What is not, is anti-what is (being), but this is categorically different from becoming, which is activity. So being is not anti-becoming, it is anti-not being. — Metaphysician Undercover

Being and becoming must have some relationship. You can't have it both ways - that as "different categories" they are related and they are not related.

It is pretty clear that if something can change to become something else, then something can stay the same by not becoming that something else.

But that's not what "change" means. It means that one state becomes another. To have two distinct states is to have two distinct states, and this does not imply change. — Metaphysician Undercover

If there is a causal relationship between the two states then they will obviously share common features. Think of yourself; as we conceive it, now you are in one state and the next moment you are in another state. You have changed but you have not become totally different; the change may be very insignificant or of greater significance, and the most significant change would be that you died and then 'you' would cease to exist and only your decomposing body would remain to eventually dissipate, returning to the environment.

The point I made before, though, is that there may be no intermediate state between two contiguous states of yourself. Indeed, how could there be; if there were intermediate states or even just one intermediate state between contiguous states then there would have to be infinitely many intermediate states between any contiguous states. This would mean that contiguous states are impossible. If that were true then we cannot understand the world at all, because our understanding of the world is always in terms of contiguous states and the causality that is implied in that understanding.

This is why I believe that dualism provides the only coherent approach toward understanding reality. — Metaphysician Undercover

Re this see my conversation with Agustino: https://thephilosophyforum.com/discussion/comment/137449#Post_137449

As Spinoza said the apparent dualism is not substantive, and as Agustino points out there, and in line with what apokrisis has been saying here; mind and matter are two poles of a continuum, not separate substances. So, mind and matter are not really separate, but in our accounts and explanations we just cannot look at both poles simultaneously. Only God can do that.

mind and matter are not really separate, but in our accounts and explanations we just cannot look at both poles simultaneously. — Janus

It is nevertheless the case that the predominant conception of the nature of reality in a secular-scientific culture is that it is something ultimately physical and therefore understandable in terms of the natural sciences; 'immanent' is the word often used on this forum. The mainstream consensus is that the mind has evolved in the same way as other organs, i.e. through the processes of evolution, as an output of the physical brain. Those who don't agree would necessarily uphold some form of dualism regarding mind.

And that conception is the consequence of an historical process the development of which can be traced over the centuries. The hallmark of the scientific attitude is that the ultimate constituent/s of the Universe can be understood in objective terms, as objects or as forces or a form of physical energy. But where I take issue with that, is that it over-estimates human intellectual and sensory capacity; it basically takes science as potentially capable of being all knowing.

Men could know God just as directly and surely as they knew the world. — apokrisis

I don't think scholastic philosophy ever said that. Orthodox Christians (in the sense of the mainstream congregations, not 'Eastern orthodox') said that knowledge of God was primarily given by revelation. 'Theosis' as a state of union with the Divine, was indeed possible, but at the cost of complete self-abnegation; it was not 'profane knowledge' in the way that natural knowledge could be. And there is a long-standing prohibition in all forms of Christianity, that God can be known in the way that the objects of perception are known - 'no man can look upon My face and live (Ex 33:20)'.

A philosophy of the supernatural replaced a philosophy of nature. — apokrisis

And that is what 'naturalism' has attempted to reverse - by seeking to understand the world as self-originating, as grounded in processes that are ultimately understandable from the naturalist viewpoint. But it seems to me that within this attitude, there is no room for any intelligence higher than the human, is there? The human sciences are then the sole arbiter of reality, of what is real and what isn't, according to their understanding of nature.

The mainstream consensus is that the mind has evolved in the same way as other organs, i.e. through the processes of evolution, as an output of the physical brain. Those who don't agree would necessarily uphold some form of dualism regarding mind. — Wayfarer

I think it is unarguable that minds have not evolved just as other organs have. You only have to look at the apparent differences between the minds of different animals and humans to see that.

But the point is, if the mind can be understood solely as the output of the evolved brain, then why is it necessary to regard it in terms other than the physical? Isn't understanding it just a matter for neuro-science and evolutionary biology? Why metaphysics and not meatphysics?

Boy, Christians would not like that. — Mitchell

Most Christians think they can talk about God, so they wouldn't understand the Wittgenstein quote.

It's only theists who define God in purely negative terms for which that would apply.

You make a distinction between Christians and Theists that I don't recognize. On my understanding Christians are Theists.

Not all theists are Christians. Christians generally have well defined concepts about God, the afterlife, etc.

It's because reality has (at least) two attributes: the mental and the physical. Consequently, we understand things in two ways; in terms of causes and in terms of reasons. Nature is mostly understood in terms of causes, although animal behavior may be understood in terms of reasons as well; and notably, there is much intentional language used in biological explanations. It has not been possible to combine these kinds of explanations, but semiotics may achieve a synthesis. It don't think it will ever be possible to fully understand matters of quantity in qualitative terms or matters of quality in quantitative terms, though: but who knows?

It's because reality has (at least) two attributes: the mental and the physical. Consequently, we understand things in two ways; in terms of causes and in terms of reasons. Nature is mostly understood in terms of causes, although animal behavior may be understood in terms of reasons as well; and notably, there is much intentional language used in biological explanations. — Janus

That's a good way of putting it - the distinction between 'cause' and 'reason', especially. I think overall that in Aristotelian thinking, 'cause' and 'reason' were unified by the 'four causes' doctrine, so that the formal and final cause of something, was also a reason for it.

Whereas in modern thought, efficient and material causes are regarded as sufficient. As we have learned, semiotics has sought to re-introduce the notions of formal and final causes, although I am still a bit unclear how you can have a 'final cause' in the sense of a 'reason for existence' with respect to sentient beings, without something towards which they are evolving (which is generally ruled out by the antipathy towards 'orthogenesis'.)

These statements I take to be contradictory:

Hence the Peircean process view. Now being is emergent and so an eternal state of becoming. You only have degrees of definiteness.
...
So the Peircean view fixes things with a hierarchical structure. — apokrisis

The problem being that Peirce's "eternal becoming" as you describe, renders definiteness incomplete. Therefore things cannot be properly fixed, and the claim that the Peircean view "fixes things" must be contradictory. This is the problem with vagueness as a first principle, it is intelligibility compromised.

Then definite being emerges as the concrete action that arises between these two bounds. — apokrisis

You've already constrained definiteness to degrees, such that you cannot validly refer to anything as "definite being", you only have degrees of being.

Being and becoming must have some relationship. You can't have it both ways - that as "different categories" they are related and they are not related. — apokrisis

I don't deny that there is a relationship between being and becoming. What I would say is that the nature of this relationship is not well known. It is a deficiency in our knowledge, just like the relationship between the past and future is not well known, it is a deficiency. Still, we know enough to say that the past is categorically different from the future, like we know that being is categorically different from becoming.

It is pretty clear that if something can change to become something else, then something can stay the same by not becoming that something else. — apokrisis

This is not as clear as you might think. If that thing doesn't become that particular "something else", it might still change to become a different something else. That is the nature of possibility. So "not becoming that particular something else" does not necessitate that the thing stays the same. Therefore staying the same is not related to becoming something else in the way that you suggest here.

In reality, when we talk about a thing staying the same thing, it does so despite changing. So I stay the same person despite undergoing changes. It might be that some aspects of me stay the same while others change. This brings us back to the issue of unity.

The point I made before, though, is that there may be no intermediate state between two contiguous states of yourself. — Janus

Yes, I understand this point that you made. The point I made is that if there is no intermediate state, then there is no such thing as "change", as we commonly use, and understand the word. That is because two distinct states, does not constitute change.

The point I made before, though, is that there may be no intermediate state between two contiguous states of yourself. Indeed, how could there be; if there were intermediate states or even just one intermediate state between contiguous states then there would have to be infinitely many intermediate states between any contiguous states. — Janus

Yes, we went through this, that's why Aristotle concluded that it is necessary to consider that change is something completely different from a describable state, to avoid having to describe it as an intermediary "state", thus leading to the infinite regress. If we remove the nature of "intermediary" from change, then how can we relate it to being?

The problem being that Peirce's "eternal becoming" as you describe, renders definiteness incomplete. Therefore things cannot be properly fixed, and the claim that the Peircean view "fixes things" must be contradictory. This is the problem with vagueness as a first principle, it is intelligibility compromised. — Metaphysician Undercover

Rest easy, MU. As usual, dichotomies rule. Stability is relative to plasticity. So we are talking here about the approach to a limit. If there is vagueness, then already there is also its "other" of the crisp.

What I would say is that the nature of this relationship is not well known. It is a deficiency in our knowledge, just like the relationship between the past and future is not well known, it is a deficiency. Still, we know enough to say that the past is categorically different from the future, like we know that being is categorically different from becoming. — Metaphysician Undercover

Well we do know the nature of the relationship. It is a dichotomy. We arrive at it via dialectical reasoning. Metaphysics has been operating this way since it began.

A categorical difference is one in which two categories stand absolutely opposed. To each other. And so therefore they are also absolutely related.

In reality, when we talk about a thing staying the same thing, it does so despite changing. So I stay the same person despite undergoing changes. It might be that some aspects of me stay the same while others change. This brings us back to the issue of unity. — Metaphysician Undercover

Again, already accounted for. Constraints regulate dynamism. The purpose of a thing maintains its identity despite all material changes it might undergo. You can't pretend this is a great mystery.

Well we do know the nature of the relationship. It is a dichotomy. We arrive at it via dialectical reasoning. Metaphysics has been operating this way since it began. — apokrisis

A dichotomy is a separation. The nature of how the two separate things relate to each other is not described by "dichotomy". To arrive at the conclusion "there is a dichotomy" is only the beginning of the metaphysical procedure. It is how the two elements of the dichotomy are related, which becomes the basis for classifying the type of metaphysics. For instance a monist would say that the two elements of the dichotomy are fundamentally the same, while a dualist would say that they are fundamentally different.

A categorical difference is one in which two categories stand absolutely opposed. — apokrisis

I believe that this is incorrect. two things which are opposed are necessarily in the same category, as defining each other. You'll see this with any opposing terms negative/positive, hot/cold, large/small, etc. The two opposing terms may be seen as defining the limits of the same category. A categorical separation is a separation between types of thing.

Again, already accounted for. Constraints regulate dynamism. The purpose of a thing maintains its identity despite all material changes it might undergo. You can't pretend this is a great mystery. — apokrisis

Actually that's very mysterious to me. You have a categorical separation here between "constraints" and "dynamism", two distinct categories. You have mentioned a relationship between the two with "regulate". This is mysterious in the sense of what is a constraint. Is it a rule, is it a law, is it a physical barrier, or is it just another dynamism? If it is the latter, then the dichotomy dissolves. I don't think it is a physical barrier, because I do not sense such a barrier to activity. I don't think it's a rule or a law, because then dynamism would have to know how to interpret rules to be able to act according to constraints. So "constraint" appears to be something you just made up, a word which has nothing underneath it, no substance, just mystery.

Then, you enhance the mystery by mentioning "purpose". Are you suggesting that constraint is equivalent to purpose? Then you would be saying that dynamism is regulated by purpose. The problem I have with this is that I see purpose in the activities of living things. And, I see a categorical separation between living and inanimate such that the inanimate is excluded from acting with purpose. But the inanimate is still active, it still contains dynamism, and that dynamism is also constrained. So I can understand how purpose regulates dynamism in human beings to the extent that they use their power of free will. And also to some extent it is evident that there is purpose in the activities of other living things. But I think the claim that purpose controls dynamism in inanimate things is unjustified. So this appear to be nothing more than a monist attempt to dissolve the categorical separation between animate and inanimate. Mystery solved.

That whereof we cannot speak, thereof we must be silent. — Mitchell

The Tao that can be spoken is not the eternal Tao.

For instance a monist would say that the two elements of the dichotomy are fundamentally the same, swhile a dualist would say that they are fundamentally different. — Metaphysician Undercover

And nicely, a triadicist would say a dichotomy is how the same becomes the different. That is why the relation can be described in terms of a reciprocal action.

The discrete becomes different from the continuous by breaking it. The continuous becomes different from the discrete by connecting it. So a differencing that begins in the sameness of an indeterminancy - a vague potential which is neither the one nor the other - proceeds in mutual fashion towards its naturally opposed extremes.

You'll see this with any opposing terms negative/positive, hot/cold, large/small, etc. — Metaphysician Undercover

These examples you've chosen are weak and easily reversed differences. They are symmetry-breakings of the same scale - anti-symmetries - and so can quickly erase each other. A metaphysical dichotomy is a full-blown asymmetry. The outcomes look to be orthogonal and as unrelated as possible. The relationship is reciprocal or inverse, not merely additive/subtractive.

So to move from hot to cold, you just have to subtract some heat. But to get from the continuous to the discrete, you must understand the continuous as "absolutely broken". It is the antithesis which is the least possible amount of continuity, or 1/continuity.

A categorical separation is a separation between types of thing. — Metaphysician Undercover

No. It is the separation that produces the familiar list of metaphysically opposed types.

Aristotle's categories were a bunch of dichotomies - quality~quantity, active~passive, time~space, symmetry~symmetry-breaking, particular~universal.

So "constraint" appears to be something you just made up, a word which has nothing underneath it, no substance, just mystery. — Metaphysician Undercover

It's just a normal, well understood, word in science.

And, I see a categorical separation between living and inanimate such that the inanimate is excluded from acting with purpose. — Metaphysician Undercover

Another thing I've explained so many times now. There are semiotic grades of telos. Minds have purposes, life has functions, and physics has tendencies.

That whereof we cannot speak, thereof we must be silent.

— Mitchell

Never stops anyone here :-)

Minds have purposes, life has functions, and physics has tendencies. — apokrisis

Schumacher advocates the traditionalist view that there are four kingdoms: Mineral, Plant, Animal, Human. He argues that there are critical differences of kind between each level. Between mineral and plant is the phenomenon of life. Schumacher says that although scientists say we should not use the phrase 'life energy', the difference still exists. Schumacher points out that though we can recognize life and destroy it, we can't create it.

For Schumacher, a similar discontinuity exists between plant and animal, which is differentiated by the phenomenon of consciousness. We can recognize consciousness because animals exhibit at minimum primitive thought and intelligence.

The next level, according to Schumacher, is between Animal and Human, which are differentiated by the phenomenon of self-consciousness or self awareness. Self-consciousness is the reflective awareness of one's consciousness and thoughts.

Schumacher realizes that the terms—life, consciousness and self-consciousness—are subject to misinterpretation so he suggests that the differences can best be expressed as an equation which can be written thus:

'Mineral' = m
'Plant' = m + x
'Animal' = m + x + y
'Human' = m + x + y + z

In his theory, these three factors (x, y and z) represent ontological discontinuities. He argues that the differences can be likened to differences in dimension; and from one perspective it could be argued that only humans have 'real' existence (hence the designation 'beings') insofar as they possess the three dimensions of life, consciousness and self-consciousness. Schumacher uses this perspective to contrast with the materialist view, which argues that what is 'real' is inanimate, denying the inherent reality of life, consciousness and self-consciousness, despite the fact each individual can verify those phenomena from their own experience.

From Wikipedia entry on E F Schumacher's book, A Guide for the Perplexed, slightly edited.

Yes, I understand this point that you made. The point I made is that if there is no intermediate state, then there is no such thing as "change", as we commonly use, and understand the word. That is because two distinct states, does not constitute change. — Metaphysician Undercover

But this contradicts experience and common sense. We are all constantly changing and may be said to be in different states from one moment to the next. This is enough to satisfy the definition of change. Whether or not there are intermediate states in between what I am at one moment and what I am at the next has no bearing on the fact that I have changed from one moment to the next.

As we have learned, semiotics has sought to re-introduce the notions of formal and final causes, although I am still a bit unclear how you can have a 'final cause' in the sense of a 'reason for existence' with respect to sentient beings, without something towards which they are evolving (which is generally ruled out by the antipathy towards 'orthogenesis'.) — Wayfarer

Final cause could be understood as something not imposed from "outside" but as the most universal 'global' conditions that determine what possible forms can evolve. If nature were deterministic, then everything that comes to be would be absolutely necessary in just the way it comes to be. This is Spinoza's view, thinking as he did under the aegis of Newtonian mechanics.

On this view, if you 'replayed' the cosmos from the Big Bang, everything down to the minutest detail would happen precisely as it has. If nature is indeterministic, probabilistic, then it would not replay the same, but differently each time you replayed it, nonetheless the same most general formal constraints would still determine the alternative ways in which it could evolve. I think this would fulfill the idea of final cause.

What I am getting at is the idea that there is a telos towards life and mind implicit in cosmic evolution, as a kind of final cause.

Thomas Nagel floated a similar idea in Mind and Cosmos, where he explores the question of whether life and mind arise solely as consequence of material necessity (i.e. materialism) or because of an agency (i.e. theism) or - his tentative view - that there is a natural tendency towards them:

The existence of teleology requires that [those] successor states [which lead to the formation of more complex systems, and ultimately life] have a significantly higher probability than is entailed by the laws of physics alone […] Teleological laws would assign higher probability to steps on paths in state space that have a higher “velocity” toward certain outcomes.

I have often contemplated the idea that humans are in some sense the cosmos become self-aware; it's not something explicitly stated in the Western tradition, although you do find such ideas amongst the Hermetics, with the notion of 'man as microcosm'. But it is very different to the typically neo-Darwinian view that the mind is the output of the physical brain, which in turn is the consequence of an essentially physical process directed by a kind of algorithm. There's no purpose other than to survive in any of it, and to ask the question 'what is the meaning of survival' is regarded as a sign of sentimentality or weakness; it's very Nietzschean in that respect.

This is Spinoza's view, thinking as he did under the aegis of Newtonian mechanics. — Janus

Something that is easily overlooked in all this was that Spinoza and Newton both were ardent believers in God - it never would have occurred to them to entertain the idea of a self-originating cosmos. 'Indwelling in every dewdrop as in the innumerable host of heaven, in the humblest flower and in the mind of man, Spinoza found the living spirit of God, setting forth the Divine glory, making the Divine perfection and inspiring with the Divine love'. Why Spinoza was depicted as being atheist (which he strongly denied) was because he sought to understand the Divine outside the strictures of orthodox religion, not because he didn't believe in God.

Spinoza's God did nothing by fiat, but by the necessity of his own nature. And in fact Spinoza equated God with Nature. Spinoza's works were anathematized because it was readily seen at the time by intellectuals who adhered to the Faith that Spinoza's ideas lead inexorably to atheism.

I have often contemplated the idea that humans are in some sense the cosmos become self-aware; it's not something explicitly stated in the Western tradition, although you do find such ideas amongst the Hermetics, with the notion of 'man as microcosm'. — Wayfarer

IF we are the only self-aware beings then this would be self-evidently true.

What I am getting at is the idea that there is a telos towards life and mind implicit in cosmic evolution, as a kind of final cause. — Wayfarer

Under either determinsim or indeterminism life can be understood to be physically inevitable. As I already said if the universe was replayed it would be exactly the same down to the most minuscule detail under determinism. Under indeterminism it is arguable that forms of life would have inevitably appeared; albeit in different forms. This is would be a telos from global conditions as I already pointed out.

These examples you've chosen are weak and easily reversed differences. They are symmetry-breakings of the same scale - anti-symmetries - and so can quickly erase each other. A metaphysical dichotomy is a full-blown asymmetry. The outcomes look to be orthogonal and as unrelated as possible. The relationship is reciprocal or inverse, not merely additive/subtractive. — apokrisis

I was just going by what you said. You said a categorical difference is a difference of opposition. Orthogonal is a completely different concept from opposition.

No. It is the separation that produces the familiar list of metaphysically opposed types.

Aristotle's categories were a bunch of dichotomies - quality~quantity, active~passive, time~space, symmetry~symmetry-breaking, particular~universal. — apokrisis

But none of these differences are differences of opposition. How can you call quality~quantity a difference of opposition?

This is your typical mode of argument. You'll insist that dichotomies are oppositions. I'll say that they are categorical differences. You'll give a bunch of examples, like this, which clearly demonstrate that dichotomies are not oppositions, but categorical differences. Then you'll go back to talking about dichotomies as oppositions, ignoring everything we've discussed.

Minds have purposes, life has functions, and physics has tendencies. — apokrisis

So if minds have purpose, and physics has tendencies, how do you get to your principle, that purpose regulates dynamism?

Orthogonal is a completely different concept from opposition. — Metaphysician Undercover

LOL. You don't say. You mean like ... an opposition so complete it is a total and complete asymmetry, not merely a weak-ass negation? >:O

So if minds have purpose, and physics has tendencies, how do you get to your principle, that purpose regulates dynamism? — Metaphysician Undercover

Hmm. I dunno. [Scratches head.] Grades of purpose?

It's very difficult to have any rational discussion with you because you show no respect for conventional use of the English language. Asymmetry is as simple lack of symmetry. It is nothing other than a weak-ass negation. Therefore, there's no sense to your claim that you're talking about something other than a weak-ass negation.

This is what I meant in the last post. You keep insisting that you are talking about something beyond negation. So I suggest category difference. You agree, yes we're talking about category difference. But instead of proceeding to talk about category difference, you attempt to bring category difference down and stuff it into a category of negation (asymmetry). So you continue to insist that you're beyond weak-ass negation, while all you do you is use terms of weak-ass negation, instead of using terms of category difference, which would indicate that you actually were thinking beyond weak-ass negation.

Maybe a diagram would be easier....

+ | -

vs

+| ------------------------------------------------------------------

Maybe a diagram would be easier....

+ | -

vs

+| ------------------------------------------------------------------ — apokrisis

Nope, doesn't help.

Sigh. What can you say when folk can't get the difference between a mirror symmetry and an actual symmetry breaking?

+ | -

vs

+| ------------------------------------------------------------------ — apokrisis

I caught your reference to "weak-ass" symmetry and symmetry breaking. I also saw it as a dichotomy between distinctness and continuity (to refer it back to some earlier discussions).

Symmetry-breaking is clearly categorically different from symmetry. But asymmetry is nothing other than a weak-ass negation of symmetry.

So the metaphysical question would be what kind of a thing is "symmetry-breaking", which allows it to be completely different from both symmetry and asymmetry?

Sigh. What can you say when folk can't get the difference between a mirror symmetry and an actual symmetry breaking? — apokrisis

Well, you could... I don't know - maybe explain better.

Well, you could... I don't know - maybe explain better. — T Clark

The argument is over what kind of mathematical relation defines a logical dichotomy - a dichotomy being a relation that is mutually exclusive and jointly exhaustive.

MU wants to treat is as simple negation. A and not-A. The presence of some thing, and then its absence or its erasure. But that is question-begging as it doesn't go to any mutuality that could form the two poles of being, nor to the way the two poles then demonstrably exhaust all other possibilities.

So a dichotomy is about taking a difference - an asymmetry or symmetry-breaking - to an extreme. It must begin in sameness and wind up looking orthogonally opposed. You don't just have chance and its absence, you have chance and necessity - an opposition of two poles of being that then encompass everything else that could be "somewhere in-between" these complementary extremes.

So likewise every metaphysical-strength category. You don't just imagine discreteness and its absence. You can only imagine discreteness in terms of the absence of something else, its exact opposite of continuity. Stasis makes no sense unless understood in terms of being antithetical to flux. Oneness is not a meaningful concept except to the degree it contradicts multiplicity.

Then seeking a mathematical model of this relation, the best understanding is an inverse or reciprocal one.

MU's weak-arse negation is like addition and subtraction. Count up three places, then erase those three places to end up back where you started. It is like a mirror symmetry. Flip the image over to break the symmetry. Then flip it again and you are back where you started. It is a symmetry-breaking, but nothing much has really changed as it is so easy to return to unbrokenness by a single step reversal of your path. An A-sized step gets negated by a second A-sized step - just now in the other direction.

Mathematically, it is the symmetry-breaking of a zero. 1 + -1 = 0. It is about the least amount of symmetry-breaking you can get away with. It is the symmetry breaking that remains as close to nothing actually happening as possible.

A dichotomy then represents the opposite end of the symmetry-breaking scale - one that is as extreme or asymmetric as possible. And a reciprocal relation models this well as each move in one direction causes a matching move in the other. If one end of the relation grows, the other actually shrinks to the same degree. Two poles of being are in play, each acting on the other in mutual and exhaustive fashion.

Now the mathematics is a yo-yo around 1, not 0. It is a relation anchored on an actual unity - a foundational sameness - that then gets broken in two complementary directions. Hence it is a triadic or developmental relation being modelled.

So consider the development of a reciprocal in the form of a fraction - a numerical inverse.

We start with 1. This 1 is 1/1 (Aha, the latent symmetry breaking which so far has changed nothing!) Then we get 2, and so 1/2. Then keep counting. We get 3, and thus 1/3 as its reciprocal. Guess where this is going next. We get 3 and its formal inverse, 1/3. Every time one number gets bigger, it forces its partner number - anchored by this particular form of opposition - to get smaller. The values are being driven apart.

Extremetise the relation and we get infiinities and infinitesimals. The infinitesimal is 1/infiinity. The infinite is 1/infinitesimal. Every actual number - fractional or whole - is then contained within the limits of this canonical relation. The infinite and the infinitesimal emerge as the limits on the breaking of the symmetry represented by the ur-somethingness of the 1.

A relation has to relate things. A self-relation is tautologous. Just counting up or down is simply to add the minimal claim that "a something" exists to break the ultimate symmetry of a zero-ness. There is at least 1 thing now, and you can then imagine 1-1 to recover the initial symmetry from which this one-ness must have mysteriously arisen, or 1+1+1+1... as the operation to keep breaking this zero-ness in the vain hope of finding its other limit.

You can see all the usual metaphysical dilemmas that flow from this sound of one hand clapping. How did something arise from nothing? How could we have creatio ex nihilo?

But a reciprocal/dichotomistic logic derives complementary limits of difference from an initial absolute sameness. Now we do start with something - but it an undefined oneness, a vagueness, a firstness. It is as much everything as it is nothing. It needs no stronger definition than the claim that it is a unity, an unbroken symmetry.

And then we can imagine a fundamental division in mutually definitional directions. If this symmetry starts to show some discreteness, some discontinuity, then matchingly, there is the new-found definiteness in the continuity that it claims to be moving away from. If the action reverses its course, it will be heading back towards its actual opposite, not simply negating its existence.

If we say something is becoming more fractional - 1/3 is now 1/333 - then it is not just shrinking towards nothingness as one of the limits on oneness. It is moving ever further away from its own inverse, 333/1. It is expressing its tendency towards infinitesimality in terms of the countering possibility of the infinite.

So it boils down to monism vs triadicism.

Monism claims there is either nothing or, instead, the one thing. (So it is in fact reliant on a metaphysical dichotomy, but understands it as a dualism - a simple presence vs absence distinction).

Triadicism fixes this by seeing presence and absence as relative to the third thing of a vagueness or apeiron. There is the unity of an unbroken symmetry which is neither A nor not-A. The principle of contradiction does not yet apply. 1 = 1/1. And turn 1/1 upside down, multiply it how you like, and you see no difference.

But as soon as you allow the possibility of a difference, a symmetry-breaking, then you get a separation to opposing poles of being. If you can have 2/1, then you can have 1/2. A single step now causes a break in the actual scale of being. Growth is matched by shrinking, not merely by not-growth. The difference is a real one, not merely the unplaced notion of one hand clapping - an event with no context by which to measure itself against.