Thought is essentially processual — Count Timothy von Icarus

It is. Cardano's point is that the earlier part of the process might belong to a subject other than the one that says "I am".

It just doesn't seem very convincing. The experience of being aware of an experience is phenomenologicaly concurrent with it. Certainly, it's true that we don't have an experience "in no time at all," but it seems like a mistake here to take experience as being decomposable into smaller and smaller intervals, with certain parts having to follow others in serial order. Understanding seems to occur as a sort of parallel, composite process (which makes sense given our cognitive architecture).

I think the issue might be conflating the process of developing a thought into a propositional form, and the experience of self-awareness itself. For example, in the passage from Augustine above he spends a paragraph unpacking inferences made from an experience of knowing and willing that occurs in an instant. These two are divided in propositional thought, yet if a line drive is hit to us while playing baseball, our experience doesn't seem to involve first knowing that the ball has been hit, then willing our body to move to catch it. We do all of these together, seamlessly knowing, willing, and acting. Likewise, in introspection we experience and experience our own experiencing together.

but it seems like a mistake here to take experience as being decomposable into smaller and smaller intervals, with certain parts having to follow others in serial order — Count Timothy von Icarus

Beyond the possibility of a mistake, the task of decomposing thoughts on the axis of time is very troublesome, and I would be interested to know if there was ever a philosopher to undertake this task. For example, when we think "red car", does that take less time than if we were to think "the happy swimmer dove into the shallow lake"? Surely one has many more concepts than the other, but ultimately — at least for me —, both give one single mental image that can be realised at a given instant of time. So is it a single thought when we say "X therefore Y" because we are uniting these concepts or is it the thought of X followed in time by the thought of Y? I expressed this worry before in the thread:

Furthermore, "someone thinks therefore something is" is a phrase, it is hard to articulate (and perhaps that is the issue) how that phrase translates to thoughts, ¿is it a single thought or 2+ thoughts one after the other? If the latter, perhaps the first "something" is not the same as the second "something".
If the former, when we say "I think" in "I think therefore I am", we can be talking about "I think therefore I am" itself, then it can be taken as self-fulfilling. — Lionino

In any case, though Cardano's criticism is very much welcome and healthy, I pointed:

But then Descartes states not "I think therefore I am" but "'I am, I exist,’ is necessarily true whenever… it is conceived in my mind." — Lionino

Descartes' idea starts with an immediate intuition. And it may not even be that we need to know thinking implies existence to have this immediate intuition (the inference we were talking about). Everytime we think we are making sure that we exist, every thought comes with the experience of being there, of existing — a "da-sein" if you will.

Professor Hintikka in a letter put it as "ego cogitans existo" instead (I, who thinks, exists).

Understanding seems to occur as a sort of parallel, composite process — Count Timothy von Icarus

Surely it is a process, but going too far with this idea would have implications on our view of personal identity, which might not be something that I want to commit myself to. Specifically, if our understandings cannot be at any point analysed from one other, we are committing ourselves to a psychological continuity. I think that it is desirable to be able to separate the thoughts from Tuesday from the thoughts of Sunday, even though they are ultimately linked — the first and last link of a chain are distinct even if ultimately connected.

I think the issue might be conflating the process of developing a thought into a propositional form, and the experience of self-awareness itself. For example, in the passage from Augustine above he spends a paragraph unpacking inferences made from an experience of knowing and willing that occurs in an instant. These two are divided in propositional thought, yet if a line drive is hit to us while playing baseball, our experience doesn't seem to involve first knowing that the ball has been hit, then willing our body to move to catch it. We do all of these together, seamlessly knowing, willing, and acting. Likewise, in introspection we experience and experience our own experiencing together. — Count Timothy von Icarus

Yes, I think this is an important distinction. Even if an experience, physically, neurologically takes place in time, it is still a experience; putting it in words is a translation of the experience, aiming for communication with people whose minds we assume are like ours; but yet language does not exhaust thought. If I had to make a wild guess, I would say this conflation is more common in people who think in words rather than images.

But as before, I still think the experiences/thoughts of Tuesday and the experiences/thoughts of Sunday can and ought to be separated.

the task of decomposing thoughts on the axis of time is very troublesome, and I would be interested to know if there was ever a philosopher to undertake this task. — Lionino

“…. The schema of substance is the permanence of the real in time…..
……The schema of possibility is the determination of the representation of a thing at any time….
……The schema of reality is existence in a determined time….
……The schema of necessity is the existence of an object in all time….

……It is clear, from all this, that the schema of the category of quantity contains and represents the generation (synthesis) of time itself, in the successive apprehension of an object….
……the schema of quality the synthesis of sensation with the representation of time, or the filling up of time….
……the schema of relation the relation of perceptions to each other in all time (that is, according to a rule of the determination of time)….
……and finally, the schema of modality and its categories, time itself, as the correlative of the determination of an object—whether it does belong to time, and how.

The schemata, therefore, are nothing but à priori determinations of time according to rules, and these, in regard to all possible objects, following the arrangement of the categories, relate to the series in time, the content in time, the order in time, and finally, to the complex or totality in time.…”
(CPR A143-145/B182-185)

Maybe not exactly what you asked for, but does show there was/is a philosopher tasking himself with decomposing that which is thought about, to its necessary relation to time.

Beyond the possibility of a mistake, the task of decomposing thoughts on the axis of time is very troublesome, and I would be interested to know if there was ever a philosopher to undertake this task. For example, when we think "red car", does that take less time than if we were to think "the happy swimmer dove into the shallow lake"? Surely one has many more concepts than the other, but ultimately — at least for me —, both give one single mental image that can be realised at a given instant of time. So is it a single thought when we say "X therefore Y" because we are uniting these concepts or is it the thought of X followed in time by the thought of Y? I expressed this worry before in the thread:

I have been kicking around ideas on this for a while. In Eddington's "The Rigor of Angles: Kant, Borges, Heisenberg, and the Ultimate Nature of Reality," he discusses a philosophical treaties by Heisenberg that tries to apply his famous uncertainty principle to language. His basic thesis is that words (and so propositional, syntactical thought) can have more or less dynamic or static meanings. In science, we try to speak very precisely and rigorously, using many words to be clear. This ultimately makes our language less dynamic, causing it to cover less cognitive ground. The more we try to focus them on to just one thing and fix that thing, the more the words lose their purchase on what we are describing.

I think we can tie this back to limits on the "cognitive bandwidth," conciousness has. R. Scott Bakker has written some good stuff reviewing studies on the quite limited bandwidth/bit rate of human propositional/linguistic thought (inner monologue being a prime example). Long descriptions essentially get too long and the flood of precise detail makes us lose the thing being described. For us to understand complex propositions about complex topics, e.g., some proposition about "Hegelian dialectical," we cannot stop to unpack our propositional knowledge of all the terms. We must have studied the terms and internalized them so that we have a "grasp" of their intelligibility such that they can be "present" to us simply, without unpacking.

We might liken this simple grasp to Aristotle's second sort of knowing, adiaireta. It's more a noetic awareness of the thing. It might be cultivated and informed by propositional knowledge, but it's opposite is ignorance or lack of awareness of a term, not falsehood as in propositional thought.

So for instance, we might paint a word portrait of the Mona Lisa quite well in a paragraph. If we try to be super detailed and start listing precise dimensions, hex codes for the colors used, etc., we can have a description with way more precision that we nonetheless read and have no idea what it is describing. By contrast, Keats' "Ode to a Grecian Urn," captures the substance of an art work in a dynamic way that a very static description cannot.

This is also why I think we can get endless milage out of some of the more poetic, vague philosophers. They don't fix their subject to the same degree, and this allows their words to cover a more dynamic range.

I think there is a good convergence here with some more phenomenolgical works on knowledge (e.g. Robert Sokolowski) and also St. Aquinas' understanding of the "God's eye view," where intelligibilities are present "all at once." The "view from nowhere," or "view from anywhere," errs by failing to account for how knowing occurs over time and how more and more abstract and rigorous formulations lose their grip on intelligibilities. The "view from nowhere/anywhere," really wants to be the God's eye view, where intelligibility is simply present, but the desire to excise God from an explanation led to excising mind as well, leading to incoherence where "objectivity approaches truth at the limit," and so the true view of things is "how they are conceived of with no mind."

The goal of understanding then is a sort of contemplative grasp that can then be used in the dividing and combining of discursive thought (e.g. Aquinas' description in his commentary on Boethius' De Trinitate).

The existence of bodies, aka res extensa, is far from certain. — Lionino

But how do I know that He has not brought it to pass that there is no earth, no heaven, no extended body, no magnitude, no place[...] — Second Meditation

It's more a noetic awareness of the thing. It might be cultivated and informed by propositional knowledge, but it's opposite is ignorance or lack of awareness of a term, not falsehood as in propositional thought. — Count Timothy von Icarus

Interesting. This seems to be where the knowledge of my own existence falls into.

This is also why I think we can get endless milage out of some of the more poetic, vague philosophers. — Count Timothy von Icarus

I wouldn't say being vague is ever a good thing in philosophy, I would say it is terrible indeed. Though the other side, limiting philosophy to language and philosophising by analysing propositions and syllogisms is also far from ideal, even if useful sometimes.

The goal of understanding then is a sort of contemplative grasp that can then be used in the dividing and combining of discursive thought — Count Timothy von Icarus

Instigating. There seems to be a useful anatomy of intellect entrenched in this idea.