The Ninth Axiom

At the very beginning of this blog, I laid out a set of eight axioms with which I was to derive my philosophy. Since they were spread out across several posts, I will collect them here for convenience:

Axiom 1: Axioms are valid starting points.

Axiom 2: The fewer axioms you need, the better.

Axiom 3: There is some underlying consistent reality that is made up of things.

Axiom 4: I (or the thing that I think of as “me”) exist in some form in that reality.

Axiom 5: Things in reality interact, forming temporal and causal relationships.

Axiom 6: My senses provide me with information that is functionally determined by the underlying reality.

Axiom 7: My memory is usually a reliable and valid guide to my past experiences.

Axiom 8: Logic is a valid form of reasoning.

Although I would probably word them differently now, and there are certainly quibbles to be had, I still think the general intent of these eight form a solid foundation for truth-seeking and philosophy.

However, they are incomplete in terms of actually determining how to live your life. I sort of already came to this conclusion in my previous post based on the Charles Taylor essay, but I want to draw some more explicit conclusions from that:

  • My core eight axioms provide sufficient grounding for determining reality and truth, but not for values or decisions.
  • I currently live my life by a so-far-unexpressed set of values which includes truth, consistency, and something akin to secular-humanist values (though that needs much more elaboration).
  • Arguing one set of value axioms over another is impossible as long as they are all reasonably simple and compatible with the core eight.
  • Adhering rigidly to any single declarable value seems to be a recipe for disaster.

With all that said, I present my ninth axiom:

Axiom 9: I value truth and beauty, not necessarily in that order.

The wonderful thing about “beauty” is that it is deliberately vague. Music can be beautiful and I value that. Human life is beautiful, and I do value that. There is beauty in some efficiencies, and I value that. Truth, though sometimes harsh, is always beautiful. The right lie may also be considered beautiful.

The beauty of beauty is its pragmatism. Plus, who can resist the implicit quark-naming joke?

The Scientific Method

All of this wandering around thinking about rationality, empiricism, truth and knowledge now finally culminates in the scientific method. This is, of course, not the be-all and end-all of truth or knowledge, but it is one of the most useful tools we have for probabilistically determining the shape of reality.

If you’re not familiar with the scientific method (really?) it goes something like this:

  1. Have a question.
  2. Take a guess.
  3. Find something that your guess predicts.
  4. Measure it.
  5. If your measurement doesn’t match your predication, go back to 2. If it does match your prediction, go back to 3.

You will note, of course, that the scientific method never stops. It is an endless loop of trying out new ideas and finding ways to verify or disprove them. Every successfully predicted measurement is another step on the way towards statistical certainty, but of course like any empirical practice we can never get all the way there.

It is also relatively important that your guess (step 2) must be capable of making predictions (step 3). Every so often someone will claim to have solved a great complicated scientific problem by the means of some mysterious new substance. When asked about this substance, they are incapable of predicting its properties other than that it makes their theory magically work. These ideas do not fit in the scientific method – if it doesn’t generate predictions, it’s not science.

Truth and Knowledge


While empiricism gives us some probabilistic, mediated access to reality, our senses are frequently consistent with each other. However, we have no way of knowing if this input represents what is or simply some high-level abstract interpretation of a transformed version of what is. Both of these could be called “true” however in normal speech, so I will borrow the Buddhist doctrine of Two Truths. In very brief summary:

Absolute Truth: The underlying reality of what is.

Relative Truth: What we consistently and reliably interpret of what we sense.

Absolute truth might, for example, be nothing more than atoms interacting according to physical laws. Regardless, it is exactly the reality accepted in axiom #3. Relative truth is the world that we construct from empiricism, with tables and chairs and clouds and cats and dogs. Even if, fundamentally, there is nothing that is a dog in the underlying reality, the existence of dogs is still a relative truth, a practical concept useful for navigating the world.


The definition of knowledge is another big problem in philosophy, and is more-or-less the defining question of epistemology. Based on our axioms so far and the previous two posts, I use the following three definitions of knowledge:

True Knowledge: This is the definition of knowledge that pedants like to trot out when arguing for epistemic scepticism. “True” knowledge is knowledge of absolute truth (see above) which is impossible because of the circular trap. We have no way of knowing that the axioms we have taken are correct, thus no way of knowing anything else which we might derive from them. However, this meaning is almost never used in non-philosophical debate.

Strong Knowledge: Strong knowledge is knowledge based on “inviolate empiricism”; facts like gravity1 that consistent across a truly enormous number of observations (all relative truths, of course). Logical derivations from axioms (and from inviolate empiricism) also qualifies of course: knowledge of algebra is strong knowledge. This usage occurs outside of philosophy debates, but mostly in scientific and other formal contexts.

Weak Knowledge: Weak knowledge is knowledge based on probabilistic empiricism. It is reasonable, for example, for me to say that I “know” certain things about Shakespeare, but the chain of actual facts and observations between myself and him is quite long and tenuous. Nonetheless, I say I know these facts because they are still far and away the most probable explanation of all the various things I have experienced. This is the most common usage of knowledge in informal conversation.

(1) Yes, I know, quantum theory and relativity etc etc. What goes up must still come down.

Empiricism and Probability

We have a set of tools now for operating on truths, but we lack the raw materials to operate on. Fortunately there is another common epistemic view.

Empiricism is the view that knowledge of reality comes from the senses. As I mentioned in my brief discussion of axiom #6, we are not assuming quite this much, though we are taking at least a partially empirical view. Our senses provide some sort of access to reality, although that access may be mediated, transformed, inconsistent, etc. Due to this caveat, we cannot simply make knowledge-claims based on our senses: “I see _ therefore I know that _.” could be perfectly wrong.

However, we can say “I see _ therefore I know that something in reality is causing me to see _”. This is somewhat more accurate though less useful. More importantly, we can appeal to the consistency of reality (axiom #3), the partial consistency of memory (axiom #7), and the validity of logic (axiom #8) to make probabilistic empirical claims, such as the following:

I see _, and I have many memories of seeing the same _, therefore it is probable that I will continue to see _.

Of course sight is not the only possible sense here; one can make similar claims using hearing, smell, touch, etc. By adding an appeal to causality (axiom #5) one can also make probabilistic claims about correlation:

I see X and have a memory of just seeing Y. I have many memories of seeing X shortly after seeing Y, and no memories of not seeing X after seeing Y. Therefore it is probable that if I again see Y, I will shortly see X.

(I use X and Y instead of _ above because I have two blanks to fill in that I wish to distinguish between). Just like any probabilistic claim, the more samples you have (in this case memories) the stronger the claim.

It is these probabilistic claims that we can use as raw materials, feeding them into our rational tools to produce an understanding of reality. Of course we cannot use this to achieve whatever constitutes “real” truth with any certainty, but strong probabilities are better than nothing. With rationalism and empiricism in hand we will use the next post to delve more deeply into the concepts of truth and knowledge.

Rationalism and Certainty

The question of what constitutes or defines knowledge is another big problem in philosophy, and on its own forms what is called epistemology. I leave aside (for now) the central epistemic questions in order to discuss certain common perspectives; I will return to the broader questions in a few days.

Rationalism is, loosely defined, the view that knowledge and truth come from reason, logic, etc. By taking axiom #8 we are inherently taking what is at least a partly rational view, although there is somewhat more to it than that. However, on its own this axiom gives us a whole slew of tools in the category of what I shall call “logical systems”. These include propositional and predicate logic, algebra, set theory, probability and all the other systems that are strictly abstract (despite possible practical applications).

The critical point is that these systems are only tools. While they do produce internal “truths” (such as the statement that two plus two is four), all of these internal truths are at root truths by definition. Even the weirder, hard-to-prove theorems do eventually fall out of starting definitions; that’s how we define a proof. More interestingly, these tools give us methods of “lossless” operation on existing truths. The logical form of modus ponens takes two truths and produces a third truth of the same strength; no certainty is lost in the deduction.

These tools are extremely powerful, but like all tools they are useless on their own; they require raw materials to operate on. There are perhaps certain interesting facts that can be derived from our core set of eight axioms, but I would be very surprised if you could get very far (I have not even bothered to try). Instead, we need some other system to generate facts for us to operate on. Next time I will look at where we can get raw materials to feed our rational tools.

Building Reality, Part Two: The Mind

Five axioms down, three to go. I’ve titled these three “The Mind” because they’re more to do with that kind of concept, but I don’t mean to imply that the mind is necessarily distinct from metaphysics (which was the title I gave the previous three axioms). As I mentioned in the discussion of axiom #4, that’s not true by definition.

Axiom 6: My senses provide me with information that is functionally determined by the underlying reality.

Reliability of the senses is a big question in empiricism, philosophy of science, etc. This axiom doesn’t go quite that far; all it gives us is the fact that our senses are somehow determined by the underlying reality. Maybe we only have access to a subset of reality, or maybe there’s some random transform in place between reality and what we sense. Regardless, we have some sort of access, however mediated or probabilistic.

Axiom 7: My memory is usually a reliable and valid guide to my past experiences.

Reliability of memory is another big question in a lot of disciplines, and there are obviously cases where it doesn’t hold. Amnesia and memory implantation both seem to be real phenomena after all. As with reliability of the senses, I’ve chosen to split the difference. We assume our memory is usually reliable, for some definition of “usually”.

Axiom 8: Logic is a valid form of reasoning.

This one is kind of cheating since I’m explicitly leaving the definition of “logic” so vague. I am intending it here to include forms such as propositional and predicate logic, and all the other weird mental constructs without which it’s hard to make sense of things like mathematics. Perhaps this would be better phrased as something more to do with thought processes in general than pure logic. Hmm…

Regardless, we now have our eight core axioms from which we can start to build up an understanding of the world. I expect to revisit these occasionally as I think up better wordings, or discover missing concepts, but as per axiom #2 I hope not to add to them unless absolutely necessary. Let’s at least see how far we get with just these eight!

Building Reality, Part One: Metaphysics

We now have two relatively basic axioms that let us pull ourselves up by our bootstraps without falling into a nihilistic or explosive view. However, that’s about all they do on their own, so beside these we will lay out a bare handful more axioms that will let us really get going. While several of these axioms may be hard to articulate, none of them should be controversial.

Axiom 3: There is some underlying consistent reality that is made up of things.

There are a couple of different ideas wrapped up in this, most importantly those of existence, consistency, and divisibility. Consistency is hard to define at this level, but is something like “follows unchanging rules”. It should be easy to understand though; if reality isn’t consistent then there’s no way to make sense of anything. It is also worth noting that the “things” which make up reality in this axiom are intentionally vague. They could be atoms, quarks, Platonic ideals, Cartesian egos… What kind of things they are doesn’t really matter at this point.

Axiom 4: I (or the thing that I think of as “me”) exist in some form in that reality.

This one is more straight-forward, it gives us a reference point to work from, although it tells us nothing at all about that reference point. The key here is that the self is part of or contained in reality, not separate from it. It isn’t necessarily a fundamental “thing” in the third axiom’s sense, but it does belong to reality. This is purely definitional: reality is all the things that actually exist, so the self is either one of those things or made up of them.

Axiom 5: Things in reality interact, forming temporal and causal relationships.

This one just works in the concepts of time and causality. Nothing tricky, though lots to argue about if you feel like it. Importantly, this axiom makes causality a property of things in reality, not reality itself. There is no commitment here to the peculiar idea that reality itself must have a cause. Something about the wording of this one still bugs me, but I haven’t been able to pin it down.

These three axioms give us some metaphysical meat to work with, but we need a just a little bit more. I have three more axioms planned for the next post, at which point we’ll hopefully be able to move beyond this low-level mucking about.

Occam’s Razor and Epistemic Explosions

We have started by accepting a single axiom on the validity of axioms, and all of the relevant circularity. We must now be careful though, for we currently have no criteria for which other axioms we accept. In fact in our current state we are allowed to choose any and however many axioms we want. Don’t feel like arguing for a particular proposal? Just take it as an axiom!

This epistemic explosion puts no limits on what we can claim as fundamental truth. As with it’s opposite, epistemic nihilism, there isn’t actually anything philosophically wrong with this. We are still effectively too close to the circular trap to be able to make that kind of judgement. However, like epistemic nihilism, accepting an epistemic explosion just doesn’t seem practical or useful. If we take a nihilistic view then there is nothing more we can say, whereas if we take an explosive view then we can say literally anything. Either way, we’re philosophically finished. Frankly, that kind of view is just boring.

So, we will add a second axiom to our set, based on a fairly well-known principle: Occam’s Razor. A lot of the historical details around Occam’s Razor are rather fuzzy, including who said it first, whether Ockham said it at all, and why the spelling has changed. These questions are not really relevant to the underlying principle though, which is often stated as “entities must not be multiplied beyond necessity” (Wikipedia claims this formulation is due to John Punch).

The razor has many other various formations, but they all basically boil down to “simpler is better”. Or more practically, if you’ve got two possible explanations for a thing that do an equally good job on all points, pick the explanation that only requires a paragraph, not the explanation that requires thirty pages, two diagrams and an appendix. With all that in mind, we can formulate our second axiom:

Axiom 2: The fewer axioms you need, the better.

An Axiom is an Axiom is an Axiom is an…

In my previous post, I discussed the so-called circular trap and how it does not really seem escapable. I also mentioned that that wouldn’t stop me – I’m basically going to ignore it because otherwise nothing can be discussed. Onwards.

This, of course, leaves us with the question of where to start if we want to move beyond the circular trap. I’m a logician at heart, so I’m going to start with an axiom. And since the circular trap is still fresh in our minds, the axiom I’m going to start with is intentionally circular. In fact, it is:

Axiom 1: Axioms are valid starting points.

Nice and circular, and generally implied whenever somebody reasons from axioms in any situation. Because of this, and because of its extremely tight circularity, I tend to regard this axiom as the beating heart of the circular trap, so it seems like a reasonable starting point. This is where I plant my flag, flying proudly unsupported by anything but itself.

Of course, this axiom on its own doesn’t really get us much in the way of practical consequences. We’re going to have to add a few more axioms to our collection before we can coherently think about all the things we want to think about. But the validity of all the other axioms we use will rest on the validity of this axiom, so I shall leave them for the next post.