Will and Free Will: Philosophical Lambda, Hume and Godel (but not really, just a little)
“Freedom is the freedom to say that two plus two make four.
If that is granted, all else follows.”
— George Orwell
“I find that I am ‘intermediate’ between God and nothingness, between the supreme entity and nonentity.”
— Rene Descartes
“The optimism of present-day rationalism sounds just as shallow to the fact-loving mind. The actual universe is a thing wide open, but rationalism makes systems, and systems must be closed. For men in practical life perfection is something far off and still in process of achievement. This for rationalism is but the illusion of the finite and relative: the absolute ground of things is a perfection eternally complete.”
— William James
“This theory will make no pretensions to being knowledge; only to being a good guess, which we may strongly and confidently hope will be confirmed.” — Charles S Peirce
This is a different take on this post. In the original post, I tried to tackle existentialism through Lambda Calculus and, along the way, made references to Descartes, Hegel, Kant, CS Peirce and some of other philosophers. I tried to tie some of their ideas together in a thing I called Philosophical Lambda. This time around I’ll go by way of Hume.
Since it seemed a bit daunting last time, I’m going to skip some of the hairier topics since it can be referenced in the link above. My aim is for a general audience and as such I try to be as thorough as possible but last time it was a bit tedious. Since it’s for a general audience, please take everything with a grain of salt and follow the links I post. My interpretation of this topic is just my interpretation.
I did try to demonstrate my understanding of Lambda Calculus last time by going through a few problems and highlighting the principles behind them. That can be found in the link above as well as HERE. I’m not a mathematician, philosopher or historian, although I pretend to be.
As such, you should be skeptical of what I write and say as much as I am, but to make up for my conspiracy-like search for Pepe Silva, I will try to highlight my references so that they can be further scrutinized by you.
ii. Introductory Recap
Last time I tried to make a link between Lambda Calculus and computational thinking and what that means in regards to our existence. This triangulation led to Philosophical Lambda. Which is supposed to be a system like Lambda Calculus but a bit informal and with different symbols; but with similar principles behind everything and with SETS!
The link between Lambda Calculus and existentialism was made through Descartes’ cogito.
In cogito, Descartes states, “I think, therefore I am.” Thus, tying his ability to think logically to his existence. Meaning that thinking, including the ability to doubt, is used by Descartes as proof of existence. Or rather, proof of certainty. Descartes can doubt or trivialize everything as inessential except for the ability to think. He seeks to establish certainty with this line of thinking.
I would also say that playing music is a form of proving your existence. And dancing. And drawing. But thinking proves existence in a more intellectually-stimulating kind of way. And I want to get intellectual with you ;)
In this intellectual courtship I’m going to perform for you, or on you(in a consenting way, of course)…, I’m going to interweave prose (these words here) with symbols that carry within them the logic behind prose (philosophical lambda (ΦΛ)). Thus, “I think, therefore I am” can be said in symbolic logic as:
Since we can frame it in formal logic, we can also frame it in Philosophical Lambda. This is the portal we use to connect everything.
When Descartes reached the conclusion “I am,” he basically said “I can doubt everything and anything except that I am.” Again, trying to reach certainty.
Let’s dip our toes in some of that Philosophical Lambda with introducing sets.
Let’s create a set of what is “I”:
I = [culture, language, race, sex, gender, arms, legs, torso, brain, thought….etc]????
We can say that “I” is a set that is composed of our culture, our language, our race, our sex, our gender, our body parts, out thoughts, etc.
Descartes said that we can remove everything from inside that set except for thought, since “I” only exist as a thought. Thus:
I = [thought(thinking)]
We can only be certain that I think.
Looking at “I think → I am” again, we can see that, symbolically, “I am” is left alone. Pushed to the side from everything else. Once alone, Descartes’ “I am” needs the guidance of God’s good will to experience the world (according to Descartes, since he (Descartes) can’t be certain of anything else since he got rid of everything).
In a very genius way, Descartes comments on certainty. He says, “imagine that the world is made to appear a certain way by a deceiving demon.” I can say that everything is a lie except that I think everything is a lie. With Descartes’ “I am” in such a vulnerable position, Descartes must imagine a God that does not deceive but rather takes care of the “I am.” And the “I am” must have faith in God in every circumstance that has uncertainty; because the “ I am” could be being deceived in a world of appearance.
Once we find the place of Descartes’ “I am” through this deduction, we find the “I am” left alone in the ether of Reality; everything else we find in this Reality should be considered with skepticism. Everything we believe, then, requires a faith in a good-willed God that leads us towards Truth.
Nietzsche’s response to that abyss?
For believe me! — the secret for harvesting from existence the greatest fruitfulness and the greatest enjoyment is: to live dangerously! Build your cities on the slopes of Vesuvius! Send your ships into uncharted seas! Live at war with your peers and yourselves! Be robbers and conquerors as long as you cannot be rulers and possessors, you seekers of knowledge! Soon the age will be past when you could be content to live hidden in forests like shy deer! At long last the search for knowledge will reach out for its due: — it will want to rule and possess, and you with it!
and in the Knight of Faith we see certainty through doubt, like Descartes:
It is quite certain that at the bottom of all skepticism there is an abstract certainty that is the foothold of doubt and is like the line one draws as the base upon which the figure is sketched. Therefore it is quite certain that nothing is accomplished even by the most rigorous attempt of Greek skepticism to round off the hovering of skepticism by emphasizing that the statement about doubt must not be understood as a position, but it still does not follow that doubt overcomes itself. The basic certainty that sustains doubt can at no moment hypostatize itself as long as I am doubting, because doubt continually leaves it in order to doubt. If I want to keep on doubting, I shall never in all eternity advance any further, because doubt consists precisely in and by passing off that certainty as something else. If I hold on to the certainty as certainty for one single moment, I must also stop doubting for that moment. But then it is not doubt that cancels itself; it is I who stop doubting. Therefore a mediocre doubter will be most likely to succeed in gaining certainty, and next a doubter who merely joins categories in order to see how they look the best without bothering in the least to carry out any of them. — I cannot stop returning to this point, because it is so decisive. If it is the case that doubt overcomes itself, that by doubting everything one in this very doubt wins truth without a break and an absolutely new point of departure, then not one single Christian category can be maintained, then Christianity is abolished.
— Concluding Unscientific Postscript
And Knight of Faith connects beliefs through leaps over doubt:
Everyone knows that the most difficult leap, even in the physical realm, is when a man leaps into the air from a standing position and comes down again on the same spot. The leap becomes easier in the degree to which some distance intervenes between the initial position and the place where the leap takes off. And so it is also with respect to a decisive movement in the realm of the spirit. The most difficult decisive action is not that in which the individual is far removed from the decision (as when a non-Christian is about to decide to become one), but when it is as if the matter were already decided.
Believe it or not, we’ve already encountered Nietzsche’s abyss and Kierkegaard’s uncertainty in Philosophical Lambda (the etc in the “I” set).
v. I, or me, or you, or it
Going back to the question “what is ‘I’?” or “what am I” or “what is there”, let’s go back the set of “I”:
I = [culture, language, race, sex, gender, arms, legs, torso, brain, thought….etc]
In set notation, we can translate the unknown in existentialism as a set that has infinite possibility. Since the concept of “I” is a bit abstract, let’s ask a more direct question.
Let’s imagine you’re at a park and suddenly you’re aware that you and your friend have 4 apples. You then can ask “How did we get 4?”
Translating this question to set notation, as soon as we ask the question “How did we get four?” We must entertain the idea that we can get 4 by various means:
4 = [1+3, 2+2, 5–1, 19–15, .05+3.95, -9+5, 9–5 ….. INFINITY]
Hopefully it’s clear to you why this is the case. The set contains all the possible ways we can get four. And as such, there is infinity ways to get four.
Without knowing anything else about the questions other than “How did we get 4?”, we can’t be certain how we got four but we can create certainty through assumptions:
- What the question means; how did we get this number? Probably through math
- What is 4? 4 = [1+3, 2+2, 5–1, 19–15, .05+3.95, -9+5, 9–5 ….. INFINITY (all possible mathematical routes that lead to 4)]
- 4 could possibly also be code for “red,” “watch out!, there’s a golf ball going your way,”. Again, 4 can represent anything and everything without proper context. But we’re talking about apples, so it’s probably not this.
We could also ask “How did we get 42?” but we’re not ready for those types of questions just yet.
The way we get to the meaning of 4 is determined in the context it’s used. This argument sounds similar to Wittgenstein’s “language games.”:
The concept is based on the following analogy: The rules of language are analogous to the rules of games; thus saying something in a language is analogous to making a move in a game. The analogy between a language and a game demonstrates that words have meaning depending on the uses made of them in the various and multiform activities of human life.
We won’t delve into Wittgenstein; just highlighting similarities. I’ll get to Wittgenstein’s concept going through way of CS Peirce and pragmatism.
But having Wittgenstein on the line, let’s bring up our friend Donald Davidson, who argues something similar to Wittgenstein:
Davidson points out that beliefs and meanings are inseparable. A person holds a sentence true based on what he believes and what he takes the sentence to mean. If the interpreter knew what a person believed when that person held a sentence to be true, the meaning of the sentence could then be inferred. Vice versa, if the interpreter knew what a person took a sentence to mean when that person held it to be true, the belief of the speaker could be inferred. So Davidson doesn’t allow the interpreter to have access to beliefs as evidence, since the interpreter would then be begging the question. Instead, Davidson allows that the interpreter can reasonably ascertain when a speaker holds a sentence true, without knowing anything about a particular belief or meaning. That will then allow the interpreter to construct hypotheses relating a speaker and an utterance to a particular state of affairs at a particular time.
Davidson argues that because the language is compositional, it is also holistic: sentences are based on the meanings of words, but the meaning of a word depends on the totality of sentences in which it appears. That holistic constraint, along with the requirement that the theory of truth is law-like, suffices to minimize indeterminacy just enough for successful communication to occur.
Reading this excerpt from Wikipedia, we can evaluate the “compositional” aspects of the question “How did we get 4?”
If you’re reading this and understanding everything so far, then we probably share a belief system and we share a concrete understanding of what “how”, “did”, “we”, “get”, “four”, and “?” mean.
In Philosophical Lambda, this means that all these symbols are associated with a set that contain that same content in your “I” and my “I”. We can see this as our minds, but let’s keep the “I”’s.
This understanding that we have is compositional because each part of the sentence plays a role in how the sentence is understood.
For example, “4” the number is defined by its composition: [1+3, 2+2, 5–1, 19–15, .05+3.95, -9+5, 9–5 ….. INFINITY]
If we said that 2+6=4, then the composition of “2+6=” leads us to expect an 8. Since it is our belief is that “2+6” is within the composition of 8, not 4. But if we change our belief and make it so that 2+6=4, then we have to foster other beliefs where “2+6=4” makes sense.
What’s cool about this is that you can see that if we change the composition of number and algebra so that 2+6=4, there’s an existential ripple effect where all other compositions change (4 can not longer be said to be [1+3, 2+2, 5–1, 19–15, .05+3.95, -9+5, 9–5 ….. INFINITY] but it will be composed of other conclusions)
We can’t use the symbol “4” without knowing its composition.
Going back to Descartes, you can’t use the symbol “I” without knowing its composition. And any change to “I” has a ripple effect that can affect “I” in infinitesimal ways so that the composition of “I” is as vulnerable as 4; if our meanings and beliefs change.
Jumping to Hume, Kant and the importance of composition, the problem with Descartes’ “I think, therefore I am” is that he’s not “I am” because he “thinks.” Hume and Kant argue that Descartes thinks before he is and thus he is something determined by his context; of which thinking is a part of it.
When we ask “How did we get 4?” referring to apples, there are certain things you need to consider before you answer the question. And the things you consider happen almost instantaneously. The existential ripples of assumptions echo throughout the “I” and the 4 to reach certainty.
This internal composition of your thoughts can also be seen in Meno’s paradox. But we won’t go down that road.
So you’re asked “How did we get 4?” Without knowing anything else, someone says that they had 8 apples but 2 rotted and were thrown away. Your belief system knows that the question is not answered fully because the symbol 4 doesn’t correspond to any composition [8–2].
There exist “8–2” instances inside 4 but it would be something like:
4 = [8–2–2, 8–2+9–11, 8–2+5–7, …. INFINITY]
Here we see again that there are unlimited and infinitesimal ways “8–2” can lead to 4 but we also know that 8–2 alone doesn’t lead to 4.
What Hume and Kant said in response to Descartes was that “I am” and “thinking” were part of a composition that was predetermined before Descartes started to think. Although the compositional part is my interpretation. They probably said something in Latin or German.
Here it gets a little tricky and I hope to illustrate the “compositional” part.
Pragmatically, we can’t include 4 inside the composition of 4, ever.
Philosophical Lambda, being pragmatic, and taking its influence from Lambda Calculus, can’t have definitions refer to itself when defining itself.
Because, really, who likes it when they read the definition of a word and the word is inside the definition? Also, it would be like a snake eating itself. It’s just not pragmatic.
As such 4+0 is not within the composition of 4 in Philosophical Lambda because it has no pragmatic value. When you ask what is 4? You don’t ask what it is, but how it works, and it works in relation to other numbers; not to itself. We get “4 +0” through seeing how 4 relates to other numbers other than itself and how 0 relates to numbers other than itself. Although I’m not entirely convinced of this.
Here’s another feature of composition and it illustrates the aspect of “interwoven”ness and the ripple effect of existential compositions:
5 = [4+1, 9–4, 2+3…. INFINITY]
4 = 5–1 if and only if 5 = 4+1
If we change the meaning and function of 4, then all of the composition of 5 changes and 4 is no longer 5–1.
Although we got to this point by Donald Davidson’s commentary, this is also similar to WVO Quine’s view on confirmation holism.
We can say that a system is holistic in that everything you need to know about a system is within the system. The significance of a system and how it functions is determined by what it’s composed of. And the relationship of this composition is necessarily relative and subject to change.
An example of this is the existential ripple effect mentioned above. If the value of 4 changes, all numbers relative to 4 change as well and thus the whole system changes holistically.
Back to Descartes, in Meditations Descartes wanted to establish certainty about the world. He could deny everything except that “I am” a “thinking thing.”
Hume and Kant argued that before Descartes started to think, he was already something. If “I am” is a “thinking thing” and this thing was already something, then it becomes a sort of Russian doll thing.
We can illustrate this embedded system in Philosophical Lambda by using the brackets to designate a function and get rid of the equal in.
Since Philosophical Lambda aims to appear and act like Lambda Calculus, we’ll only use functions to express definitions. And, again, the inherent pragmatism of this cannot be ignored. 4 is only 4 because it functions within a system were 4 is 4. If you remove the function and system, 4 can’t be existentially coordinated.
Note also the beauty of this system:
4 is within itself but only after going through it’s relation to 5 and 1. The number 4 sees itself, or rather references itself indirectly, through 5 since it’s within it’s composition (eg 5[4+1…infinity) and the same thing for one.
It’s wrong to talk about 4 outside of context expecting it to retain its value and meaning. And all other numbers are established using the same pragmatism.
Some things such as numbers are infinite. However, the addition, subtraction and other similar functions are strictly defined.
4[…] denotes the functions that define 4 insofar as the functions insides the  result in 4.
viii. Back to Philosophical Lambda
Going back to Descartes’ I and it being within an embedded system, let’s make some assumptions: The “I” exist within a Universe, a Galaxy, a Solar System, on Earth, it’s Human, it has awareness:
I[ Universe [ Galaxy[ Solar System[ Earth[ Human [ Awareness [ I [..]..]..]..]..]..]
Again, we see how the I can define itself (you and me) but going through the systems Universe, Galaxy, Solar System, Earth, Human, Awareness…
Hopefully you can see how this self-consumption is also self-nutritious.
I don’t think the Universe can be defined with regards to Infinity:
Universe[ galaxies, stars, chemical reactions, beliefs, thoughts, function, ….. END]
The same with subtraction, addition, Solar System, etc. Their function is strictly defined as they are not variables but guide the way variables move within the system. Although I’m not entirely convinced of this.
One thing to note is that the function of a things comes only after it’s been observed. So the product of a function is what defines a function. It’s the inherent nature of pragmatism: what is pragmatic is predetermined by what is produced. You may just not be aware of it.
The function of the “I” in I am is that of a sub-system inside other subsystems. Like the number 4 is a subsystem of other, greater subsystems that are defined holistically. Regardless, the 4 and the I as sub-system are as integral to the pragmatic nature of the whole system as any other part of the system. Once the 4 or the I change in meaning and value, then the entire systems which depend on it also have a ripple effect that changes everything else.
The I, however, is a very special type of subsystem in the Universe.
For example, there’s only one I and that’s yours. And that’s a problem you need to overcome through will.
Another reason why the I is special is because it’s only one of two things in the Universe that can be said to have a will: the Universe and I.
The problem with induction is that we can only be certain of our own personal “I am” will. Everything else has to be abducted, induced or deduced.
From the link above:
Hume asks on what grounds we come to our beliefs about the unobserved on the basis of inductive inferences. He presents an argument in the form of a dilemma which appears to rule out the possibility of any reasoning from the premises to the conclusion of an inductive inference. There are, he says, two possible types of arguments, “demonstrative” and “probable”, but neither will serve. A demonstrative argument produces the wrong kind of conclusion, and a probable argument would be circular. Therefore, for Hume, the problem remains of how to explain why we form any conclusions that go beyond the past instances of which we have had experience
We can show what Hume means by this in Philosophical Lambda by assuming that we’re systems (“I”, dreamer of dreams, music makers) within systems (Universe, Galaxy, Earth, Humanity) that give rise to further systems (math, ethics, restaurant menus).
And thus What the Universe is composed can be summarized as:
As the Universe coalesces into what we may recognize as the Present Moment, so does the I coalesce into the “I am”; the body’s culmination of awareness.
Now you have 4[….] defined by functions and interwoven by relationship to other numbers and relative context. What about love? Don’t let it slip away!
Universe[…..Body[ Limbs[..], Organs[..], Awareness[ I[ love[ happiness[..], warmth[..], care[..]….INFINITY]…]….]…]
Inside the “I” we see stuff that our will is familiar with but can’t be said to exist outside the will. You can always be skeptical of someone pulling a fast one on you. And to a degree, you acceptance of happiness, love, warmth and care depends on a certain trust and will that, while guided by logic, cannot depend on logic to always be true.
This is where we see the significance of Godel’s incompleteness theorem:
Gödel said that every non-trivial (interesting) formal system is either incomplete or inconsistent:
1. There will always be questions that cannot be answered, using a certain set of axioms;
2. You cannot prove that a system of axioms is consistent, unless you use a different set of axioms .
So a question like “Do they love me?” when pressed against the skepticism of Descartes and Hume cannot be justified outright through deduction.
Remember, Descartes and Hume both were in search of certainty. And “Do they love me?” cannot be made certain.
BUT IT CAN BE MADE PRAGMATIC! For a system can have assumptions and use those assumptions as premises. Whether right or wrong, existentially for the self it doesn’t matter, because it’s pragmatic! (and unsure if not true or inconsistent)
Now you go on with the belief that they love you and all the pragmatic implications this has:
This is very existential and this is where the Kierkegaard quotes go talking about how love builds up; you can reach greater heights with love then without it; your existential self is the one that puts that concept of love with positivity inside the “I” function, it’s on you to love, etc….
When the body coalesces, it gives rise to the existential “I”, but through the coalescing of the body, there’s factors such as the coalescing of the enteric nervous system, sympathetic and parasympathetic nervous systems, atoms, neurons, blood distribution, etc.
When the “I” coalesces it gives rise to ideology. Which includes the beliefs, impressions, ideas, and truths you hold to be pragmatic when interacting and coalescing with the Universe. And this is what makes up your will.
Although Universe (physical) as the “ultimate” thing can be superseded by Reality (maybe non physical) if someone were to pragmatically choose. I’m not entirely convinced of whether Universe can be said to be the end all be all; I suspect Reality is.
Regardless, the pragmatic nature of the Universe doesn’t require certainty but pragmatic results. Because these results are what fuel the pragmatism the Universe requires to function. And this Universal will counterbalances the will of the I.
Let’s illustrate the Universal will:
Universe[ Galaxies[ Solar systems [ … Apes [ Humans [ You [sexual reproduction[..], health[…], genetic makeup[..], height[..]…]…]….]…]….]
Now that’s the Universal will. It’s complete and defined. You’re in it right now. It’s going through you, over you, into nooks and crannies you didn’t even you had.
The existential “I”, like the 4, is made up of the circumstances found within the Universe (including the “I”, but through indirect reference; like the 4),
Body[ Awareness[ I[ good[…], height[..], bad[..], language[..], appearance[ body[..]..], love[..]…INFINITY]..]..]
Since Body and Awareness is something founded in the Universe, it’s complete and defined. But the existential “I”, like the 4, has many possibilities for its definition. Since its definition is based on pragmatism and what the “I” can or can’t do is not known until it is done; since pragmatism is determined after the product from an action is seen.
The freeing aspect of the will of the “I” is that the “I” reaches definition and completeness through a choice:
Whereas the the 4 is defined impragmatically, in theory, as:
4[1+3, 2+2, 5–1, 19–15, .05+3.95, -9+5, 9–5 ….. INFINITY]
so is the “I” defined with such grandeur. But when the 4 and I are compressed into context, the possible answers shift and shrink.
Against the pragmatic will of the Universe, the “I” must choose from the possibilities of “I” and live it with. This is the leap and bridge found in Nietzsche and Kierkegaard.
Whether the “I” defines itself as 1+3, 2+2, 345274–345270, or whatever is within the will of the “I” as seen through its relation with the will of the Universe.
Like 4[…] is only seen through its relation with it’s context and definition.
Looking at the set of “I” again:
I = [culture, language, race, sex, gender, arms, legs, torso, brain, thought….etc]
We see that this is totally incomplete. It should look like this:
I[culture[..I[..]..], language[..I[..]..], race[..I[..]..], sex[..I[..]..], gender[..I[..]..], arms[..I[..]..], legs[..I[..]..], torso[..I[..]..], brain[..I[..]..], thought[..I[..]..]….INFINITY]
Where the significance of everything is interwoven and determined by the I, which in turn, holistically, determines the I. The I sees itself through its relationship with different concepts and helps determine itself through those concepts. The same way you’re looking at yourself now through language.
The “I” in this sense is post-organic. Like the 4. The “I” stays consistent and eternally the “I” while simultaneously changing its compositions.
The freedom comes from the ability to define what the “I” is through its relationship with the Universe. The will of the I is embedded in the Universe, and thus it’s embedded with the will of the Universe; but only insofar as the I recognizes and actualizes this will through pragmatism.
Thus, the uncertainty of the mental law is no mere defect of it, but is on the contrary of its essence. The truth is, the mind is not subject to “law,” in the same rigid sense that matter is. It only experiences gentle forces which merely render it more likely to act a given way than it otherwise would be. There always remains a certain amount of arbitrary spontaneity in its action, without which it would be dead.