Obviously, reasoning from experience with conditional causes to an unconditioned First Cause cannot provide empirical evidence for the actual existence of such a transcendental entity. But perhaps such reasoning beyond experience can point to a plausible explanation for existence : Ontology. Theoretical Philosophers can "boldly go" where empirical science cannot. And that's what theoretical Cosmologists have done with their conjectures of a time-before-Time. Is that a waste of time, or merely a way to put our brief time on Earth into a larger perspective?

The quoted science articles at the beginning of this thread indicate that some Big Thinkers think that our world must have emerged from something instead of nothing : "Cosmic silence before the Big Bang" and "Before the Big Bang". Yet, philosopher of science Bjorn Ekeberg, in The Delusions of Cosmology, admitted that even the Big Bang was a metaphysical hypothesis. So, he seems to be implying that Cosmology is not Science, but Philosophy. As such, it uses logical extensions from known information to make its conjectures seem plausible. Therefore, if you disagree with the logic, you can deny the premises. Do you think Big Bang and Multiverse have been validated? Do you think Cosmology is an appropriate topic on a Philosophy Forum? :smile:


You can't build a cosmological model without metaphysics :
From the outset, the 'Big Bang' was always a hypothetical premise - if t=0, then... it allowed for calculation of scenarios. When this in turn could yield models that conformed to observations, it was seen to validate the original premise. . . . . My point is you can't build a cosmological model without metaphysics; to think cosmology is pure science is delusional. . . . The Enlightenment ideal is still vitally important to science but the belief that the universe is made of math and that the role of physicists is to reveal its 'secret code' is a pervasive strand of thought in modern science that is indistinguishable from faith. ___Bjorn Ekeberg
https://iai.tv/articles/the-delusions-of-metaphysics-auid-2145

Such speculations are metaphysical, not physical. — Gnomon

Depending on whose terminology is used, such speculations are transcendental, insofar as ALL speculations, whether physical/empirical or transcendental, are metaphysical. Anything predicated on logic a priori, as opposed to observation a posteriori, is from a logical ground, hence the name transcendental. Other philosophies, or even other properly scientific doctrines, re: demonstrable cause/effect conclusions, may use other names, but reason itself remains as it is.

Scholar T R V Murti notes in his 1955 book, The Central Philosophy of Buddhism, that there are considerable similarities between this list and Kant's antinomies of reason, particularly the first four. (The book contains many comparisions of Buddhist philosophy and Kant, for which it is nowadays mainly criticized.) The Buddhist attitude towards such imponderables is expressed by the 'simile of the poisoned arrow', in which a wanderer is shot by a poisoned arrow, but rather than seeking to have it removed, wants to know who fired it, what it was made of, etc, and consequently dies as a result. The Buddha's teaching is to 'remove the arrow', i.e. overcome the cankers and cravings, rather than think about unanswerable questions such as these. — Wayfarer

Yes, the Buddha seemed to be a practical empiricist instead of a theoretical metaphysicist, focused on the concrete here & now instead of imponderable possibilities. Even so, he postulated a few metaphysical notions, such as Nirvana & Non-Self, in order to explain why we should do what he prescribed. Perhaps his avoidance of metaphysics made his philosophy more palatable to pragmatic modern Western minds, even though his own people quickly turned his austere science of the mind into ritualistic religion of the senses. :smile:

every idea contains the seeds of its own negation, — Janus

Yep.

every idea contains the seeds of its own negation, — Janus

Then presumably there is an idea that negates "every idea contains the seeds of its own negation"...?

I was using quibbleable "atomic" in the original Greek sense of irreducible. — Gnomon

:up: I guess I can't steal that word "quibbleable" now. You own it.

This thread was inspired by the Big Think article, which mentioned "Kant's First Antinomy". The rest is just me babbling about Transcendence --- about which, according to Kant, I know nothing. — Gnomon

I see.

Then presumably there is an idea that negates "every idea contains the seeds of its own negation"...? — Banno

Of course. Such would be the dogmatic ideas. Could not any of those be negated? Do you think it is possible that any idea exists which could not be negated? Isn't there always a 'no' to any 'yes?

It would be a shame to mistake such grammatical observations for metaphysics or epistemics.

I was using quibbleable "atomic" in the original Greek sense of irreducible. — Gnomon
:up: I guess I can't steal that word "quibbleable" now. You own it. — L'éléphant

That's easy for you to say. :joke:

I apologize for the muddled message. It was not intended as a formal mathematical definition, but more like a poetic metaphor of mirrored universes — Gnomon
Thanks for the reply. Neat image. :cool: — jgill

Yes, the cosmic sausage-link image does neatly encapsulate the "Big Bounce" theory of cyclic universes pinched-off from previous 'verses. But such information leakage models require some exotic physics. And the accelerated expansion models seem to turn the bounce into a "Big Rip". Those one-way models assume a single line of linear time. Yet other Cosmological models envision multiple miniverses budding-off from a singular central Multiverse. However the point of the original post is that all of these math-supported speculations, while internally logical, are not scientific theories, but philosophical conjectures that attempt to deny the unique creation-event implications of the Big Bang theory..

Meanwhile, other thinkers limit their speculations to the knowable world. And a popular cosmological model (Tipler 1995) begins and ends with a Singularity, sometimes labeled "Alpha & Omega Point" theories. Ironically, both Singularities are defined as "God". Others label the future Omega Point as a Technological Singularity (Vikoulov 2020). How can these confusing Ontologies be simplified into a plausible Epistemology? :smile:

Alpha and Omega: The Search for the Beginning and End of the Universe
by Charles Seife, author of ZERO : Biography of a dangerous idea
https://www.amazon.com/Alpha-Omega-Search-Beginning-Universe/dp/0142004464

ONE & DONE : EXPANSION + ACCELERATION

BIG RIP

BIG BUDS from Multiverse

RHIZOME (rootlike) Multiverse

It would be a shame to mistake such grammatical observations for metaphysics or epistemics. — Banno

Metaphysical ideas can be negated; Kant's antinomies are based on metaphysical speculations. It's not merely a grammatical matter; the grammar reflects what is imaginable. Any speculative idea may be true or false, or at least so we might think. Scientific theories themselves are never proven; they can, and often do, turn out to be wrong.

It's not merely a grammatical matter... — Janus

Underestimating grammar's capacity to mislead is the source of metaphysics, don't you think?

I think we are misled mostly by reification; the "fallacies of misplaced concreteness". I think we are misled when we think that our dualistic abs tractions can capture the real. We are also misled when we become preoccupied with philosophical discourse. and concerned with propositional correctness, at the expense of the kinds of transformative philosophical practices in which much of ancient philosophy consisted. I suppose grammar plays a minor role in this, but I think it mostly reflects our illusions rather than creates them.

This is all just my opinion, containing the seeds of its own negation, of course.

I think this discussion too abstract to be of much use.

I agree it is. like most philosophical discussion, of little use; it's just a way to pass the time.

it's just a way to pass the time. — Janus

More a way to passing wind.

Perhaps we can tie things down a bit more. Any proposition can be negated. All the supposed mysterious, metaphysical antinomies amount to is a statement and its negation. What appears to some as profound is no more than prefixing a tilde. Hence:

Thesis : The world is limited with regard to (a) time and (b) space.
Antithesis : The world is unlimited with regard to (a) time and (b) space. — Gnomon

is just (P v ~P) with the content to be filled out.

The issue then becomes what "limit" might mean, in regard to space-time. And it's not going to be the same now as it was for Kant.

This by way of showing that there is nothing in the antinomies themselves.

"Limited" as I read it just means something like "of finite extent". That is probably what Kant meant. Can you propose an alternate usage?