Category: Philosophy

Start with the question of what a truth even is. Philosophers have circled this for a long time without settling it. Some say a truth is whatever matches reality. Others say it’s whatever holds together inside a system of beliefs. Pragmatists say it’s whatever works. Deflationists shrug and say the word adds nothing. They say calling something true is just another way of asserting it. None of these positions has knocked the others out, which is probably the first clue that we’re dealing with something harder than it looks.

If we grant that truth exists, certain constraints fall out of the idea almost on their own. A truth can’t contradict itself. It can’t shift depending on who’s asking. It must be about something. It must be checkable in principle. These are minimum demands. Even philosophers who disagree about what truth is, tend to accept that whatever it is, it must obey something like these rules. Otherwise, the word loses any usefulness.

But constraints aren’t the same as foundations. Knowing what shape a truth must have doesn’t tell you how to get one. So, the real question becomes; what does it take to prove something?

The trouble with proof

Strip proof down to its bones and you find this: a demonstration that, given what we both already accept, the claim follows. Shared starting points, legitimate moves, a chain to the conclusion. That’s the minimum. It works inside mathematics, inside courts, inside everyday argument.

The trouble is that every proof needs premises, and those premises themselves need proving, and so on. You can only escape this in three ways: by going on forever, by going in a circle, or by stopping somewhere and calling it self-evident. The ancient skeptics noticed this and called it a trilemma. None of the three escapes is satisfying. The infinite regress never lands. The self-evident foundation just relocates the problem. And the circle looks like cheating.

Can a circle do real work?

It depends on the circle. A tight one “God exists because the Bible says so, and the Bible is true because God wrote it” is plainly empty. It proves nothing because it presupposes everything. But there’s a different kind of circle, a wide one, where many beliefs prop each other up across a whole system. Coherentists think this kind of mutual support is the best we can do, and that the strength of the web is what makes knowledge possible.

The weakness of that picture is that a perfectly consistent system can also be perfectly wrong. A conspiracy theory holds together. An ideology holds together. Internal consistency doesn’t prove contact with anything outside itself. Circles can lock in falsehoods just as well as truths, and once locked in, they’re hard to escape because everything inside the circle vouches for everything else.

So circularity alone won’t do. It might keep a truth stable once you have one, but it won’t generate one. Something else is needed — something that pushes back from outside.

Triangulation

When I was young I spent many hours with my father mapping. I would stretch out a chain for him, and he would measure the angles of a distant point at each end of the chain.  So, mapmakers had solved a version of the truth problem long ago. You can’t measure a distant mountain by walking to it. But you can stand in one known place and take a bearing. Then move to a second known place and take another. The two lines cross, and where they cross is the mountain. Add a third bearing as a check. If all three lines meet at one point, you’ve found it. If they don’t, something is wrong and you know it.

What’s powerful about this is that none of the individual measurements is foundational. Each is fallible. But because they’re independent, their errors don’t compound, they cancel. The convergence is more reliable than any single bearing could be. And the method announces its own failures: when the lines don’t meet, you can’t pretend they do.

This is roughly how science works at its best. Different lines of evidence from fossils, genetics, biogeography, direct observation all meet and point at the same conclusion. The age of the universe is triangulated from several independent calculations that didn’t have to agree but do. None of them is certain on its own. Their agreement is what makes the result hard to dismiss.

But the mapmaker is one person

Here the analogy nearly breaks. The mapmaker chooses where to stand, reads the instruments, judges when the lines converge. Every step passes through one mind. So in what sense is this really triangulation between independent viewpoints? Three measurements taken by the same person, filtered through the same assumptions, might just be one perspective wearing three coats.

This is a serious objection and it doesn’t fully go away. A lone observer with a quiet bias could produce a map that looks internally perfect and is still wrong. The convergence would be an artifact of the mind, not of the world. The philosopher, Davidson, thought the only fix was another mind. He thought that objectivity requires a second perceiver looking at the same world, because only then can you tell the difference between how things seem and how things are.

The instrument as another viewpoint

I worked for most of my life with scientific instruments and I believe they offer a way around the lone-observer problem that doesn’t require a second consciousness. The instruments themselves do part of the work. A compass doesn’t share the mapmaker’s hopes. A theodolite doesn’t care what answer would be convenient. I catch a 13 inch trout, but my ruler tells me is 12 inch. My ruler contradicts me; tells me I am wrong. These tools encode the discoveries of people the user never met. These experts were opticians, geometers, clockmakers, generations of testing and correction. When the mapmaker looks through a lens, they’re already in collaboration with absent minds, and with the physics those minds figured out.

More importantly, the instruments can disagree with their user. A thermometer reading 40 when you expected 20 has contradicted you. It has acted, in a way that another person disagreeing would act. It doesn’t matter that it isn’t conscious. What matters is that the resistance came from outside the mind, not from inside it. The instrument is a delegate of reality; a piece of the world rigged to report on the rest of it.

This changes the picture. Objectivity doesn’t strictly require another consciousness. It requires another source of constraint — something the mind doesn’t control, that can push back on it. Other minds are one such source. Instruments are another. The world’s own behaviour is a third. A child learning that stoves are hot is triangulating against reality without needing a peer. The criterion isn’t how many minds agree. It’s how many independent sources of resistance the belief has survived.

Where this leaves truth

Truth, on this view, isn’t something we derive from certain foundations. There aren’t any. It’s what survives convergence from as many independent angles as we can muster — different people, different methods, different instruments, different eras. Each angle is fallible. None is foundational. But their intersection narrows reality down further than any of them could alone.

This sidesteps the old trilemma without pretending to solve it. We don’t need an infinite regress, or a self-evident starting point, or a vicious circle. We need a wide enough parliament of fallible witnesses that the particular biases of any one of them stop mattering much. The history of human knowledge, looked at this way, is the history of building better otherness. Constructing new instruments, gathering wider communities, finding new ways to let reality talk back.

None of this gives us certainty. A perfectly triangulated belief can still be wrong if every viewpoint shares some hidden bias. What it gives us is something more modest and more honest: belief that has been tested from many directions and hasn’t yet broken. That’s not nothing. It might be everything we ever actually have.

The mapmaker is a single mind. But the map, by the time it’s been used and corrected and redrawn across centuries of sailors who needed it to work, is no longer the product of any single mind. It’s a sediment of many — and of the world’s stubborn refusal to be whatever they wanted it to be. That sediment is, in the end, what we mean by something being true.

“For us believing physicists, the distinction between past, present and future is only a stubbornly persistent illusion.”¹

A month before his death, and two months before my birth, Albert Einstein wrote a letter of condolence to the family of his close friend, Michele Besso. In it, he offered the view above.

Outline of the essay

I begin with my own phenomenological experience of the “now” and a story based on Valtonen’s three-body work², which I use to develop a candidate for a finite time-window τ based on the Lyapunov time. I then examine McTaggart’s classical argument³ for the unreality of time and Broad’s alternative⁴. I present my main argument, which relocates change from metaphysical A-properties to non-linear, irreversible, asymmetric physical relations, and show that my experience is consistent with this argument. I conclude that the “now” is real.

Experience of the “now”

I am aware of the “now” moment, I reflect on the past, I imagine the future but the “now” is an experience. I can remember a frightening moment and feel the fear, but not in the same way as being in it. If I awake from a nightmare, the moment I realise it is a dream, I still retain the emotions, but the realisation moment changes the “now”.

The “now” does not feel like a point in time; it has a feeling of width, it extends a little back, it lingers, and it extends a little forward, it anticipates. Some describe time as a flowing state, with the “now” moving. I think of time more like the fly fisherman at the head of the salmon pool seeing the water toss into the pool and settle. The direction of flow is from fixed memory towards a settling “now” towards an unfixed future of potential.

A classical independent view is expressed in the poem – The Waterfall⁵ by Mary Oliver, and I particularly refer to the final lines;

“finally, in the deep and green and utterly motionless pools after all that falling? “

The Race of the Asteroids – a story

Zeus was bored, so one day he held a race. His two best asteroids, M and N were set in motion racing across the outer reaches of the solar system. He asked the Sun to arrange a beam of light from beyond the planets. When the light beam reached the racecourse, 30 AU from Neptune, the race began, and when the light reached Neptune, the winner was announced.

The race was a great success. The first asteroid M was out in front, but N was in hot pursuit. Gravity pulled back M but accelerated N so N finally passed M, angular momentum was the jockey that kept them from colliding as they passed. The race continued with both asteroids passing each other until the light reached Neptune, one was declared a winner.

After a few races the planets realised that by measuring the initial conditions of the asteroids as the light approached the racecourse in the darkness beyond Neptune, they could predict the see-sawing motion accurately.

Zeus, annoyed, decided to enter a third asteroid, O. Now the motion of the three became chaotic with M rushing ahead for long periods, the three quickly changing positions, the results appearing truly random. But again, the planets found that by measuring the initial conditions accurately, which they called the form, they could predict the winner.

Zeus imposed an uncertainty limit: no matter how carefully the planets measured the form, they could not know the result exactly. The uncertainty was tiny but in a chaotic system it would grow exponentially. He extended the light path to Earth where its arrival would signal the race end. The planets could predict outcomes faithfully for a while, but beyond a certain time, Lyapunov timescale, the certainty dissolved into probability. They, like Zeus, would know the odds, but not the outcome.

The planets complained that if they could not measure things accurately, how would they ever know if Zeus was cheating. Zeus agreed to write down the rules of the race for all time; the planets could calculate and check that every motion followed those rules exactly. He called his rules the General Theory of Relativity in the Race of the Asteroids and placed a copy at the light beam end point, Earth, so that all could see. The race is run every day, and no one has yet been able to consistently beat the odds set by Zeus.

Thought experiment

Thinking about the story, I imagined an experiment performed by one of the planets. In his laboratory analysing the trajectory of the three asteroids, he would take his real measurements and run a simulation on his computer, writing into three columns: A column, the winners, B column, the logical order of winners, Column his model’s predictions. I realised that he need not fill in the B series. No information leaves the volume. We define V throughout as a local volume of 4D spacetime. There is much happening in an unpredictable way but no new information, no real change. The planet cannot predict the outcome of the race past the Lyapunov time, yet for every conceivable experiment that he might run, the B series is logically fixed, known, but just not to him. The outcomes are indexed to time.

I ran a second thought experiment replacing the Newtonian physics with general relativity. In GR, gravitational interactions are dissipative through radiation. The motion of the masses creates information that needs to be shared with the rest of the universe. Every Lyapunov time scale, the system must radiate a unit of entropy. The race outcomes are now indexed to time and conditional on entropy having increased along the history.

This is fundamentally different. In the classical case the system is informationally closed: everything is determined, and our inability to predict is purely a limitation of measurement, not of reality. The B-series is fixed, and ignorance of its order is an epistemic fact about us, not an ontological fact about the world. In the general relativistic case, something physically different happens. The radiation of gravitational energy is lost information from the closed volume. The local system that existed before the radiation is not the same one that exists after it. A new physical state has come into being, and the old one is gone. This is a claim about what exists in V. It is an emergence of a new state in a period τ, a new becoming.

It is important to note that this does not make the “now” observer dependent. The τ associated with a specific V is a property of that region’s chaotic physical processes, not of whoever happens to be looking. When I later discuss experiencing the “now”, I am speaking of a different τ which is the experiential coherence window of a cognitive system, which varies between observers. This is an integration period τ not a formation period τ, but they both apply a non-linear process, one to agree paths and the other to agree truths.  But a every volume of spacetime has a unique τ, the becoming window of the region.

McTaggart, Broad, and the classical debate

The three familiar positions in the philosophy of time are: eternalism, where past, present, and future are equally real and the “now” is merely a perspective within a fixed four-dimensional block; presentism, where only the present exists; and the growing block, associated with Broad, where the past and present are real but the future is not yet real. In my view the future is real, because my future could be an observers present, or past. The process of becoming consists in the local information update within V as information (gravitational radiation) arrives or leaves.

I therefore propose a fourth view, the block is eternal. It is like a library, truth are books on the shelf, but a librarian is using not one but two non-linear rules, the first is logic to decide if it is a book, and a second non-linear rule chaos to decide the path to its location. The truth exists at a co-ordinate in spacetime, but the exact location is being, or will be decided. 

The classical challenge comes from McTaggart³. In his famous argument he constructs the A and B series. In the A series events pass through the present, from future to past. Events in the B series are a permanent record of their ordering. He argues that events must have the mutually exclusive properties of futurity, presentness and pastness all at once, which is contradictory, and therefore time is not real.

McTaggart sees the “specious present” as a further problem: if the present has a width, that width depends on the observer and is not objective. To address this, we need to deal with two situations: two distant observers, and two observers in the same space. The first is handled by defining a local coherence period τ in each V, local time is chosen based on general relativity. The second is handled by defining an integration τ of an observer: a professional tennis player will experience a different “now” when addressing a ball serve than an amateur. While τ may appear arbitrary, it is not. It is the Lyapunov time period of the observer’s integration process combined with the Lyapunov time period of process in the local volume.

McTaggart also says that there is no independent way to validate time other than experience. It is subjective. But the arrow of time does emerge, as Valtonen² argues: “as soon as the universe has at least three particles, the Arrow of Time appears; it is the direction into which the Kolmogorov-Sinai entropy increases.” This points to entropy as an arrow of time.

Finally, McTaggart assumes that pastness, presentness, and futurity are intrinsic properties an event simply has, independently of its relations to other events or processes. This is what generates the contradiction: an event must have all three, yet they are incompatible.

But the entropy reformulation challenges this at the root. If what distinguishes past from future is not a primitive A-property but rather the direction of irreversible physical processes such as entropy increase, information dispersal, the asymmetric radiation of energy. Then past and future are not intrinsic properties of events at all. They are relational and processual. An event is “past” relative to V not because it possesses some metaphysical property of pastness, but because its information has irreversibly dispersed beyond V’s local domain. The co-ordinates of truths are fully known.

On this account, McTaggart’s contradiction simply does not arise. There is no single moment at which an event must simultaneously bear all three properties. Instead, there is a physical process, a Lyapunov-scaled becoming, through which states transition from informationally unresolved (future) to locally integrated (present) to informationally dispersed (past). These are stages in a process, not simultaneous property-attributions.

A closely related approach is found in C.D. Broad⁴ ’s Scientific Thought. Broad says: “The past and present are real, but they are not real in the same way. The present consists in events undergoing becoming: the ontological transition in which possibilities become actual. The past consists in events that remain real as fixed achievements of that process. The future is unreal, not because it is unknown, but because it has not yet been generated.”

Broad avoids the McTaggart paradox by refusing to attach properties to what does not exist. Time is real, but the future is not. I find this a powerful and radical move. It recalls, in reverse, the story from Antoine de Saint-Exupéry, where Bedouin visitors, brought to see a waterfall, wanted to remain until the water stopped. The possibility that the waterfall might stop captures real change.

Here, as a Physicist, I part company with Broad. Future propositions are true or false. What is missing is their exact location in V, not truth-value. Broad says future propositions are neither true nor false because the future does not exist.  I agree with Broad that the future cannot serve as a local truth-maker, but unlike Broad I maintain their truth-makers are not locally accessible to V. 

In a system governed by general relativity, the physical state that would serve as truth-maker exists in the future region. Becoming resolves the path. It is not hidden from us; it is present in reality (future region) but hidden from local V ontologically.

Broad’s position is that events emerge, become in the present, then exist in the past. They do not exist in the future. In my view, when a relationship refers to an event at time, t with t > t_p + τ, where t_p is the current time on a local clock, then the truths are contingent on facts not locally settled. If t < t_p + τ and t > t_p truths are contingent on facts being settled, and if t < t_p all truths are fixed. They are fixed ontologically in the past, but because the information has dispersed the settlement is not known: entropy is not zero.

Broad also argues that the laws of identity, contradiction, and excluded middle assert that any proposition is either true or false but do not assert that the number of propositions is eternally fixed. They only assert it cannot be diminished; it may be increased by the process of becoming, which continually augments the sum-total of existence. I take this as my starting point.

As a physicist my instinct is to look at entropy. The time t can advance without change, as with the inert rock. Reversible processes show motion without irreversible change. The fundamental laws of physics do not show an arrow of time; it must come from entropy. Relativity forbids a global “now” based on t but permits local change related to entropy. The “now” correlates with Lyapunov time and entropy production, not with coordinate intervals of time. I argue that events are not indexed to local time they are contingent on local entropy. There co-ordinates in spacetime are not settled. The B series has future uncertainty.

Broad’s coat analogy is useful here. He said he might need his coat at night but not during the day: the truth of Coat or not-Coat is based on local conditions of temperature and dampness. A local equilibrium distribution gives a factual and real account of the conditions. Temperature is not defined globally but within a local space. Entropy can also be defined locally and is the ratio of heat transfer to temperature, it is what has happened in a directional sense, and what is happening in the “now”.

The main argument

My argument is that McTaggart’s contradiction does not show that time is unreal. It shows only that time cannot be understood if change is modelled as the acquisition of incompatible intrinsic A-properties by events. Once change is relocated to a non-linear process, which is irreversible, the contradiction no longer applies. Thermodynamic and informational asymmetries explain why past, present, and future are differently accessible from within a V,  without implying that logic itself changes.

A committed eternalist might object that none of this escapes the block universe. Physical irreversibility, on this view, is simply a feature of the block’s internal structure: the arrow of time is baked into the four-dimensional manifold, and entropy increase is an asymmetry within it. My claim is about what irreversibility means when taken ontologically, not just descriptively. If the exact 4D location of a state remains uncertain, the physical state exists but has not yet been informationally integrated into V.

My claim is that a physical state becomes locally determinate when it is informationally integrated into V. It goes through a process filter, which is a non-linear filter when τ < infinity, to determine its location.  This alters a one-dimensional ordering, even time. The process filter is chaos, and it has a Lyapunov time period, τ.  It achieves this through irreversible processes, when the gravitational radiation has been emitted, the entropy produced, the quantum superposition resolved. Before that process has taken place, truth has no exact location, not because we can’t see it, but because the physical process that would determine it have not yet occurred.

A stone falls into a pool, and a frog sits on a rock nearby. The stone enters the water, but the frog does not register this immediately. The sight reaches him quickly, the sound reaches him later, and the wave later again. What counts as “now” for the frog is not the moment the stone hits the water, but the moment the information arrives. This shows that the present is not a sharp boundary shared by all events, but a locally constructed window shaped by when information becomes available.

The frog example illustrates the experiential now, the integration of information and the application of a non-linear filter to establish truths. It also shows the need for a coherence window, τ. It does not, by itself, refute eternalism. The ontological claim is that the information goes through another filter: a chaotic filter, a decoherence period τ. Before arriving at the frog, the ripple in the water might cause the bank to collapse.  This is a 4D uncertainty resolution process occurring in every V.

This defines a natural time window τ. Within this window, distributed information can be brought together and treated as one event. Beyond it, the signals no longer cohere, and what was a single event fragments. The present is therefore not a point, but a coherence window determined by the structure and timing of information flow through the system. There are two windows, a process path or decoherence window and an integration, or coherence window. I refer to both as τ, they are non-linear filters, and I do not always separate them, I treat them as the same information process. We take information and reconstruct or deconstruct the spacetime coordinates. One is an observer process the other a spacetime volume process.

I propose that if irreversible change exists inside a specific volume of space-time V, then in the past, within V, the physical state is fixed, while the information required to reconstruct that state has propagated beyond V and is no longer locally accessible. I do not claim that information is fundamentally erased but that it has dispersed into the wider universe. In the future, information has not yet arrived, local microstates are unresolved, and entropy limits reconstruction of physical states that make propositions true.

A critic might say that the Lyapunov time merely marks the limit of our predictive power, and that the future state is fully determined even if we cannot calculate it. In a purely classical system, that objection holds. But in a system governed by general relativity, where the Lyapunov time decoheres the location of events. It is not just a limit on our predictions; it jumbles the path to truth.

This is not a claim that logic itself changes. The logical operators remain classical. The objection that the future is determined but unknown only holds if the physical process is reversible. Irreversible entropy production means that V after the emission is a different from the V before it. The determination of truth is therefore constrained not just by our thermodynamic situation but by the thermodynamic structure of reality itself.

When two spacetime volumes V and V+dV interact, they exchange information through gravitational waves, electromagnetic radiation, and quantum correlations. This exchange updates the information available within each volume, and the determination of truth within each region becomes increasingly aligned as shared information grows. The apparent drive toward consensus is not a logical requirement but a physical result of how information flows, constrained by causality and entropy. This is what produces the observed arrow of time. As general relativity demands, there is not one “now” — every volume V has its own τ(V). As defined earlier, τ(V) is the timescale over which information about V’s microstate becomes irrecoverably dispersed beyond V. This τ(V) is the ‘now’ of that volume.

The Lyapunov time, a change-based duration rather than a coordinate interval, is a suitable finite time-window for τ. It defines a natural coherence or decoherence time over which information about the system can be reliably determined. For a classical system τ is infinite, for a rock almost infinite, for the asteroid race 4 hours, for a mind 3 seconds. It is worth being clear that τ plays three related but distinct roles. The first is τ as a dynamical timescale: the Lyapunov time of a given physical system. The second is τ as an experiential coherence window: the depth of the phenomenological “now” for a cognitive system, constrained by the dynamical timescale of the brain. The third is τ(V) as a local regional now: a decoherence or becoming window associated with a spacetime volume V under general relativity. The experiential τ is bounded by the dynamical τ of the underlying physical system, which is itself a particular instance of the more general τ(V). I have grounded this irreversibility primarily in GR and entropy, I avoid quantum decoherence which offers an independent and, in some respects, stronger basis, but is beyond the scope of this essay.

How does the argument align with my experience

My brain is a complex chaotic system. In such a system the uncertainty grows exponentially from a scale defined by the uncertainty principle to the local area scale on a timescale called the Lyapunov time, which for a human brain is of the order of 100 cycle times. I can hear a 16th of a note at 120 beats per minute and see a stable TV picture at 50Hz, which indicates a cycle time of the order of 30ms, making my Lyapunov time of the order of 3 seconds. This aligns well with my experience of the “now” as a process with depth.

The feeling that there is a “now” and only the “now” really exists, that it is where change happens, is exactly what is proposed here. The width of “now” varies with the complexity of events — as with the tennis player versus the amateur, or an earthquake versus lying in bed. It aligns with the feeling of anticipation and imagination as separate: the first fleeting, the second more stable based on memory. It aligns with surprise when something I thought had happened turns out not to have, as things resolve — a rank outsider won.

My sense of a flowing present moment is a response to my mind gradually resolving the causality and order of events. I am not creating a false belief about the nature of the world. I am recognising that my immediate prediction horizon is limited by chaos. My certainty often collapses at a hard border which is “now”. This is a real “now”, not just a process of my own mind but one of the many processes I sense.

I conclude my essay by stating that the “now” is real and returning to Oliver’s beautiful image, which is not only a decoration in this essay but captures the reality of the falling water, the process, and the always falling and always staying — the argument that a process and an identity are never in conflict.

References

1. Einstein, A. (21 March 1955). Letter to the family of Michele Besso. The original German: “Für uns gläubige Physiker hat die Scheidung zwischen Vergangenheit, Gegenwart und Zukunft nur die Bedeutung einer wenn auch hartnäckigen Illusion.”

2. Valtonen, M. & Karttunen, H. (2006). The Three-body Problem. Cambridge University Press; and Valtonen et al (2016) The Three-body Problem from Pythagoras to Hawking. Springer Cham.

3. J. Ellis McTaggart “The Unreality of Time” (1908) p457-478 the journal Mind (New Series, Vol. 17, No. 68)

4. Broad, C. D. (1923). Scientific Thought. New York: Harcourt, Brace and Co.

5. Oliver, Mary. “The Waterfall.” New and Selected Poems, Beacon Press, 1992.

Movement Happens in a two step dance

We are taught to think of motion as simple.

An apple falls because of gravity.
A planet orbits because spacetime is curved.
A particle follows the path that minimises the action.

In each case, the story is the same, it follows the laws of nature.

But there is something quietly missing from that picture.

At any moment, the present state of the world does not know a single future. The future is underdetermined.

So how does motion ever stabilise into a smooth trajectory?

Why doesn’t the apple jitter between multiple admissible futures?

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The usual answer is that the actual path is the one that minimises the action. Out of all the possible trajectories, the system follows the best one.

But that assumes something extraordinary: that all possible paths are somehow available for comparison, and that one is selected.

It quietly assumes that the trajectory already exists as a globally evaluable object.

Real physical processes cannot do this. They operate locally, incrementally, with finite information. The universe cannot step outside itself, survey all futures, and choose the optimal one.

So movement must happen differently.

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What if motion unfolds in two steps?

1. Projection:
   The system commits to a locally compatible continuation of its current state. Not the best one — just one that does not immediately violate its relational constraints.
2. Accommodation:
   Future updates must now remain compatible with that prior commitment. They cannot start from scratch. They must grow around what has already happened.

Once the first step is taken, the second step follows automatically: the history of commitments begins to constrain what can come next.

Each new moment must remain compatible with the last.

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A trajectory is therefore not selected in advance.

It is stabilised over time.

Smooth motion emerges not because the system has found the optimal path, but because it cannot afford to reconsider all possibilities at every step. The cost of abandoning prior commitments grows as they accumulate.

Inertia, on this view, is not simply resistance to change.

It is the persistence of unfinished consistency.

A moving object continues along its path because each new update must remain compatible with a growing history that cannot be undone.

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We often describe motion as the unfolding of a law.

But it may be closer to the progressive stabilisation of a commitment.

The apple falls not because it knows where to go, nor because the best path has been chosen in advance, but because once a locally compatible future is projected, the rest of reality must accommodate it.

Movement may not be a single step from cause to effect.

It may be a two-step negotiation between what is possible now and what must remain compatible later.

To most people, relationships are secondary—interesting relationships happen to exist between individuals. As an Irishman, I find relationships much more important, fundamental in fact. We are defined by our relationships. Philosophically, this is considered inaccurate, as it suggests relationships can define things.

In classical metaphysics, things exist and relationships exist between things. Things exist first, then relate. Relations are external or accidental properties of things. There is no reason why relations must be symmetric or reflexive.

Turning this on its head, what if relationships are elemental and things arise from relationships? We have an elemental relationship between nodes A and B that are not things. We don’t care what the nodes of our relationships are, and we choose not to name them, but we can apply our framework to any system that can be described by logically constrained relationships as fundamental elements.

Let us assume there is no outside observer, so no perspective from which asymmetric relations can be established. Let us also assume that we can give each node an identity and that identity is not altered by its relationships. Even if A relates strongly to B and C, A’s self-strength stays at unity. A doesn’t lose “A-ness” by spreading its influence. Identity is preserved regardless of relational activity.

In this framework, we can say that the relationships will obey logical symmetry. We call this a logical constraint on what relationships can exist; other relationships are prohibited. If A relates to B, that relation must simultaneously constitute both A and B. A can only exist through its relations to B, which requires B to relate back to A.

Are there other logical constraints that we can impose on our relational ontology? We can assume that reality will not allow inconsistency, so that if A relates to B, and B relates to C, then the relationship A to C must be fundamental. Strong indirect paths force corresponding direct paths to be strong. If A to B or B to C weakens, then that weakens A to C.

We now have three key logical constraints on our relationships: symmetry, transitivity, and reflexivity. These can be expressed mathematically with respect to a matrix B containing our relationship strengths.

1. Symmetry: B = B^T
   If node A relates to node B with strength 0.7, then node B must relate back to node A with the same strength 0.7.
2. Transitivity: B² ≤ B
   If A connects strongly to B, and B connects strongly to C, then A should connect to C at least as strongly. Here B² means matrix multiplication (paths of length 2), not squaring individual elements.
3. Reflexivity: diag(B) = I
   Every node has maximum strength relationship with itself equal to unity.