Contrariwise, if it was so, it might be; and if it were so, it would be; but as it isn't, it ain't. That's logic.
Lewis Carroll
What is possible?
The metaphysics of science says that everything is physical.
Yet there is talk of physical phenomena obeying laws. These laws being partially constrained by logic, and partially also by empirical evidence or calibration, as in the free parameters of a physical theory being fixed upon testing it against reality.
There is also talk of possibility. Of the actual being drawn from a partially constrained space of the possible, halfway between being completely arbitrary and completely determined. Probability can even assign a measure to this degree of possibility, with corresponding rules of statistical inference.
It is said that the physics of our universe is drawn from a space of possible physical theories, because a law of physics cannot contravene the rules of logic.
But if all there is is physical, everything must be actual. Where does possibility come from? How does logic come to dictate what could be? How is the possible, well, possible?
What is actual?
Before we deal with the matter of what could be the case, how did we even arrive at what is the case? How can we come to know things at all?
Dualism
Actually, we need to take a step back. What does it even mean to know? Dualism rears its head: we imagine ourselves apart from reality, our physical environment, and we imagine we hold a picture or representation of it in our head, which we can say is either faithful or not with respect to a certain correspondence with reality.
But this is a fiction: there is no representation, no magical aboutness in our theorising, only ourselves imagining there is. We simply interact with reality, and with the sense data coming in we construct a hallucination of a picture of what is out there.
Game
This however does not mean such a process of interaction with reality is arbitrary. It is an activity we engage with in accordance with certain rules. In other words, the natural process we partake in comes in the form of a game: a policy for engagement with the rest of reality outside of ourselves.
How does this work? As a child we played the game of naming the highest number. More naive than someone who does not yet know that it is possible to play a perfect game of tic-tac-toe, we imagine we can beat our opponent definitively: by naming a high-enough number. But of course, at some point we realise there is a general policy: just add one to your opponent’s number. With this we come to realise a certain property of the natural numbers: that given any number we can always add one. The game2 has been found.
And here is the funny thing: with only this modest newfound ability, the policy of always adding one, we imagine a certain picture of an infinity of natural numbers. This picture of course is never complete: if there is some picture of all the natural numbers, we can always again apply our policy of being able to add one, and then again we need a new picture. The game and the policy of moves is real, whereas the fictional representation of reality is not.
Logic
Notice too that this belief in always being able to add one is not true: we could be mentally disabled, or if you imagine a program which takes an input number and produces its successor number, the program could run out of memory, or simply your laptop could run out of battery. Or we could simply refuse to continue to add one, stopping perhaps arbitrarily at some point. Logic is after all just a game, and we can choose to stop playing at any time.
Nevertheless this belief in always being able to add one attests to a certain commitment, even if not quite faith or belief, to the policy of behaving-as-such. We elevate the policy to a principle of logic: the principle of mathematical induction. The commitment says we will always apply the logical principle where we can. The faith says that we can always do it.
Yet whence this faith?
Co-survival
History is written by the victors. But that is only obvious in hindsight, when those once-victors are no longer. Right now, the fact that you are here, that you have survived, means that you are yourself a victor, and that your narrative is no better than that of the previous victors which came before. Yet you are pretty sure it is correct aren’t you? You cannot imagine otherwise. Why is this?
It does not necessarily entail that you are deluded. In fact it may be an admirable thing: it could simply mean that you stake your very existence on your narrative. Actually, this often does happen, where a new dominant power invents a narrative to justify their position. Cynically promulgated at first, funnily enough in some sense they eventually come to believe it. This is when rent seekers within their ranks take the calcified word for gospel, cynically optimising on the narrative to gain power for themselves. To some extent this can corrode the institutions which perform the actual non-narrative work of sustaining the collective. Despite that, the narrative does perform some coordination function, since otherwise the rent seekers would not be able to get away with it, because those doing the actual work would not put up with it. In a strange way, the collective finds its existence staked on the “truth” of the narrative.
Now this is not quite “truth” in the ordinary sense of verisimilitude, because as we have seen that is but a fiction. Nevertheless, in perhaps what seems a perverse way, this “truth” remains a non-representational active governing principle in the functioning of the collective.
Well, with logic it is the same: you cannot imagine doing without it. Which is to say, you stake your very existence upon it. Again you protest: logic just seems necessarily true, how can it be otherwise? But that is exactly the point, you cannot imagine otherwise, precisely because you have staked your very existence upon it: if logic doesn’t work, then you cannot exist.
No certainty
When a river changes its course, entire ecosystems it once nourished are no longer.
Certainty exists along a spectrum. Deposit insurance depends upon the solvency of the insuring body. Property and land titles depend upon the integrity of the legal system. The soundness of the currency depends on its stewardship by the central bank. We can conceive of all of this as a form of co-survival: your survival is guaranteed, provided that the guarantor also survives.
In logic too there is a notion of relative consistency. Due to Godel incompleteness we often cannot prove a theory to be absolutely consistent, i.e. free of contradictions. But we can prove it consistent relative to other theories. For example, we may not be able to prove the absolute consistency of spherical geometry (a non-Euclidean geometry) but we can embed it within the sphere in Euclidean space, thus yielding a relative consistency proof: if Euclidean geometry is consistent, then so is spherical geometry. We trust Euclidean geometry, and so this increases our faith in non-Euclidean geometry. Similarly, many theories reduce to the consistency of arithmetic. But the consistency of arithmetic we cannot absolutely prove. It is an article of faith, or should we say, we co-survive with our belief in arithmetic.
God created the integers, all else is the work of man.
Kronecker
It is selection all the way down
But it is a fair question: you did not choose to believe in logic, how did you come to do so?
In Darwinian selection by natural selection, a species adapts to its environment by variation and selection. The variation is random, but what gets selected is beneficial to the propagation of the species.
But wait a minute, there is no species. Only some persistence of pattern we observe in natural phenomena, after-the-fact of brute selection. Natural selection is a tautology: what survives is what survives. And whatever so happens to survive we call the lineage and the species.
Of course what survives isn’t arbitrary, and neither is logic. Logic is the way it is, being laws of thought, because it is somehow deeply functional in the operation of the human organism.
To take a less widespread example, some RNA sequences involved in transcription and protein synthesis are highly conserved7, meaning they are identical or similar within a genome or even across species. This is of course owing to their crucial function in the organism. With DNA you can express genome sequences which give rise to all the possible species on earth, but all that rests upon a certain framework of DNA transcription and protein synthesis. Similarly, you can think all the possible thoughts in heaven and earth, but you cannot escape having to do so under the framework of logic.
Logic may or may not be only a legacy inheritance. There is the question of whether an intelligent alien species would share the same mathematics or even the same logic as us. Would such an alien species also share the same framework of DNA/RNA coding and protein synthesis? Closer at hand, we do know that convergent evolution8 happens, where for example unrelated lineages independently arrived at a certain design for the eye. The eye being the common solution to the problem of taking in sensory data in the form of light. Logic may be just such a common solution to the problem of reasoning in general.
But of course, there are other ways of processing light, and other means of taking in information from the environment. And so there are other logics10 as well! And also of course, other forms of reasoning not involving logic at all.
Like how nature sometimes chooses to dispense with eyes altogether.
Leverage
It seems the further we go back in evolutionary time, the more fundamental or entrenched is what was selected, in the sense that what was subsequently selected is selected within the context of what was already selected before. There is a co-survival relationship between what is later selected and what was previously selected.
To take an economic example this time. Your job depends on the structure and operation of your company. Your company depends on the industry it is embedded in, with its network of suppliers and producers. That in turn depends on a certain legal framework for conducting business, and also a certain financial framework for how to deal with money. Which depends on a certain belief in the value in a capitalistic organisation of society. That yet again depends on a certain myth of the inherent worth of the individual and their freedoms.
Notice here that the further back or up we go, the less we are able to justify why things should be the way they are. This is because we can only justify what is more recently selected against what was already selected before. At some point, some existential thing, like DNA transcription or democracy, was selected simply because.
When a company borrows money to fund its operation, we say that it is leveraged, because the value of the additional available money is conditional on existing funds, situated within the existing structure and function of the company. But as we have seen, it is leveraged on more than this. It is leveraged also on the industry, on the economy, on the legal and financial systems, on capitalism, on individualism, on dualism, and of course, on logic.
The hitchhiker's guide to the entire universe
The analogy of genetic hitchhiking13 is instructive. When a trait is being favoured by natural selection, the beneficial mutation which codes for it gets selected for, i.e. its frequency increases. But not only this, its genetic neighbourhood gets swept along too, in what is called a selective sweep14. This is because recombination has a certain granularity to it, and it is a certain segment of the genome, containing multiple sites of genes, which increase in frequency, tagging along on the work done by the beneficial mutation.
There are some pretty dramatic examples: there is a single version of the Chr1a gene for all strains of Toxoplasma gondii across Europe and North America, meaning at some point in the past, a novel genotype containing Chr1a appeared and swept the entire Toxo genome, dragging it along and eliminating all other variation. All current variation in the Toxo genome, or we could say, Toxo itself, can be said to be co-surviving with that single Chr1a gene variant.
Taking a step back, the entire universe may be in a false vacuum15 state, with the existential threat of vacuum decay, changing the fundamental forces and even the possibility of life itself. Or it could be the cosmological origin of the universe itself, where the quantum fluctuations of a metastable false vacuum with no time, space, or matter, gave birth to the true vacuum of our universe as we know it today.
Zooming out, a picture emerges where the dynamics of the universe gives rise to an arbitrary regime of selection. At the highest and coarsest level, a selective sweep which sets the existential conditions upon which everything else, being merely hitchhikers along for the ride, rests. And this pattern repeats at a lower and finer level. And repeats again. So that there are selection regimes within selection regimes.
Possibility is local
Raison d’etre is local.
Notions of what is possible are local, since what is true or useful depends on the context. Your professional values may mean nothing in the wider world of business within which the company you work for rests, where an innovation can disrupt your once cherished means of delivering productive value. Business values, such as non-zero sum entrepreneurial competition, mean nothing in the realm of political power where regulations which favour certain firms or industries are passed which run roughshod over what actually generates wealth. Power in a democratic institutional context is in turn meaningless when it comes to war, where military power simply ignores the political rules for engagement. Nature doesn’t care for military might, as Napoleon found out in his invasion of Russia.
Crime and punishment
Within some circumscribed domain we seek to establish rules, and lay down the law. But even if you may try to coerce something, you cannot decree its behaviour, since it can simply decide to disobey. You can only set up incentives, and punish disobedience.
Physics
Physical law is no different. Remember, our attempts at restricting its behaviour must work with the material we have at hand: there is nothing outside of the universe, only what is already within the universe itself.
Take for instance again vacuum decay leading to a change in physical constants (nevermind for the moment that we are again really with our theorising, merely reflecting the behaviour of some small part of what’s within, to model the whole, in our attempt to decree the behaviour of the universe). Upon decay, physical law changes and phenomena will behave differently: they become unruly, they disobey.
Of course, there may be some higher order, like how there are reinsurers for insurers, or bigger gangsters or warlords offering protection or government services. But I suspect, at the highest level, all is chaos.
Logic
Even beyond physical law, our dualism leads us to decree the space of possible physical laws. Such conceit!
When you are learning to play a new game and you are still figuring out the rules. Based on what you have observed within the game itself so far, you have some idea of what the ultimate rules are, but you are not completely sure. Do you simply declare that the space of possible game rules is encompassed by the candidate rules not yet ruled out? No! And this is because you have experience of other games, and you know that the space of possible game rules is far greater.
It is only because some part of the universe, in reflecting the whole, underdetermines17 it, that we have the phenomenon of possibility at all. This is akin to syntax under-determining semantics, but having the gall to decree that it exhausts all possible meaning.
Logic exists within reality, and does not operate outside of it.
Mathematics
What about mathematics? That supposedly non-physical game we play par excellence?
Even there the attitude of structuralism18 has been on the rise, and seems to have become mainstream. Mathematicians used to worry about the ontology of their universe, larger than our physical universe, of the mathematical objects they work with, at one point trying to base everything on set theory. These days they are happy to work with things only up to isomorphism. That is, they have stopped decreeing all that can possibly exist, and content themselves working locally with whatever is around them.19
The notion of an isomorphism itself is based on two mutually inverse morphisms. A morphism is often thought of as a structure-preserving mapping. But the operation can also be conceived of dynamically: performing a sequence of actions in the source domain, then translating the result to the target domain, is equivalent to translating first your starting point to the target domain, and then performing the analogous sequence of operations in the target domain. That is to say, the physical game we play is preserved by the act of translation, also physical.
All there is
Within the universe, possibility is possible, nay, in fact actual: it is the case that some things might be the case. This is but a useful game we play inside of reality, and it helps us make sense of things, come to decisions, and take action. Outside of reality, there is nothing. Not even what reality could be. Reality just is. Possibility is ultimately impossible. Everything simply is.
https://www.behance.net/gallery/63695833/Penrose-Triangle-Illustration/modules/373519441
https://en.wikipedia.org/wiki/Game_of_the_scene
https://gusandco.net/2013/01/15/le-meilleur-jeu-de-tous-les-temps/
https://www.vox.com/2015/2/5/7986829/river-meander
http://www.drmarkliu.com/noneuclidean
https://en.wikipedia.org/wiki/Conserved_sequence
https://en.wikipedia.org/wiki/Convergent_evolution
https://www.phos.co.uk/journal/the-evolution-of-sight
https://en.wikipedia.org/wiki/Non-classical_logic
https://www.nationalgeographic.com/animals/article/150911-blind-cavefish-animals-science-vision-evolution
https://inequality.org/great-divide/some-leveraging-inspiration-from-old-archimedes/
https://en.wikipedia.org/wiki/Genetic_hitchhiking
https://en.wikipedia.org/wiki/Selective_sweep
https://en.wikipedia.org/wiki/False_vacuum_decay
https://gameofquotes.blogspot.com/2014/03/petyr-baelish-knowledge.html
https://en.wikipedia.org/wiki/Underdetermination
https://en.wikipedia.org/wiki/Structuralism_(philosophy_of_mathematics)
https://golem.ph.utexas.edu/category/2015/02/introduction_to_synthetic_math.html