What do religion, truth and lies, philosophy, fiction and math, intellectuals, journalism, propaganda, cults, politics and mental health have in common? The humble axiom. And its errant son, the profound insight.
All rationality, all reasoning, all the things that a human mind can think– are bounded. That means that they work only within the boundaries set by their axioms or central assumptions, and break down outside of it.
Axioms are the fundamental basis of every field of human knowledge. They are statements that you consider as self-evidently true, that cannot be broken down into smaller parts and so do not require direct proof. For example, the entire field of Euclidean geometry is derived from just 5 of them:
1. A line can be drawn from a point to any other point.
2. A finite line can be extended indefinitely.
3. A circle can be drawn, given a center and a radius.
4. All right angles are ninety degrees.
5. If a line intersects two other lines such that the sum of the interior angles on one side of the intersecting line is less than the sum of two right angles, then the lines meet on that side and not on the other side
Axioms are used to generate “models” of a world. “Models” are abstractions or close-enough-approximate ideas of the world, of its systems or sub-systems, because humans can’t imagine worlds and systems in their full complexity. Models are always incomplete, waiting to be improved or replaced.
Models can be explored and tested in many ways, and the best ones produce insights, which are ideas that might provide a deeper, more accurate or more useful understanding of some aspect of the “real world.”
For example, the usefulness of the insights of Euclidean geometry were judged by how well it corresponded to and competed with other models of the world, and ultimately reality.
It helps you understand shapes and angles, it helps you farm, craft things with your hands, build homes and fight wars. Its insights were valid and useful across various ideas and real-world fields; so, they were “real” insights, now superseded by more advanced forms of geometry. These kinds of “real” insights are often found when mathematics is applied to material and practical problems, leading to the feeling of “the unreasonable effectiveness of mathematics”, as the physicist Eugene Wigner called it.
Science, more aptly called “the scientific method”, stands out in that it does not have “axioms” in the way that other fields do. That’s because it’s not a field. However, the philosophical and psychological frameworks that allow the scientific method to be considered valid and worthy do have foundational axioms. So, what the method results in are statements of probability about sets of data and observation, with numerous related ones coming together to form a “biology”, a “chemistry” or a “physics”.
Perhaps more controversially, the reason that atheists cannot “prove” that god does not exist is because their model of the world competes with religion’s model of the world; they can disprove specifics, they can highlight abuse and incoherent conclusions and practices, but they haven’t convincingly replaced the ancient, accessible axioms that lead to a religious mind and worldview. So one can deconstruct it to its foundations, but it will simply build itself back up again– differing only in those specifics that are dependent on the trends and whims of the time.
Philosophy, practical morality and law are also “man-made”, in the sense every moral and principle you know is derived from a few axiomatic morals or principles which may or may not be considered divine. As GK Chesterton put it, “Art, like morality, consists of drawing the line somewhere.”
A writer uses axioms to construct the world you will inhabit when reading their work, especially when it comes to magic systems in fantasy.
So, when they pull something out of their hat to write themselves out of a corner, we notice and are pulled out of the flow, even if the rules were only implicitly established.
We also see it in our narratives of ourselves. We have certain assumptions of our personality, borrowed in significant quantities from what others have told us, and we derive our opinions of our own actions from those core assumptions, for better or for worse. They all come together to form a complex gestalt of mental health.
So, it is the basis for sanity as individuals and societies— without it, reason is unbounded and circular.
Axioms do not prevent you from making mistakes in your reasoning. They are also not proof. Instead, they are a way to bring multiple parties on the same page, create a set-point in uncertainty, and illuminate some kind of path forward.
For example, “All men are born free and equal” is a foundational statement for modern democracy, and ““the universe was created by God” is one for many religions.
Humans are drawn to coherence and meaning, to finding the rules and principles that generate cause and effect and establish something as real, and to finding the metaphors that relate it to our actual lives and ambitions.
When you are following instructions or being guided by someone or something, the components of the world those instructions live in are constructed for you, and your imagination follows the rules set by the guide.
Within any such model, no matter how good or bad, a sudden insight can “feel” like enlightenment, revelation or mastery over a concept. However, that feeling is not reliable, it’s just the brain’s reward mechanism for figuring something out. And novelty, especially. In truth, if that insight and the model in which you derived it does not compete with other models, does not match reality, and is not useful, it’s probably a fake insight, to rudely co-opt a phrase from David Chapman’s Meaningness metablog. Your brain provides you with a heady little injection of “wow, this feels profound” juice, but we can call it “fake” because the actual insight is often utterly useless outside of that artificial model.
The mechanism of this “feeling” can be compared to Déjà vu, where one is certain that the present has happened before down to every detail, and though we know that is not possible, our mind convinces us that time is really repeating “again”. Then after the moment passes, we notice all the obvious differences, occasionally discover the trigger, and soon forget the entire experience.
The mind has many such glitches in the matrix.
These features and flaws of the mind are exploited by sociopaths, narcissists, gaslighters and well-meaning suckers: to construct questions in a way that guide you to the answer they want, and create an emotional world around you that results in you feeling the emotions that they choose for you. This is so easy to do that it has almost certainly already happened to you in at least one subject of your interest, and probably more, and the subject most probably relates to your ideas about yourself and your political, religious or social beliefs. It is so easy to do that it has probably happened to these very people who do it others.
For instance, in a meditation, yoga session or Satsang conducted by such a teacher, or an article written by an intellectual or journalist skilled at fooling you and themselves, you might find yourself having multiple “breakthroughs”– shocks of clarity or profundity, recovered memories, release of tension, unexpected connections, the thrill of new knowledge slotting into place, a glimpse of god or chaos, overwhelming empathy, sadness or connectedness. It may shock and awe, and send shivers down your spine or spread prickly warmth along your limbs.
In truth, there’s a pretty good chance this is just “wow this is profound” juice. Many such insights are useless, evidenced by the fact that they almost never lead to tangible changes that others can perceive (and it is only others you trust who can judge whether you’re a “better person”) or tangible solutions that appeal to the various competing parties affected by a problem.
The more “complete” the experience around getting them– covering design, rhetoric, statistics, mystery, build-up and pay-off, charisma, emotion, eloquence, unusual sensations– the less likely we are to notice that the underlying model itself is nonsense.
A disturbing-yet-freeing realization is that we often do it to ourselves even when it is actually under our control: a hallmark of the depressed or anxious mind is to produce unsound axioms leading to unsound worldviews, resulting in even more unsound insights to which it attaches unwarranted importance.
Some of frustration and ill-temper debates cause in the last century or so are based in our skillful use of multi-layered abstract concepts and models built up over millennia, and in ignoring or being misled on their axiomatic foundations.
So naturally, our next questions are: “what do we do about it? How do I identify a real spiritual phenomenon, real knowledge and flashes of real insight?”
Well, if the universe could be generalized to a few “scientific” axioms, we wouldn’t be having this discussion. What we’re left with is a few different answers– a need to take things case-by-case, as much as possible.
The lofty, overarching answer is probably something like “honesty”. The practical one is perhaps to cultivate trusty, stable-minded, self-aware peers and rigour in testing our models across fields.
We aren’t talking about mistrusting every insight or idea. But if it came from a very specifically designed and constructed setting, it vastly increases the chance that it is useless outside that setting, and that you were prepped to get it.
But the same artificial construction becomes much more interesting if its self-aware and honest:
Can we build a world where even crazy experiments feel encouraged to be localized, honest and voluntary?
Would we have as much of a problem with a cult that tells us it’s a cult because it wants to see how far it can take its theories and does not harm its members on purpose– or are cults inherently incapable of such a structure? Would we object to a crazy dietary ideology whose members are voluntary self-experimenters and are famously transparent and rigorous about their results? Would we mind hardcore communism or unfettered free market capitalism if it is attempted as a voluntary experiment in a localized area?
Don’t talented teachers use artificial thought experiments and simplified situations to build up to understanding a bigger subject? The key differentiator in actual results seems to be honesty. On the part of both the teacher and the taught, or the experimenter and the experimented on.
Luckily, sometimes we don’t even need to go as far as testing each and every thing.
Sometimes if we stare at a supposedly profound insight long enough, then like Déjà vu’s and mirrors in lucid dreams, you’ll find it simply melting away, revealing what you might have wrought upon yourself had you taken it to heart.
I borrowed the term “fake insights” from David Chapman’s excellent Meaningness article about them. You can read it here: https://meaningness.com/metablog/fake-insights
Great post, fellow Complice user!
The Nobel laureate Gerald Edelman said “we don’t have goals. We have values”. Your notion of axioms sounds a lot like “values” to me. Innate notions that guide our behavior and goals, although they do not really constrain actions (I am capable of acting in a way opposed to my values, although I’ll feel bad about it).
Thank you! And will look him up. 🙂
This is less of a comment and more of an essay in response.
I would like to engage with all your writings and respond to them to see how much I can flesh out as a learning experience for myself. I hope my efforts would not seem obnoxious and an unnecessary distraction. I am an armchair philosopher – a human being with no qualifications, but I love to read and think. Haven’t really begun writing yet. I would also like to point out that I am neither a specialist in mathematics nor philosophy, but based on my historical leisure readings, I would like to add certain points on both fronts. Do correct me if I am wrong.
I believe our interests match in nature, quantity and intensity, approximately, if metatags are anything to go by. And due to the sheer magnitude and difficulty of the concepts we engage with, I’m sure that this is just the tip of the iceberg and our interests are much broader in scope.
I would like to begin by disagreeing with you. It is impossible, or extremely difficult to say anything of objective significance without admitting contextual and historical influence. Similarly with Axioms. I do not know for certain, but axioms seem to be embedded within a reductionistic worldview. I believe the concept of knowledge is fuzzy, and one can come to understand a concept by a process of discussion of case by case basis in different contexts and seeing if any general principle emerges. And if we were to talk about it, the way we define knowledge depends on which methods of analysis and discourse we engage in. In other words, the process and underlying philosophy/values of the conversation is implicit and paramount. People stumble and discover things on the way. When a person comes with a stroke of insight and is able to formalize some relations in a vocabulary that belongs within some paradigm of science, then they are heralded as geniuses or founders of some new field (distractingly and misleadingly so). Take Euclid for example. He is often referred to as the “founder/father of geometry,” inspite of the fact that the practice of geometry predated Euclid by a large span of time. Euclid may have been the first to recognize some sort of axiomatic relation between certain “elementary facts” and what knowledge may be generated from the axioms. This furthered the practice of Mathematics which then became more regulated because it could now follow an apparently rigorous process of proof. But this did not mean that the the large body of knowledge of geometry did not exist prior to this realization. If I may go on to state a more general statement that I feel is correct without being able to prove so: A body of knowledge and set of optimal practices within a community which depends on that knowledge due to certain essentials like survival or the like, will most likely have embodied the behavioral practices and oral history of that practice, or may alternately be an implicit attitude before it is sufficiently articulated and documented explicitly. It’s more like people muse on some practice long enough by observing themselves and their fellow humans and then find some descriptive they articulate.
Now, maybe the reductionistic or axiomatic/reductionistic understanding/paradigm/worldview made certain things more streamlined in terms of knowledge discovery/verification. But it also became the predominant mode of engagement. I suggest that knowledge may exist without need for an axiomatic understanding even though aciomatic relations may exist.
Moreover, I believe that it is niche mathematical knowledge that there are certain mathematical statements that cannot be proven just from the axioms of that model system, but may have to be reached from some other system. I believe some keywords are “independency”, “undecidability” and they have something to do with “Godel’s Incompleteness Theorems”. Moreover the complexity of the world has given us such crazies as zero-knowledge proof and the like. My point being, yes, let’s look at models and axioms, but maybe we don’t have to always start with them and keep insisting on their essential nature and instead mention that they’ve been things that made things a lot faster.
I would like to expand upon your description of insight by adding the word “counterintuitive.” Results in scientific models ought to be especially useful if their results go counter to what we would consider common sense, and then we would be encouraged to rely on these models which model reality better than our intuition. These models of reality would act as corrections to our naive mental models/heuristics.
As I read more, I see that the above disagreement is more of a superficial nature, and you use the word axiom in a broader social and psychological sense, thus removing it from the exact nature of the truth of the “hard sciences.”
But there’s something implicit that I disagree with. But maybe it’s just a language game that I have an issue with.
This section:
“Many such insights are useless, evidenced by the fact that they almost never lead to tangible changes that others can perceive (and it is only others you trust who can judge whether you’re a “better person”) or tangible solutions that appeal to the various competing parties affected by a problem.”
Now, I believe what you are trying so hard to say without actually saying is the word “bullshit,” which has become a philosophical study recently (thanks goes to Harry Frankfurt, I believe). Bullshit involves less of lying and cheating and more of self-deception. Intelligent people tend to buy into their own bullshit because they can construct arguments and statements that can play around with their notions of causality. Now, I think language itself is a very ambigous tool, and that’s fundamentally what leads us to several problems. The language of maths is very specific, but the english language is vague. So if I call myself a loser and then a genius, or I say some bullshit, I could be right on all counts. As you mention, it’s difficult to separate the wheat from the chaff because the explanatory powers of the different so called truths appear on par with each other, but they espouse different values. I merely wish to ask a simple question: are there certain kinds of bullshit which you would consider permissible and how would you characterize that domain?
Just some thoughts. But I liked the read. I do have some resources I would love to share, but each targeted to a specific section. Each section would be like a mindmap with branching points.
Thanks for taking the trouble to make a detailed reply and engaging with the topic.
Unless I missed some of what you said, I don’t think what you’ve outlined presents a disagreement.
Is it possible you are picking up on what I left implied, or rather, did not make explicit, whether on purpose or by accident/ignorance? (I see you yourself have hinted at this possibility with your statement: “As I read more, I see that the above disagreement is more of a superficial nature..”)
For example, what you are speaking of about knowledge is very close to Taleb’s conception of heuristics, tinkering and trial-and-error (there are others who speak of them, but Taleb is the main popularizer for many).
Which is an idea that I, and this article, fundamentally agree with. You can see this agreement with the idea in the way I characterize Science at the beginning of the article.
The agreement perhaps, I can make more explicit here: good tinkering and trial-and-error leads to good heuristics, good heuristics eventually leads one to form good axioms and in fact, sometimes, are worthy as axioms themselves– often counterintuitive ones– and therefore lead to good models of the world. Put another way, the tinkerer simply arrives at the axioms through tinkering, not thinking. And because he is a tinkerer, he may not care to ever formalize or state them, but by his very work and existence he has discovered it and made it important for others to understand them, and earned the right not to care himself.
The importance of stating them is for civilization– to understand it, to preserve it, to teach it, to build on it, and especially to bootstrap it.
Which means that stating those axioms, while unimportant to the tinkerers who discovered it by accident or don’t care for it, are important to everyone else, and also to at least some tinkerers who are working in other fields, who can use them as shorthand to gain an interdisciplinary insight, without going through the original tinkering process in a field outside their own. Even the famous “first-principles-thinking” is enabled by having a solid grasp of the foundations around the field you are boiling down and starting from scratch with.
If I can be allowed a moment to speculate, as a tinkerer myself (though I’m not sure if I’m any good), the unshakeable, axiomatic belief in a tinkerer’s mind is often the belief that whatever he’s working on is the right or only place to be working and there’s no place else to be. They go so far as to model the entire world around that axiom, therefore building beautiful things, like most of the things we see around us.. and sometimes wrecking important things, everything from their marriages to the ozone layer of our planet.
I would go so far and really stretch my luck here when I say that philosophers are tinkerers of language, and mentally-ill folk (whether situational or clinical) are tinkerers of their thoughts, and their tinkering leads to axioms and models, and proceeds from axioms and models, just as I’ve outlined. From when we start out as babies, our personality is enabled by axiomatic/in-built behavioural logic (“x feels nice therefore keep doing” “y feels bad therefore avoid”), actually created by using those in-builts to explore the world through trial-and-error, arriving at heuristics, and the heuristics are directly converted to axioms, which then creates an axiomatic personality (“I am a judicious/balanced/good person, therefore I will do x in y situation”) whose conclusions we play out even if it hurts us.
We’re all tinkerers in something, our entire mental narrative is axiomatic, it is just that certain subjects are more dangerous, unethical or irresponsible to tinker with than others.
You have correctly picked up that I’m using the word “axiom” in a way much more generalized than its mathematical constraints, I’m doing the same with “tinkerer”, for similar reasons.
You said: “Now, I think language itself is a very ambiguous tool, and that’s fundamentally what leads us to several problems.”
A Wittgensteinian statement, and also arguably an indigenous Indian one– in those schools of thought that eschewed verbosity and discussion for heuristics and direct experience (like Yoga. Meaning the philosophy, not Hatha Yoga) or early forms of empiricism (ex.Vaisheshika). I think we’re running into a similar problem here, as it turns out. Because this is the issue also with the question you posed: “are there certain kinds of bullshit which you would consider permissible and how would you characterize that domain?” — this, I feel, is a language game.
The trouble is that you’re asking me about an abstract that cannot and will never be defined.
So, the answer is, boringly, aggravatingly: I don’t know.
But I know from collating experience, my own and others, that bad axioms often lead to spiralling mental health and increasingly self-destructive decision making or as you put it, “bullshit”. And there’s no scientific or rational way to arrive at that realization, no way to formalize/subset/superset that bullshit with the rest of its x-shit family. There is a verbose, deductive way but it will lead to nothing, because there is no cause to take the words seriously, no irrational emotion (I almost always mean this in a good way) driving/motivating the act of reasoning.
In many ways, my article seems almost bipolar, doesn’t it? On one hand I say, “axioms are important in the world we’ve chosen to build” and on the other I seem to say “beware of axioms, they’re a trap”.
But I see no conflict– I mean both, at once, because they are a function of each other.