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.

(source)

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.

vii. Holism