acknowledging the various debates of Hume and Kant. — schopenhauer1

Yes, and thanks for the summary. Is it clear to you that either Hume or Kant has the better explanation here? Are Jha et al. Kantians? (Note, too, that Kant did not think math was analytic, like logic. He thought it gave us synthetic knowledge about the intuitive concept of "magnitude" -- that is, number per se. This makes me wonder if he would allow math an explanatory role, as in the above discussion.) — J

The current volume of Philosophy of Science has a paper on mathematical explanations in the sciences that I realize is talking about something very similar. The paper is “Are Mathematical Explanations Causal Explanations in Disguise?” by Aditya Jha et al. The question raised is whether a distinctively mathematical explanation (DME) for physical facts truly exists – whether “the facts under question arise from a degree of mathematical necessity considered stronger than that of contingent causal laws.”

Does it matter, for this parallel, whether math is a branch of logic, as many philosophers (and scientists) believe?

Yeah, but remember Kant thought math was synthetic a priori. In other words, our minds are still structuring time and space and experience. The math wasn't "in the world", that would be violating his phenomenal/noumenal distinction — schopenhauer1

evolution does provide a certain flavor of answer whereby our brains could not but do otherwise. — schopenhauer1

Pancomputationalism . . . would make cause (i.e. how past states determine future states) a sort of stepwise logical entailment. — Count Timothy von Icarus

Math, like language, is a tool of logic with rules. If we use it with the idea that our abstraction is trying to match reality, and we are correct in matching our abstractions to reality, it works because that's how we perceive identities, and our identities are not being contradicted by reality — Philosophim

This seems to make the LNC, e.g., contingent on the way the world is. But don't we want something much stricter than that, some way we can talk about necessity and impossibility? Can we arrive at what you're calling "a necessary understanding of the world"?

I'll bring in Nagel any post now! :smile: — J

I see what you mean, but we can construct an infinite number of worlds with different abstract entities highlighted (see "grue and bleen", Sider, p. 16) and most of them won't "work" at all, if by "work" you mean "give us a useful conceptual basis for navigating the world." Yet there is nothing wrong, logically, with the way these abstractions are being matched to reality. So can you expand on what it is to "perceive an identity"? -- that seems crucial.

Does the "pan" part of pancomputationalism provide a response to Jha et al.'s objection? That is, are the background assumptions which Jha et al. call "the very facts that make a purely mathematical result applicable" also generated computationally? I'm out of my depth here, but is there meant to be a beginning to this process of entailment -- some first premises?

[Husserl] tries to show how the formal, logical structures of thinking arise from perception; the subtitle of Experience and Judgment is Investigations in a Genealogy of Logic. The “genealogy” of logic is to be located not in something we are born with but in the way experience becomes transformed. Husserl describes the origin of syntactic form as follows.

When we perceive an object, we run through a manifold of aspects and profiles: we see the thing first from this side and then from that; we concentrate on the color; we pay attention to the hardness or softness; we turn the thing around and see other sides and aspects, and so on. In this manifold of appearances, however, we continuously experience all the aspects and profiles, all the views, as being “of” one and the same object. The multiple appearances are not single separate beads following one another; they are “threaded” by the identity continuing within them all. As Husserl puts it, “Each single percept in this series is already a percept of the thing. Whether I look at this book from above or below, from inside or outside, I always see this book. It is always one and the same thing.” The identity of the thing is implicitly presented in and through the manifold. We do not focus on this identity; rather, we focus on some aspects or profiles, but all of them are experienced, not as isolated flashes or pressures, but as belonging to a single entity. As Husserl puts it, “An identification is performed, but no identity is meant.” The identity itself never shows up as one of these aspects or profiles; its way of being present is more implicit, but it does truly present itself. We do not have just color patches succeeding one another, but the blue and the gray of the object as we perceive it continuously. In fact, if we run into dissonances in the course of our experience – I saw the thing as green, and now the same area is showing up as blue – we recognize them as dissonant precisely because we assume that all the appearances belong to one and the same thing and that it cannot show up in such divergent ways if it is to remain identifiable as itself. [It's worth noting the experiments on animals show they are sensitive to these same sorts of dissonances].

[Such experience is pre-syntactical, nevertheless] such continuous perception can, however, become a platform for the constitution of syntax and logic. What happens, according to Husserl, is that the continuous perception can come to an arrest as one particular feature of the thing attracts our attention and holds it. We focus, say, on the color of the thing. When we do this, the identity of the object, as well as the totality of the other aspects and profiles, still remain in the background. At this point of arrest, we have not yet moved into categoriality and logic, but we are on the verge of doing so; we are balanced between perception and thinking. This is a philosophically interesting state. We feel the form about to come into play, but it is not there yet. Thinking is about to be born, and an assertion is about to be made…

We, therefore, in our experience and thoughtful activity, have moved from a perception to an articulated opinion or position; we have reached something that enters into logic and the space of reasons. We achieve a proposition or a meaning, something that can be communicated and shared as the very same with other people (in contrast with a perception, which cannot be conveyed to others). We achieve something that can be confirmed, disconfirmed, adjusted, brought to greater distinctness, shown to be vague and contradictory, and the like. All the issues that logic deals with now come into play. According to Husserl, therefore, the proposition or the state of affairs, as a categorial object, does not come about when we impose an a priori form on experience; rather, it emerges from and within experience as a formal structure of parts and wholes...

This is how Husserl describes the genealogy of logic and logical form. He shows how logical and syntactic structures arise when things are presented to us. We are relatively passive when we perceive – but even in perception there is an active dimension, since we have to be alert, direct our attention this way and that, and perceive carefully. Just “being awake (Wachsein)” is a cognitive accomplishment of the ego. We are much more active, however, and active in a new way, when we rise to the level of categoriality, where we articulate a subject and predicate and state them publicly in a sentence. We are more engaged. We constitute something more energetically, and we take a position in the human conversation, a position for which we are responsible. At this point, a higher-level objectivity is established, which can remain an “abiding possession (ein bleibender Besitz).” It can be detached from this situation and made present again in others. It becomes something like a piece of property or real estate, which can be transferred from one owner to another. Correlatively, I become more actualized in my cognitive life and hence more real. I become something like a property owner (I was not elevated to that status by mere perception); I now have my own opinions and have been able to document the way things are, and these opinions can be communicated to others. This higher status is reached through “the active position-takings of the ego [die aktiven Stellungnahmen des Ich] in the act of predicative judgment.”

Logical form or syntactic structure does not have to issue from inborn powers in our brains, nor does it have to come from a priori structures of the mind. It arises through an enhancement of perception, a lifting of perception into thought, by a new way of making things present to us. Of course, neurological structures are necessary as a condition for this to happen, but these neural structures do not simply provide a template that we impose on the thing we are experiencing...

-Robert Sokolowski - The Phenomenology of the Human Person

We’ve talked a lot on TPF recently about thinking and being – not just Irad Kimhi’s book of that title, but the larger issue of how thought mirrors reality. Does the Law of Non-Contradiction state a logical truth? a truth about how things must be in the world? or, somehow, both? neither? — J

Q2. Why are 23 objects not evenly divisible into three collections of whole and unbroken objects? — J

Q2 is a linguistic problem and results from a particular definition of "object".

23 things can be evenly divided into three collections of 723
7
2
3
things.

But Q2 defines an object as something that is whole and unbroken, meaning that if a thing can be divided into parts, then by definition that thing cannot be an object.

Therefore, although 23 things can be evenly divided into three collections, by the given definition of "object", 23 objects cannot be evenly divided into three collections.

However, other definitions of "object" are possible.

For example, as the object "house" is the set of other objects, such as "roof", "chimney", "windows", etc, an "object" could have been defined as a set of three other objects, in which event 23 objects is evenly divisible into three collections of whole and unbroken objects. — RussellA

I look at a notepad, and I think "notepad". A notepad is a conventional object. It is a socially created object, for all intents and purposes. But then there is various laws of mechanics that were used in the making of the machines that made the notepad. These are "laws of physics". Whilst the technological use is in a way conventional, the physical laws behind it, which we also derived, as humans reasoning, are supposedly the ones we are discussing, the "objective" ones "in nature". The "true mathematical laws" that we are not conventionalizing, but teasing out with our mathematical models, and cashing out in accurate predictions and technological usefulness. So it is those we are getting at. Yet, imposed on top of that, is the same brain that makes a conventional item like "notepad", into "something" real, something that I presuppose every time I look at a notepad. I don't just see a bunch of atoms grouped together- I see a type of object. Now this is the tricky part where Kant does come in. What is the part that is conventional, and what is the "objective"? How are we to really know? These are two very different types of capacities coming together and converging:

1) The ability to parse the world into discrete objects and arrange them and describe them.
2) The ability to parse out various empirical understandings of the world THROUGH THE PRISM of a kind of brain that does the capacity described in 1.

So Nagel might say something like, The 2 [objective laws/logic] has created the 1 [cognitive laws/logic]. There is something that connects the two.

A true agnostic or nihilist of this scheme would say 1 and 2 are not connected in any meaningful way. Kant, for example, will make the move that 2 is really a sub-species of 1 (or how I interpret Kant). — schopenhauer1

Q1. Why is the number 23 not divisible (evenly) by 3?
Q2. Why are 23 objects not evenly divisible into three collections of whole and unbroken objects? — J

What we really want is an explanatory structure that preserves both of the seemingly ineluctable realities – of logic and of being. Kimhi has his views about how we might get there. A theistic argument might posit a “perfect match” because creation is deliberately thus. Or – using a metaphor from Banno – we find ourselves with a Phillips-head screw and a screwdriver that matches, so let’s leave a designed creation out of it and try to work on the problem in evolutionary terms. (I don’t think such an approach will take us far enough, but it’s certainly respectable.) — J

My take is that the tremendous success of our efforts to understand the world, which has translated into the causal mastery embodied in techne, represents strong evidence that we do come equipped to know the world and that the world is intelligible. — Count Timothy von Icarus

I buy Gadamer's argument that it's quite impossible to make any inferences without begining with some biases. We can always question these biases later. — Count Timothy von Icarus

To say that the alignment between screwdriver and screw is an opaque and brute fact is to have abandoned the search for an overarching explanatory structure. If there is an explanatory structure that preserves both, then that explanation must encompass both the mind that knows reality and reality itself. I don't see how one could arrive at an explanatory structure such as you desire without this overarching aitia. — Leontiskos

There is a major debate as to whether there are non-causal mathematical explanations of physical facts that show how the facts under question arise from a degree of mathematical necessity considered stronger than that of contingent causal laws

“The facts allegedly explained by a DME do not obtain because of a mathematical necessity but by appeal to the world’s network of causal relations. . . . [Mathematics] is not a constraint on what the physical world must be.” — J

Q1. Why is the number 23 not divisible (evenly) by 3?
Q2. Why are 23 objects not evenly divisible into three collections of whole and unbroken objects? — J

I don't just see a bunch of atoms grouped together- I see a type of object. — schopenhauer1

But that doesn't make Q2 a linguistic problem, since we've stipulated what an "object" will be in this question. — J

But what about the problem posed by the question itself, now disambiguated? -- presumably you'd say "No, it can't be divided evenly" and so we want to know whether this is due to a mathematical fact or a fact about the world. — J

Q4: Why can’t my cat be on my lap and in Paris at the same time? (constraint: I live in Maryland) — J

So mathematics models the world because the world exhibits regularities that can be mathematically described, not because the world is constrained by the mathematical framework. — Wayfarer