On the PSR (Principle of Sufficient Reason)

Introduction

The PSR and Ontological Mathematics

In this post we will examine the Principle of Sufficient Reason (PSR) and what is exactly meant by this term in Ontological Mathematics (OM) esp. its relationship with Euler's equation (EE) (*1). While it may seem unnecessary to do this given the apparent simplicity of the PSR, and how much Ontological Mathematics talks about it, upon closer examination it's a term that I find somewhat conflated and not properly nuanced in the God Series (GS) or the Truth Series (TS).
As previously stated, the biggest distinction that needs to be made is how the PSR is similar and yet differs from Euler's Equation. We will also examine if the PSR can indeed qualify as the sole axiom upon which a math system such as Ontological Mathematics can be built.

*1 Technically Euler's Identity  

The Groundwork: Basic Concepts  

Here we briefly review the PSR, EE, and the need for a master axiom.

The PSR: A Basic Overview  

The PSR, as stated by Leibniz, basically says "everything must have a reason or cause". I will occasionally refer to this form as the "text PSR" to distinguish between the other various components of the "full" PSR, or what I'll be calling the Master Axiom or Master Principle.
A somewhat more precise definition of the text PSR comes from Wikipedia:

For every entity X, if X exists, then there is a sufficient explanation for why X exists.
For every event E, if E occurs, then there is a sufficient explanation for why E occurs.
For every proposition P, if P is true, then there is a sufficient explanation for why P is true.

I think this intuitive definition is what most Ontological Mathematicians are using for their mental model when referring to the PSR.
Okay, simple enough, right? While this is a good intuitive definition, it's really not sufficient to base a system of math on, which is where Euler's' Equation (EE) comes in (which we will discuss later).
It's important to really nail down what the PSR is since it's the central tenet of all Ontological Mathematics. Here is one of many citations where they site the centrality of the PSR:

The PSR is of course the final reason for all things. Only the PSR can be the final reason. It is God.
Stark, Dr. Thomas. God Is Mathematics: The Proofs of the Eternal Existence of Mathematics (The Truth Series Book 10) (Kindle Locations 4164-4165). The Ontological Mathematics Foundation. Kindle Edition.

And:

The PSR establishes absolute meaning and purpose. It establishes both syntax and semantics: a dual-aspect reality, a dual-aspect mathematical monism that fully explains both mind and matter.
Stark, Dr. Thomas. The Language of Reality: The Answer to Existence (The Truth Series Book 4) (Kindle Locations 4200-4201). The Ontological Mathematics Foundation. Kindle Edition.

The Need for a Master Axiom

One of the reasons the PSR carries so much importance in OM is that any math system striving to be complete and consistent necessarily must be based upon a single master axiom, per Godel's Incompleteness Theorem (GIT) (*2):

Why is it impossible to base mathematics on a set of independent axioms? It’s because the very fact that the axioms are independent means that they are incommensurate and belong to different conceptual systems. For mathematics to a unity, a whole, a completeness, all of its axioms must be derivable from a single master axiom. Mathematics must be holistic and holographic, hence no part of it can be independent of any other part of it. The whole is in every part, thus making all parts commensurate. If the whole is not in every part, as scientism claims, every part is automatically incommensurate with every other part, hence irrational and illogical.
Stark, Dr. Thomas. Extra Scientiam Nulla Salus: How Science Undermines Reason (The Truth Series Book 8) (Kindle Locations 5917-5923). The Ontological Mathematics Foundation. Kindle Edition.

And:

It has been disastrously overlooked, especially in science and mathematics, that it is impossible to create a complete and consistent system if it is not grounded in one principle or axiom, from which everything else inexorably flows. It is impossible for mathematics to be complete and consistent if it is based on multiple independent axioms. This is what Gödel proved, yet Gödel never once tried to derive the whole of mathematics from a single principle or axiom. Ontological mathematics rectifies this extraordinary omission. It derives the whole of mathematics – and thus the whole of reality – from the Principle of Sufficient Reason (PSR), which can be expressed mathematically as Euler’s Formula. This is the Master Formula of mathematics, and, thus, of all of existence.
Stark, Dr. Thomas. God Is Mathematics: The Proofs of the Eternal Existence of Mathematics (The Truth Series Book 10) (Kindle Locations 1447-1453). The Ontological Mathematics Foundation. Kindle Edition.

(*2) Although I always find it funny that OM uses GIT, or any theorem of non-ontological math, since they're all standard "analytic" math (using many independent axioms) and thus not ontological, and invalid per their own words. However, I concede the practical need that in building a pure ontological system one must initially rely upon non-ontological structures such as analytical math as a temporary bootstrapping scaffold.

Euler's Equation

Another tenet, equal in importance to the PSR, is Euler's Equation. While hard to write in simple text, this is basically the formula of e to the i-theta power equals cos theta + i sin theta (e^i0 = cos 0 + i*sin 0). It basically deals with circular motion in a complex plane.
Note how I said it has equal footing to the PSR. Indeed, in reading the TS or GS , one may be forgiven for thinking Euler's Equation is the PSR, or that EE itself is fundamental not the PSR.
Here they parenthetically suggest the PSR and EE are the same thing:

In fact, the final reason for things is the PSR (Euler’s Formula) and it is instantiated through infinite eternal necessary beings (monads), which stand outside the series of contingent things.
Stark, Dr. Thomas. The Book of Mind: Seeking Gnosis (The Truth Series 5) (Kindle Locations 4486-4487). The Ontological Mathematics Foundation. Kindle Edition.

And here again, they explicitly conflate the two again:

The universe has a single principle, a single axiom, a single formula, and that is Euler’s Formula, the PSR presented in its base mathematical form. The PSR/Euler’s Formula, is the arche, the fundamental definition of reality.
Stark, Dr. Thomas. The Book of Mind: Seeking Gnosis (The Truth Series 5) (Kindle Locations 4382-4384). The Ontological Mathematics Foundation. Kindle Edition.

Obviously EE is something different than the PSR. After all, one's a symbolic math equation, and the other is essentially an intuitive maxim, expressed in a non-symbolic human language to boot. They can't obviously be the same. So what gives? 

 
That's what we will examine next.

Analysis

Introduction

It's funny how you can read something again and again and never really stop to ask yourself if you truly understand what is being said. It wasn't until I watched a video from Diabollically Informative where he was talking about how the PSR can be used to derive actual math theorems that I realized the text form of the PSR: "Everything has a reason", was obviously not what he was talking about, since this is not even a symbolic equation and would constitute a category error if used as a math thereom. Anyway, this is what prompted me to make the distinctions I'm about to make (*3).

(*3) I recently "scraped" all the text from the GS and TS into a relational database, so now I can search across all books with something like:
select id,text from text_corpus where text like "%euler%is%god%" limit 20;
This greatly enhances my ability to find quotes on things I remember reading about. However, when I searched for any reference to the semantic and syntactic aspects of the PSR, as we will subsequently discuss, I can't find any hits. While this doesn't prove the idea isn't discussed in the books, given my inability to find any quotes in the corpus, and the fact that I can't recall reading an analysis like the one in this post, realize that this is my own interpretation.

What is the PSR really?

Basically there are three components to the PSR: the thing itself, a sematic aspect (e.g. the text PSR), and a syntactic aspect (Euler's Eq.). Note, this is just a narrower example of a more general insight OM has about language:

Syntax and semantics are two indispensable aspects of one and the same thing, and they do not reduce to each other. There is one entity, a single language, and it is neither a syntax, nor a semantics, nor a strange hybrid or merger of both. The language necessarily has two distinct aspects, a syntactic aspect and a semantic aspect. We experience the semantic aspect, and we can use reason and logic to determine the syntactic aspect. The syntax is the information carrier, and the semantics is the information carried.
Stark, Dr. Thomas. The Language of Reality: The Answer to Existence (The Truth Series Book 4) (Kindle Locations 4097-4102). The Ontological Mathematics Foundation. Kindle Edition.

Note also that I'm not going to refer to this full "trinity" as the PSR, since it will get confused with the text form of the PSR. Instead, I'll just call it the Master Principle.

The Two Side of the Coin

But before we dive deeper into this PSR triad, let's talk a little about coins since this will make it easier when we apply the concepts to the PSR Master Principle.

A coin is a single thing. It is "composed" of two sides. I have quotes around "composed" because composition normally suggest that something has parts. However the two components of a coin, its sides, are aspects, not parts. You cannot isolate a side of a coin: it's always comes in dualistic pairs. If you split a coin in two, you don't get one side and then another. You get two (thin) coins each with two sides, for a total of four sides. And since a coin is a single thing it's a monism. A coin is therefore a dual-aspect monism: a single thing with two aspects.

As previously stated, according to OM everything in the universe, has two aspects: a semantic aspect and a syntactic aspect. For example, since everything is made of waves, the carrier wave -- the wave that carries the signal, is the syntactic aspect. The information carried by the wave is the semantic aspect. A photon with a frequency of 450 THz is perceived as red by humans. So syntactically it's a wave with a frequency of 450 THz, but semantically, it's the color "red". Humans typically only experience the information carried (the semantic information) but cannot detect the information carrier (the syntactic information).

I suggest the same goes for my concept of  "Master Axiom". It's semantic aspect is the intuitive text form of the PSR: "Everything has a reason", and it's syntactic aspect is Euler's Equation: e^i0 = cos0 + isin0. So we have the following mapping between Coins and the components of the PSR:

Coin = Master Axion
heads (or side 1) = text PSR
tails (or side 2) = Euler's Equation. 

 
This analogy make several things clear. First it clearly identifies the relationship between the text PSR, and the EE: how they are similar (aspects of one thing) and yet different. Two, there are three distinct concepts, not two. The concept of Master Axiom is what I feel is not clearly distinguished in the GS or TS. I think they do use the concept but they have a have habit of referring it to simply as the "PSR", which they also use when referring to the semantic aspect of the Master Axiom (or text PSR) and the syntactic aspect of the Master Axiom (Euler's Equation). Thus they conflate Master Axiom and the text PSR, and Master Axiom and EE.

Here is an example of where I think they are actually referring to the Master Axiom portion of the "trinity" and not the text PSR:

You have to transcend human thinking, mired in the contingent, temporal, a posteriori, empirical and material, and reach the eternal, necessary, a priori, rational, logical and mental level at which reality itself thinks. That’s what ontological mathematics, derived from the eternal PSR, achieves.
Stark, Dr. Thomas. Castalia: The Citadel of Reason (The Truth Series Book 7) (Kindle Locations 4379-4380). The Ontological Mathematics Foundation. Kindle Edition.

Note the usage of the phrase "eternal PSR". To me this indicates they are referring to the more general Master Axiom concept and not merely the semantic aspect of the Master Axiom.

Ramifications to the Requirement for a Single Axiom

Now we explore how this model effects the need for a single axiom. The Master Axiom should really be referred to as "Our one big free axiom". Everything has to have a starting point. And since you're only allowed one, you better make it a good one. However, a big (i.e. complicated) assumption is more likely to be wrong, so there's an opposing force of wanting to make "your one big free axiom" extremely basic. Leibniz tried to find something that is maximally explanatory yet minimally simple, of which he concluded the (text form) PSR was the solution.
I think OM does an excellent analysis of how frail assumptions can be, even (maybe even especially) the most simple and "obvious" ones. Here are some excerpts from TS 7 where they discuss this:

Peano says that the successor of any number is a number. That seems uncontroversial, but the least controversial claims are usually where the deepest problems are found...
Peano says 0 is not the successor of any number. What about negative numbers? Isn’t zero the successor of -1? What about a circular conception of numbers rather than a linear conception? If zero is the South Pole of a circle, we will of course always return to it, hence zero must succeed a number. Moreover, if we take the holistic view and define zero as the whole that is present in all numbers then zero can be regarded as the successor of every number, no matter what...
The point of this exercise is to show that no matter how self-evident axioms might seem, no matter how logical they might seem, no matter how eagerly mathematicians might agree to them and utilize them, they are always constructed from within a specific paradigm, and if that paradigm is false then every conclusion is either outright false, or radically compromised, and certain to generate incompleteness and inconsistency sooner or later. What tends to happen in both mathematics and science is that paradigms are constructed that have a reasonable approximation to reality within a certain range, and this encourages people to believe in the paradigm, and reject alternatives. These approximate paradigms are associated with approximate axioms, and everything proceeds in this approximate manner, but with their proponents claiming they are exact and not approximate. The problems become apparent only at certain scales (no one noticed quantum mechanics until experiments started probing the world of the extremely small).
Stark, Dr. Thomas. Castalia: The Citadel of Reason (The Truth Series Book 7) (Kindle Locations 2695-2697). The Ontological Mathematics Foundation. Kindle Edition.
Stark, Dr. Thomas. Castalia: The Citadel of Reason (The Truth Series Book 7) (Kindle Locations 2710-2714). The Ontological Mathematics Foundation. Kindle Edition.
Stark, Dr. Thomas. Castalia: The Citadel of Reason (The Truth Series Book 7) (Kindle Locations 2719-2726). The Ontological Mathematics Foundation. Kindle Edition.

For instance, the assertion that the successor of a number is always greater than the original only applies in a linear system.  In a circular system, where some large number eventually wraps back to "zero", trivially invalidates this "obvious" assumption.

What exactly is the Master Axiom? I say it's simply that which has a semantic aspect that is the Text-PSR and has a syntactic aspect that is Euler's Equation. Yes, this is a circular definition. What is a coin, but ultimately its two sides. If you take away the two sides, you essentially have nothing. What is the Master Axiom proper -- the thing that's left if you remove its two aspects or sides? If there is something still left, I think you could say it's unknowable, maybe even inconceivable (*4). Maybe you can say a coin has something like a value that is still left. But that's really just a conjecture. Maybe a coin does have a value aspect and your conjecture is in fact right but you can never know this, so it's effectively unknowable and inconceivable. (*5) Just to confuse things further, I guess you could say there's a fourth thing as well -- the thing that is the combination of the Master Axiom proper, the text PSR, and EE, or the coin and it's two aspects.

(*4) Another interesting interpretation would be to consider the Master Axiom proper to be at a higher level of being a la Heidegger e.g. hyper or wild being. I don't know what the OM authors attitude is toward the philosophy of Heidegger and Husserl e.g dasein since a full scan of the corpus only reveals two brief mentions. 

 
(*5) I know what the answer would probably be: that you can know the Master Axiom proper through pure rationalism. The argument that's there nothing left after you remove both sides smacks of an empirical argument. I understand this, but I'm arguing what's left if you define a coin as being its two sides. This blog post is more of an open exploration: asking the necessary questions rather than trying to "prove" anything.

Basically, I think there are two assumptions in the Master Axiom PSR: one that the Master Axiom itself is true, the other that Euler's Equation is the syntactical equivalent of the Master Axiom (or the equivalent of the semantic aspect of the PSR aka the text-PSR). 

 
It's tempting to say all three are just part of one big assumption, thus avoiding the problem of having two assumptions. A sly person might even assert that one assumption plus another assumption still equals one assumption. While technically this is true, it's really pure sophistry. This is like someone trying to refute the truth that 1 + 1 = 2, by saying that one cloud + one cloud = one cloud (albeit, a bigger cloud). Of course, if you more precisely state the situation you have to say one cloud with a mass of 1Kg plus a cloud with a mass of 1Kg gives one cloud with a mass of 2Kg, thus the kernel 1 + 1 = 2 still holds.

 
And since I'm asserting that the actual Master Axiom -- the thing itself, the "container" independent of it's two aspects, is unknowable, you basically have to equate it with either the semantic or syntactic aspect as a proxy. And since I believe most Ontological Mathematicians are assuming the semantic aspect, we can effectively say they are assuming the text-PSR (*6). However, I think it's basically a second assumption that EE is the mathematical equivalent or expression of the Text-PSR. Basically, you can think of the GS and TS as being a 46-(and running)-tome justification, or "proof", that EE is in fact rationally derivable from the Text-PSR thus not constituting a second assumption.

I think this is problematic for two reasons, however. One I don't think you can derive a precise symbolic mathematic equation from a relatively vague human-based maxim (e.g. what does "sufficient" mean mathematically?). It's a category error: you can't derive syntax from semantics and vice-versa. Two, the sheer preciseness and power of EE, which is derived using analytic math and builds upon literally hundreds of (non-ontological) axioms and theorems over hundred of years, is just too big a step to claim as being derivable from the Text-PSR alone. So I think that the idea that EE is the syntactic equivalent of the text-PSR is a second assumption: maybe even a true one, but an additional assumption nonetheless.

(*6) In a future post, I'd like to explore the potential of assuming EE as the proxy for Master Axiom instead of the text-PSR.

Off the Rails on a Crazy Train?

If in fact this two axiom conclusion is true, then that would mean OM goes wrong almost immediately, at least as far as completeness goes. I can't help but bring up several comments made by the authors themselves on this very topic:

Big Ideas go wrong immediately. As a matter of simple logic, the objective world has only one right answer – the actual truth of reality. But if only one answer is right then it automatically follows that all other answers are wrong, and, moreover, they are wrong straight away, right off the bat, right at the instant they are first conceived. If humanity were actually interested in the truth, it would go through each defining idea, each Big Thought, in minute detail to discover where it goes wrong. Because we know that all but one must be, and are, wrong.
Stark, Dr. Thomas. Tractatus Logico-Mathematicus: How Mathematics Explains Reality (The Truth Series Book 14) (Kindle Locations 193-194). The Ontological Mathematics Foundation. Kindle Edition.

And:

Theories invariably go wrong with their defining idea. If that is false – and it always is, except in the unique case of the right answer to existence – then everything that flows from it will of course also be false. At best, you might get some use value from the system, but no truth value. Science is a use system, not a truth system. Russell’s logical atomism is just useless. Full stop.
Stark, Dr. Thomas. Tractatus Logico-Mathematicus: How Mathematics Explains Reality (The Truth Series Book 14) (Kindle Locations 1990-1993). The Ontological Mathematics Foundation. Kindle Editio

By having two independent assumptions, your system is now vulnerable to Gödel's incompleteness theorem.

In a weird way, I'm grateful for this "limitation".  It's my get out of jail card from OM, if I ever reach a conclusion that is uncomfortable.  While this hasn't ever happened yet, when dealing with a system as powerful for OM, you might potentially discover pathways that are disturbing.  Maybe even just temporarily disturbing because you don't fully understand something.  For instance, OM asserts you are eternal, and constantly reincarnating, as the universe cycles continuously between alpha and omega states, with no chance of  you or the universe ever "dying".  While on the surface this is a comforting notion, esp. if you were like me, a (former?) "nihilistic" atheist, what if you were to reach the conclusion (rationally or irrationally) that 99.9% of your reincarnated lives are "hell", not "heavens" or "neutrals", thus you would have to conclude you are destined to spend 99.9% of eternity in hell.  Yes, weird, and probably irrational.  But if you truly believe in OM and you make this additional "hell" assumption, then there's no out.  Infinity is another scary to thing to contemplate as a (naive?) understanding of infinity means that if anything has the smallest finite chance of happening, not only will it happen, it will happen an infinite number of times (BTW Leibniz' concept of compossibility is a good get out of jail card for this scenario).   The fact that there are (at least) two independent axioms in OM, means it's either incomplete or inconsistent at some level.  Under certain potential scenarios, this can be relief, a seed idea that can be used to deconstruct the system to show it may be wrong or at least potentially wrong (incomplete or inconsistent).

It seems like it's either feast or famine:  either you're a nihilistic atheist who believes you live only once, or you have to live for eternity.  Each has their associated "bleak" scenarios, so ironically, it can be nice to have a little doubt in your system. 

Conclusion

In this post we characterized the "PSR" as actually being a triad of concepts analogous to a coin and its two sides. This model helps us clarify the relationship among all three. Using this three-pronged model, we can recognize the conflation that is common throughout the GS and TS between the Master Axiom component and the semantic text-PSR aspect and this helps us make the proper distinction when necessary. We can also more clearly see that in fact two basic assumptions are being made: the Master Axiom itself and the assertion that EE is the mathematical equivalent of the text-PSR (or Master Axiom).

While it may seem like I'm attacking OM by suggesting it's hoist on its own single axiom petard, my goal is to really just explore and to understand OM more deeply: I come to praise Caesar not to bury him. Unfortunately, there's really no forums out there for discussing these ideas, and I don't personally know anyone who's into these ideas, so I'm kind of stuck reflecting in an echo chamber of one. I'm really just asking open questions, not trying to prove anything. If I'm making an error somewhere, I would be grateful to be corrected as my goal is to understand and not to "win" an argument.

I'm personally not bothered by this dual axiom conjecture anyway because as I've stated before in other posts, I'm a pragmatist at heart (which is kind of ironic because OM is very anti-pragmatist and all about absolute truth) . I will still find OM a very beautiful and useful model regardless.

In a debate with slavoj zizek Jordan Peterson raised an interesting argument in his analysis of communism stating that your idea (referring to communism) is almost certainly wrong. After all, there's an infinite number of ways to be wrong and only one way to be right. So with any new idea the default assumption has to be it's wrong. Only after it's stood up to scrutiny, including from people who are actively against your idea, can you begin to have some confidence in it. In a similar vein, in Is PI falsifiable I basically argue that OM is probably instantially wrong, but classically (e.g. at the meta or class level) more likely to be correct. 

 
Either way, as always, I find OM an extremely useful philosophy distribution (even better than Software or Systems Engineering) and helps me become a better programmer and wiser person overall.

Update History:
2020-07-06:  Add discussion about "get out of jail" aspect.