Why the Universe Exists - the Short Answer
by Zeb G. (April 2008)

[category: cosmogony]

Heartfelt but unauthoritative speculation
about the nature of time, space, mass,
force, causality, particles, and reality.

Introduction to the Answer

'If logic alone somehow required the universe to exist and to be governed by a unique set of laws with unique ingredients, then perhaps we'd have a convincing story [of why there is a universe at all]. But, to date, that's nothing but a pipe dream.' - Brian Greene, The Fabric of the Cosmos, 2004

A pipe dream maybe, and perhaps understandably when considered in the context of the rigorous demands of sound scientific theory, but can we envision an answer, by using loose reasoning and proposing probably incorrect details, in order to fill the gaping hole where any answer at all seemed simply unimaginable?

Here follows a vision of how it should happen that the universe can and must exist (one with many failings no doubt).

The Nature of Time

If we insist upon preserving our intuition that time is truly fundamental, then we will never explain it, and we destine ourselves to find it forever mysterious. To what then might our conception of time truly refer?

Let us imagine what might happen if everything in the universe suddenly froze still, say for a few seconds. How would we know, when life in the universe resumed, how much time had just passed? What if instead of a few seconds, it was an hour, or a day, or a year? What if it was a billion years, and it had happened right now, halfway through this sentence? Clearly the passage of time in such a frozen universe has no real meaning. It's not observable and so we should question whether it can have existed at all. The conclusion appears to be that time in a universe devoid of change is meaningless. Could it be then that it is change, not time, that is fundamental, and that our concept of time is only derived from our experience of all this inevitable change going on in the universe?

Such a concept of time as a phenomenon derived from change lends itself to explanation. Change might be understood in an abstract mathematical sense and might be determined to be the necessary consequence of yet more abstract mathematical truths.

The Nature of Space

Likewise, if we insist upon preserving our intuition that space is truly fundamental, then we will never explain it, and we destine ourselves to find it forever mysterious. To what then might our conception of space truly refer?

Let us first consider any entity that we might find in space. It is related to every other thing in space by its separation from every other. In fact, we might pinpoint its relationship by noting those separations. Could it be then that it is this relationship, not space, that is fundamental, and that space only derives from the character of this relationship?

To see how this might be the case, consider the following. The Four Colour Theorem is a proof that, for any pattern of regions that might be drawn on a piece of paper, four colours will only ever be needed to ensure that no regions with a common border have the same colour. This then is the two-dimensional case. For an analogous situation in one dimension, one can imagine needing to colour different lengths along a piece of string. It should be clear that in this case, no more than two colours would be needed (unless we join the ends of course). What about three dimensions? If I were to stick blobs of modelling clay together, how many different colours of modelling clay would I have to use in order to ensure that no two blobs of the same colour ever touched? The answer, in fact, is that as many colours as regions may be required! To see this, imagine a scarf with coloured stripes along its length. If we fold the scarf over on itself so that one end of the scarf makes a right angle with the other, making a neat forty-five degree crease in the scarf, then by tracing along one stripe at a time, we notice that each stripe now comes into contact with every other, and that therefore we require every stripe to be of a different colour. This seems to suggest the conclusion that the type of relationship in which every entity might be related to every other is at least three-dimensional in character, and that it is this inter-relationship, not space, that is fundamental.

Such a concept of space as a phenomenon derived from the character of this inter-relationship lends itself to explanation. It might be understood in an abstract mathematical sense and determined to be the necessary consequence of yet more abstract mathematical truths.

The Nature of Mass

And likewise again, if we insist upon preserving our intuition that mass is truly fundamental, then we will never explain it, and we destine ourselves to find it forever mysterious. To what then might our conception of mass truly refer?

The everyday items with mass that we are familiar with are composed of particles, which contribute their individual masses to the total, but the particles are nothing like the large items they compose. Whereas the everyday items are unquestionably real, it has been discovered that the existence of particles depends upon the frame of reference from which they are observed. Whereas everyday mass seems tangible and real, the reality of the particles from which the mass should derive seems less certain. Could there be something more fundamental than mass?

Heisenberg's uncertainty relation asserts the fundamental uncertainty of position and momentum of any particle, but can we interpret further meaning from this relation by turning it around? Could it be that certainty of (spatial) relationship is a phenomenon we recognize as the presence of mass and that certainty of change is a phenomenon we recognize as the presence of energy?

Such concepts of mass and energy as phenomena derived from uncertainty lend themselves to explanation. Whereas mass was mysterious, fundamental uncertainty might be understood in an abstract mathematical sense and determined to be the necessary consequence of a general demand for consistency - a universe which is finite in every sense must be finite in both the large and the small scale.

The Nature of Force

What is this mysterious thing we call force? The way things seem are not always the way things are. It seems that we look out of a window, but it is the light from outside that passes through the window and impinges on our eyes. It seems that we are thrown outwards by a merry-go-round, but it is our feet that are accelerated towards the centre of the merry-go-round by its rotation. It seems that we suck a drink up a straw, but it is the greater pressure of the atmosphere that pushes down on the surface of the liquid and forces it up the straw into the slightly lower pressure in our mouths. Things are rarely exactly how they seem. Force is not as it seems.

It is said that anything is possible, but only that which is not mathematically impossible is possible. That means that reality is constrained. It appears to obey rules. Those rules give rise to phenomena that we recognize. Force is just such a phenomenon. It is the character of that pattern of constraint.

The Nature of Causality

What is this mysterious thing called causality that compels one thing to follow another? We have noticed reliable patterns of change, and we have abstracted these patterns into the concept of cause and effect. But as with force, this is a phenomenon that results from the fact that reality is constrained by what is mathematically possible. Change is constrained and therefore change occurs in patterns. And change is inevitable.

The Nature of Particles

What are these weird things we call particles? Everything is made out of particles, but particles are nothing like objects of everyday reality. The nature of their existence, and even whether they exist at all, seems to depend upon how we observe them. And while no everyday object is absolutely identical to another - when you look closely enough, there are always fine differences - particles are not like that; they are absolutely identical. All that distinguishes them is their quantum state. Individual particles have no unique fine details; they have no texture.

Particles are as close to abstract objects as you might hope to find. They can be completely described by their quantum state - just a set of numbers. Since a set of numbers is all that distinguishes them from each other, is it such a leap to imagine that a set of numbers is all particles are?

The Nature of Reality

But all this must sound rather odd. We might one day understand time, space, and mass as the logical consequences of certain as-yet-undetermined abstract but self-evident mathematical truths, but time, space, and mass can't really be just mathematical truths themselves... or can they?

Physicists have long wondered about the unusual power of mathematics to explain and predict physical phenomena. Added to this is that, to mathematicians, it has often seemed that mathematical truths have an existence of their own, and they have been left to ponder to what extent they are creators of mathematics and to what extent they are just explorers discovering it. Might mathematics be a mixture of physical truths, which can truly be said to exist, and purely imaginary concepts, which can exist only in our minds?

To shed light on this question, let us consider a concept in mathematics that is implicit in the foundation of calculus and in the foundation of set theory. That concept is infinity, yet there has never been any observational evidence for it, in any shape or form. In calculus we imagine taking ever thinner slices without limit, something that the uncertainty principle assures us is not physically possible. In set theory, we assume the existence of sets that already contain all of an infinite number of members, the natural numbers for example, whereas if we were to realize the set of natural numbers physically, we know we would always find a new member to add to it. Infinity then, without much doubt, is a purely imaginary concept - something that is not obviously apparent for concepts such as pi.

Might pi (the mathematical constant that tells us how many times the diameter of a circle fits around its circumference) be an abstract truth with a physical existence? Certainly any time we come across a wheel, we can measure a value of pi, but we will never determine a final exact value. Mathematics proposes to offer an exact value of pi, but this exact value is pointed to by means of a power series - a series with an infinite number of terms that must be summed. Mathematics, with its blurring of the physical and the imaginary, proposes a concept of pi that is fixed, whereas, denied of the reality of the concept of infinity, we see that the value of pi must in fact be dynamic (and the same might be said of any irrational number).

This has raised an interesting point. Whereas it might so far have seemed that the only dynamism in mathematics was provided by the mathematicians themselves while engaged in the act of doing mathematics, we now see a hint that this may not be the case. Might time, space, and mass be the logical consequences of, and therefore themselves in essence be, such dynamic non-imaginary mathematical truths?

The Universe

Are we able now to propose in principle why the universe exists? If infinity is denied a physical reality, then the universe must have had a beginning. And if time in the absence of change is meaningless, then the universe must have begun with change. If change is the necessary consequence of more abstract mathematical truths, then given those truths, change was and is inevitable. So it seems that we might have reduced 'why does the universe exist?' to 'why do abstract mathematical truths exist?'

So why do they? Just before we tackle that question, let us first note that this appears to be a rather convenient situation in which to find ourselves because abstract mathematical truths depend on nothing for their existence but consistency with each other. Separate from purely imaginary concepts, which require minds in which to live, such truths don't require a universe in which to be true, and their truth is mutually self-explanatory. All that remains to be said is that the leap of thought required is to realize that they are the universe, inseparable from everything within it, the origin and essence of every phenomenon, mental or otherwise.

'The ultimate theory should take the form that it does because it is the unique explanatory framework capable of describing the universe without running up against any internal inconsistencies or logical absurdities. Such a theory would declare that things are the way they are because they have to be that way. Any and all variations, no matter how small, lead to a theory that - like the phrase "This sentence is a lie" - sows the seeds of its own destruction.' - Brian Greene, The Elegant Universe, 1999

With this understanding of the nature of the existence of the universe, we can see now why attempts to interpret, for example, quantum theory in ways that fit well with (less abstract) everyday intuitions about reality are fundamentally flawed. Each deeper level of physical understanding will be necessarily more abstract than the last until we arrive finally at abstract truths, the basis and justification for which will be not statements of principle (though that is how we presently think of such truths), but that when their non-truth is hypothesized, it is shown always to lead to unresolvable contradictions.

A Finely Tuned Universe?

It has been remarked that if the fundamental constants were even very slightly different, the universe would be a very different place, one where intelligent life would likely not have had a chance to evolve. And it has been asked how then were we so fortunate as to find ourselves in such a special universe. And in answer to this question, it has been proposed that our universe is not the only universe, but one of many in which the fundamental constants (and possibly the laws of physics) are all different, and that therefore our universe isn't so special after all but only seems that way because we are here to observe it.

But all those universes (if they existed), with their vastly different phenomena arising from their different fundamental constants and possibly even different physical laws, would have something absolutely in common. Mathematics!

Mathematics because its patterns and it constants depend on nothing in the universe. In ancient times, pi was believed to be a physical constant that had to be measured, but in the Middle Ages it was discovered how to derive the value of pi to any precision without making any measurements at all, and pi was realized to be a fundamental mathematical constant. The power series and the general and simple continued fractions of mathematical constants such as pi and e show patterns that would be the same in every possible universe - because they are not derived from measurements or anything physical at all (so the physical differences in other universes would make no difference to them).

The values of the physical constants depend upon the units in which you measure them, but there are combinations of physical constants in which the units cancel, and so the value determined is independent of the units in which its constituents were measured. These dimensionless physical constants are like pi in ancient times, waiting for us to discover how to derive their values to any precision without making any measurements at all.

So if all these other universes existed, it could be the case that no two universes were exactly alike, and that the only thing common to all of them were the abstract truths and patterns of mathematics. If mathematics is the only thing that absolutely must be common to all of them, then could mathematics be the ultimate origin of all of them?

And just supposing that this is the case, we can dispense with all the other possible but unobservable universes because in that case the so-called physical constants are not finely tuned at all but mathematically necessitated.

Why does the universe exist? Because it is the not unexpected consequence of what is mathematically necessary - and one day soon we will be able to demonstrate exactly that.

'We don't invent mathematical structures - we discover them, and invent only the notation for describing them. [...] Everything in our world is purely mathematical - including you.' - Max Tegmark, Reality by Numbers, New Scientist, 15 September 2007

What this means is that what we have so far been calling mathematics and thinking we understand is broader, more encompassing, and more awesome than anyone ever imagined. But how could this be?

Godel's Calamitous Resort to Metamathematics

In the introduction of 'On formally undecidable propositions of Principia Mathematica and related systems', 1931, Kurt Godel argues verbally, but does not prove rigorously, that a theorem which is undecidable within the formal mathematical system has been decided by metamathematical considerations, essentially that 'Theorem A: A is not provable' is a correct theorem rather than an incorrect one. But there is a problem.

It could also be conjectured rather that theorem A is inconsistent and is therefore incapable of being either correct or incorrect. Given the history of Godel's paper, it would seem that an inconsistency in theorem A, if it exists at all, must be a subtle one to have gone unnoticed for 77 years! But we decide to entertain the notion and ask whether Godel cannot prove rigorously what he has argued metamathematically.

And what do we find? We find that Godel has already answered this question. He has rigorously proved that theorem A cannot be decided in any consistent formal mathematical system. But could theorem A be decided in an inconsistent formal mathematical system? Can anything confidently be decided in such a system? Godel has shown it to be unlikely that he could ever rigorously prove what he has argued metamathematically and, by doing so, would appear to give support for the contention that theorem A and its ilk cannot be decided in any system at all. In other words, theorem A is inconsistent and therefore incapable of being either correct or incorrect.

The popular interpretation of Godel's result then becomes not that 'any reasonably expressive formal mathematical system is capable of expressing correct theorems that cannot be proved', but instead that 'any reasonably expressive formal mathematical system is capable of expressing inconsistent theorems which, since they are incapable of being either correct or incorrect, can neither be proved nor disproved'. The latter would seem to be a rather obvious result and a far from discouraging one, in stark contrast to the former.

And so as a result of this unprecedentedly subtle misunderstanding, for over three quarters of a century now, mathematics has been wrongly thought of as to be somehow lacking in something essential, and has therefore been deposed from its rightful place at the throne of ultimate truth... and ultimate reality.

Physics is Impossible without Observation

The ultimate origin of the universe may be essentially mathematical, but there is a catch. We can recognize that there must be both truths that hold only in certain special circumstances, and other more general, more abstract truths that hold in broader circumstances, but can we prove that there are truths that are so general and so abstract that they hold in every possible circumstance?

Mathematics works by identifying basic principles that are believed to be true without proof and then formally labelling them as the axioms of the mathematical system. Everything that can be proved in the mathematical system is essentially a restatement of those axioms.

The problem is that our confidence in these axioms relies upon the belief that there can be such a thing as a priori knowledge, knowledge that can be acquired irrespective of experience. But just because such knowledge might in principle be acquired irrespective of experience, this does not ensure that we will in fact acquire it without error or flaw. How then can we be sure that our mathematical axioms are not just very general truths rather than the completely general truths we sought?

The answer is that we can allow observation to guide and inform our mathematical inquiry. Observation can inspire challenges to the accepted axioms. Perhaps one plus one does not equal two in universal circumstances. Perhaps in complete generality, probably one plus probably another one equals even more probably at least one. Or perhaps neither of these has it right and the actual case is so unexpected that it has yet to occur to anyone. Can it be that the bewildering physical discoveries of the last century show to be naive the current mathematical assumptions that we have held to be inviolate? We should be wary of our propensity for dogmatic adherence to traditional assumptions.

The only thing we can say with upmost confidence in a universe whose existence resides in the abstract truths we seek is that by means of observation, the universe itself will be the final arbiter. Then one day, we might finally achieve a precise understanding of our origin in infallible abstract truths.

What is Truth?

Given the importance of abstract truths to our explanation of why the universe exists, we should if possible be absolutely clear about exactly what truth is.

Philosophers have long argued about the precise nature of truth without reaching a consensus. And to an even greater extent, the same may be said of existence, which has spawned an entire discipline formally referred to as ontology. But given the insights we have made, we find ourselves in the uniquely privileged position to solve two great riddles in one fell swoop. Truth and existence are equivalent.

Such an understanding of truth to be one and the same as existence must be disambiguated from the notion of true statements. A true statement is only a statement that is judged to refer to something that was or is true, holds true only in certain implicitly agreed circumstances, or is perhaps mistakenly believed to be true. By contrast if something exists, it is uncontentiously some form of a truth, but we have gone further and propose that such existence is no more than nor extra to the truth to which it is precisely equivalent.

The equivalence of truth and existence and the assertion that consistency necessitates certain truths provide a rational explanation for why the universe exists.

Paradoxes versus Truth

It is popularly held to be the case that a statement is either a true statement or a false statement, and correspondingly that either a statement is true or the negation of that statement is true. A certain type of paradox is typified by the perplexing situation when neither the statement nor its negation appear to be possible. Two of the most famous examples of this type of paradox are the Liar's paradox, 'This sentence is false', and Russell's paradox, 'The set corresponding to the property “being a set that does not belong to itself”', and the Liar's paradox is just one instance of a large class of similar paradoxes referred to as the insolubilia.

Our instinct that truth and existence are equivalent leads us to suspect that these paradoxes must be an artifact of statements rather than be indicative of the fundamental nature of truth. Since we have, on the one hand, truths that exist and on the other hand, statements which are intended to be about such truths (rather than mere flights of fancy), we realize that every statement implicitly includes an assumption that it is consistent with that to which it is intended to pertain.

The breakthrough comes with the realization that there may be occasions when this assumption is not valid.

A common property of the paradoxes mentioned above is that they contain a self-reference of some sort. Therefore, we realize that the general assumption that a statement is consistent with that to which it is intended to pertain is an assumption that statements that contain a self-reference are self-consistent. If we make this assumption explicit in their expression, then when we consider the negation, we are presented with the possibility that the original statement is not self-consistent. Now we find the original statement is still impossible, but the negation has suddenly become perfectly possible. The paradox is resolved.

Using as our example the Liar's paradox, the assertion of the paradox is transformed from '(This sentence is false) is true' to '(This sentence is false and self-consistent) is true'. Now when we consider the negation, we obtain '(This sentence is false and self-consistent) is false', which is equivalent to '(This sentence is false) is false, or (this sentence is self-consistent) is false', the latter of which is perfectly plausible. The paradox has vanished.

This might seem like a trick, for one might think one could substitute anything for the words 'self-consistent' and in so doing, provide a similar escape from the paradox. But the point is that the assumption that we have made explicit in order to provide this escape is one that absolutely cannot be refuted.

Every statement implicitly includes an assumption that it is consistent with that to which it is intended to pertain (or else it is by that omission just an outpouring of the imagination).

The long history of the paradoxes mentioned might demonstrate, if nothing else, what a struggle it can be to recognize what in hindsight is patently obvious. Therefore perhaps the essence of the answer to why the universe exists is in fact no simpler than we should have expected.

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