• J
    621
    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? That’s just one example.

    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.”

    If you substitute “logical” for “mathematical,” you can see that the question is very much about whether our analytical (for lack of a better term) knowledge imposes itself on the physical world in a way that is genuinely explanatory. Jha et al. think not, and present a strong argument that all purported DMEs are actually rephrased or disguised versions of the causal explanations that ordinarily obtain. I won’t go into all that in detail. Rather, I want to focus on two sample questions that Jha et al. offer, and make a connection with the larger questions about thinking and being.

    Consider these two questions:

    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?

    Obviously, the first question is about abstracta, and the second about physical stuff. Jha et al. point out that these questions also require two different sorts of answers. The first needs a mathematical proof. The second can assume the truth demonstrated in Q1, and allegedly apply it to a physical situation, arguing modally that such a division is impossible. I say “allegedly” because that would be an example of a DME, if legitimate. But is it?

    Jha et al. argue that various presuppositions such as the principle of mass conservation and the physical integrity of individual objects (this has to be assumed in order to get “whole and unbroken”) are essential to the answer to Q2. “These presuppositions do not participate as trivial, ordinary background facts merely constituting the task at hand; these are the very facts that make a purely mathematical result applicable, relevant, and meaningful in a physical context.” In other words, if the world (of the example) was not the way it was, then the mathematical facts alone would not supply an explanation. In their conclusion, Jha et al. put it this way: “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.”

    This looks like a definitive answer to the question about how math and the world relate: not necessarily, if at all. But Jha et al. recognize that this still leaves deeper problems; they end by asking how it is that math nevertheless consistently plays such a vital role in scientific explanation. “Does this mean that mathematical structures essentially represent physical structures in some deep way?” I think the argument of their paper demands a No, but I’m glad they aren’t willing to declare the question settled.

    The parallel with the more general questions about thinking/logic is, I hope, obvious. We can easily create two new questions:

    Q3: Why must the LNC hold (under the usual constraints) as a principle of thought?
    Q4: Why can’t my cat be on my lap and in Paris at the same time? (constraint: I live in Maryland)

    Following the DME idea, are we to answer this by saying that there is a “distinctively logical explanation” (DLE)? If so, what about Jha et al.’s challenges about non-trivial background conditions?

    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.)

    So, I’m interested to know if others see the same connection I’m seeing here with the cited paper. And I’m interested in any other thoughts the paper might raise about the nature of explanation as an appeal to causes. A specific question: Does it matter, for this parallel, whether math is a branch of logic, as many philosophers (and scientists) believe?
  • Philosophim
    2.6k
    All people have the ability to discretely experience. That's to take the sea of existence and form identities or 'existences'. So you can look at 1 field, or 1 blade of grass, or 1 piece of grass. The ability to form a discrete identity, is what '1' is. When you are able to say you have 1 identity, another identity, and you want to lump them into another identity that counts how many individual identities there are, that's 2.

    Math is simply the logical result of the combination and separation of discrete identities. That's why I can have 1 banana, add another one, and I have 2. Each banana isn't the same mass or size. Its about adding the concepts of what we discretely identify together. That's why it 'works'. If our discrete identities about the world "That is a banana" are true, then it is also true that there are two bananas in our grouped identity.

    But because math is about identities, we can create identities in our head that don't work in the real world. For example, each family in America has 1.5 children. The abstraction of the average is mechanically correct, but if it is trying to match reality, it fails as no one has 1 and a half kids.

    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. Thus we can have two bananas, because they are actually bananas. We can add two unicorns, but we cannot have two unicorns, because unicorns don't exist.
  • schopenhauer1
    10.9k

    I just want to add, that I don't see how this discussion can move forward without at least acknowledging the various debates of Hume and Kant. Kant, as we know, made the way humans conceive the world as "transcendental" and thus made it not only the limits, but necessary that we see this world the way it is. Our cognitive mechanisms can only engage the world in such a way, in other words.

    On the other side of the spectrum is the notion that the world is amenable to numeracy and mathematical analysis, because indeed, there is a logic there in the world. We can call these "realist" theories, and can even take from ancient philosophies of the sort like Logos, Natural Reason, and the like. Pythagoreanism is another one.
  • J
    621
    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.)
  • schopenhauer1
    10.9k
    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

    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.

    Granted, I think we can move beyond Kant. He didn't seem to have a notion of evolutionary change, and I think this might have changed his theory a bit.

    I will say, Schopenhauer was aware of evolutionary ideas (not Darwin yet as that came about around the last years of his life). Schopenhauer thought that any materialist/physicalist answer would always be discounting the way our minds presuppose the world in a sort of "If a tree falls..and no one there to hear" kind of way. But, moving those kind of debates aside, or perhaps returning to them, evolution does provide a certain flavor of answer whereby our brains could not but do otherwise. Evolution works contingently but not unconstrained. There is a bounded freedom that evolution can only allow perhaps, for so much tolerance but what survives perhaps, is a necessary kind of understanding of the world, that conforms with how it "really" works.

    And though some posters on here dismissed my claims regarding evolution and logic in other discussions, I think it now comes right back into focus. That is to say, there is a "foundation" to logical reasoning that I might call a "primitive inferencing" that through the contingencies of cultural learning, can understand and refine more accurate versions of the world. The "primitive inferencing" was necessary to survival, but the contingent part was how accurate we were able to shape it through cultural learning.
  • Count Timothy von Icarus
    2.8k


    Interesting post there, I'll have to check it out.


    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.”

    On a related topic, I've seen information processing and computational theories of causation. The Routledge Handbook of the Philosophy of Information had a good one but I forget who wrote it. It might be a bit dated now.

    It's worth noting that a great many physicists embrace pancomputationalism to some degree, which would make cause (i.e. how past states determine future states) a sort of stepwise logical entailment. Off the top of my head I can think of Vedral, Davies, Landaeur, Lloyd, Tegmark — although they have quite different views in some ways, Tegmark's "Mathematical Universe Hypothesis," (the universe just "is" a mathematical object) probably being the most divergent and most open to criticism. But this position is open to a number of critiques, in particular that it requires that the universe be computable and not contain true continua, which is an open question.

    Not all pancomputationalist literature really brings out how different it actually is from the "everything is little balls of stuff, building blocks," 19th century metaphysics that somehow remains the "default" in many of the special sciences. It would be wrong to say such a view implies things are "made of bits," for instance, and the necessarily relational character of information and the inability to carry out reductionism, at least in the manner of corpuscularism (i.e., parts defined in terms of wholes instead of whole being just a sum of their building blocks) makes for a different framing of causality.

    Just for an example, in the process of computing PRIME(7) (the functions spits out 1 for prime, 0 for not-prime) we might say there is a sense in which "what 7 is," determines the output of the whole, even though it's also true that if a thing "is what it does," "7" is not some sort of atomic entity here.

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

    IDK, it seems to me like a great deal of math and logic is fairly irrelevant to our knowledge of the world. There is uninteresting math. There are logics that let you show anything expressible is true. So either way, it seems we have to be selective using epistemic criteria.
  • J
    621
    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 distinctionschopenhauer1

    OK, I think that's right.

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

    It's appealing as an answer, no doubt. What troubles me about the evolutionary explanation is that the "arrow" seems to go from experience to thought. That is, our minds evolved as they did because of what we encountered in the world. 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
    621
    Pancomputationalism . . . would make cause (i.e. how past states determine future states) a sort of stepwise logical entailment.Count Timothy von Icarus

    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?
  • Carlo Roosen
    243
    I didn't notice this post while I was busy posting my own. It seems the topics are highly related. Also, the approach is so different that I can barely see the connection between the two. Can anyone comment on that, how these two post relate in terms of frame of reference?
  • J
    621
    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 realityPhilosophim

    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.
  • J
    621
    Could you refer us to which post? And perhaps quote some key passages here, so we can respond? Thanks.
  • schopenhauer1
    10.9k
    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 know Nagel thinks that the universe is directed in some way to reveal objective truths, or something of this nature. It's sort of a neo-Logos philosophy, perhaps.

    There are really sticky and interrelated problems here..

    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).
  • Count Timothy von Icarus
    2.8k


    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.

    Well, first I'd say that a great number of constructs seem "wrong" logically. For example, logics where one can prove anything and its negation dont seem to have anything directly to do with truth-preservation or inference.

    Grue and bleen are a bit different. Here is where the appeal to "the logic of the world," shaping "the logic of natural selection," and thus "the logic of cognition," comes in. I'd add that we should not be tempted to reduce everything to evolution here either. Developmental biology is also key; the fact is that if there is a "logic of the world," our own growth and development as individuals is constantly being shaped by this, e.g. that we experience touch isn't just "evolution," but also due to our touching things and the properties of the things we touch.

    Sokolowski has a great explanation (via Husserl) of how predication emerges from phenomenology (which of course is underlied by physical processes, but perhaps not "reducible" to them). This explanation sits anterior to the Kantian and biological ones, rather than conflicting with them, which is what makes it so interesting to me.

    Our natural faculties, perhaps our "form of life," precludes certain abstractions that might be "valid" in a sense. Grue and bleen might be examples. People will never use them because people cannot see, touch, taste, etc. how old something is. I say "might" be valid because "how old is something," is also a fraught question. On one view, everything is about 14 billion years old, no variation. Or, "how old is the Ship of Theseus, rebuilt in whole 20 times since it first set sail?"

    Likewise, while Wittgenstein notes that pointing "could" refer to what is directly behind our shoulder, it doesn't in any culture because our eyes are not on the back of our head and we could not see what we were pointing at in this manner. This isn't just about evolution, but also about the properties of light. One sees nothing to point at in a dark room. In the same way, human cultures distinguish colors with some small variation, but absolutely none developed names for colors in the ultraviolet spectrum. Presumably, this is because, while insects can distinguish these, we cannot.

    So we come fixed with a starting point, with biases. This isn't a bad thing. 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.

    But I guess what this topic often seems to boil down to is "either we are equipped to know the world or we aren't." One can always throw up road blocks, denying the validity of reason, or claiming we only ever experience ideas not the world, etc. 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
    2.8k


    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?

    That's a tough question because it really varies. Tegmark thinks the multiverse is just an abstract object that exists necessarily. This is ontic structural realism; things just are the math that describes them, so there is no separation as Jha supposes. The cleverly titled book "Every Thing Must Go: Metaphysics Naturalized," by Ross and co. is a somewhat similar project.

    Being "first" would just be in reference to the time dimension of some universe (most of these guys are eternalists, but not all). The problem of "if stuff can just start existing for 'no reason at all' at T0 and it existed in no prior state, shouldn't things just start to exist at random?" is still considered relevant in cosmology, and so the idea that the universe is "without beginning or end," (e.g. Aristotle) continues to be popular speculation.

    Some forms of "It From Bit" (John Wheeler) are participatory and have mind built into them from the outset. Thought doesn't "mirror" reality because thought and intelligibility (quiddity) is essential to reality. If "being" or "reality" are to mean anything, it has to be what is given to thought. (Another old idea, maybe better expressed in ancient philosophy TBH, but new stuff makes an empirical case for it as well). Henery Strapp is an example here.

    Yet in either case, I think the separation between mathematics and "the world" is blurred from the get go. Sure, the universe isn't all mathematics. But isn't it necessary that the universe (or its contents) be something and not everything?

    Where there is similarity is in the view that the world can be viewed as a giant quantum computer, perhaps a lattice of cellular automata. I do think this answers Hume's argument against causality to some degree, because here cause is intrinsic to "what the universe is," rather than natural laws somehow "causing" things "like a headmaster shuffling the planets around like school children," (as Hegel puts his objection).




    -----------

    Anyhow, for those interested, this is the Sokolowski explanation. For those familiar with phenomenology, it might not be as interesting. It was big for me because I hadn't even considered analyzing the emergence of logical reasoning in terms of the content and properties of experience before, just a total blind spot.

    [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
  • Joshs
    5.7k


    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

    I take it Richard Rorty’s book ‘Philosophy and the Mirror of Nature’ didn’t leave much of an impression on you.
  • RussellA
    1.8k
    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 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.
  • schopenhauer1
    10.9k
    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

    This seems to relate to what I was saying here:
    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
  • Srap Tasmaner
    5k


    Do we really expect explanations of anything to be relative to nothing , not even the whole universe?

    Ho ho, comes the answer, mathematics is eternal and unchanging and not relative to our universe! Logic too! And some other stuff. A complete explanation is relative only to this meta-universe, which, being necessarily as it is, needs no explanation. (Which is lucky.)

    Yeah.
  • Count Timothy von Icarus
    2.8k


    IMO, Rorty offers a much stronger critique of a particular modern view of "objectivity" than a positive case for considering all uses of "truth," "dependence," or even "objective" as "old epistemological honorific."

    The argument always felt to me like:
    A xor B
    Not-A
    Thus, B

    But we can simply deny the premise "A or B," because we have C, D, E, etc. Plus, there is the issue of accepting B seemingly allowing for "A and/or B, your choice!"
  • J
    621
    Oh, it left a big impression! He's an elegant, insightful writer. But I think he's wrong on every important point. This is too major a question to tuck into this OP, but I'd love it if you or someone else wanted to do an OP on Rortian pragmatism.

    (Have you seen the discussion of Rorty by Richard J. Bernstein, in connection with Gadamer and Habermas, in his Philosophical Profiles? Bernstein's view is, approximately, the same as mine.)
  • J
    621
    Yes, that's why the constraint of "whole and unbroken" matters. But that doesn't make Q2 a linguistic problem, since we've stipulated what an "object" will be in this question. We've solved that problem. 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.
  • Leontiskos
    3.1k
    This is a helpful OP.

    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

    In your other thread we touched on the Scholastic transcendentals or convertibles. Another transcendental besides being and truth is oneness (unum).

    For Aristotle mathematics is the study of what belongs to quantity in various different ways. For example, arithmetic is the branch of mathematics that studies discrete quantity.

    Now is it a causal fact that reality is bound up with oneness? Not really. Oneness is metaphysically foundational to reality, and is convertible with other foundational rational aspects of reality. Usually when we think of a causal reality we think of something that is limited to some subset of reality or some subset of substances. For example, reproduction via pair mating is a causal reality because it is differentiable from other kinds of reproduction and from other kinds of causes. To call the transcendental of unum "causal" would seem to be mistaken given its extreme ubiquity. Nevertheless, we need not say that it is necessary in some super-metaphysical (mathematical?) sense. So if the only categories are thought to be the category of the causal and the category of the mathematically necessary, then we would be out of luck. A universal metaphysical property of all reality, such as unum, is neither.

    This idea is bound up with Platonism: that there are universal forms in which all of reality participates, and in which the human mind participates in a special way through studies like mathematics. In that way I would want to say that mathematics is not prior to reality and reality is not prior to mathematics—which is perhaps an Aristotelian variant of the Platonism. But whether we think of Plato or Aristotle, in either case there must be some tertium quid in which both reality and human knowing participate.

    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

    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.
  • J
    621
    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 think we have to leave a pretty large area of "the world" open to hermeneutic interpretation rather than empirical/analytic inquiry, but as long as we do that, the world does seem intelligible, as you say. I suppose people will differ here -- for some, that's enough said; for others, me included, there's still the question of why? Maybe "it just is"? But isn't that super-convenient for us? Can some sort of evolutionary account get us to an explanation? Nagel's concerns enter here . . . plus, there remains the question of whether any of this entitles us to speak about necessity and impossibility.

    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

    Me too. I've noticed that some philosophers want to use logical principles as a kind of bulwark against the dreaded hermeneutic circle, which they fear leads to logical nihilism, and relativism in general. Would Gadamer agree that the LNC is a bias? Need a new OP for that.
  • J
    621
    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

    Yes, and in fairness, a good evolutionary explanation wants to respect these constraints. It wants to show us how both mind and the world evolved to reflect what you're calling the unum of being and truth. But as Nagel and others have pointed out, if the explanation is genuinely scientific, then it's going to have to account for consciousness (mind) before it can tackle any relation between mind and world. And of course a really thorough explanation would almost certainly dissolve this crude binary, "mind/world". Moreover, the "mind" of evolution may or may not turn out to be the same thing as whatever would be able to, for instance, participate in the Forms. In other words, we may not be able to get from "mind" understood as a singular psychological/biological phenomenon to the sort of mind that could have access to truth. Anyway, we're a long way off from any workable theories about all that.
  • Leontiskos
    3.1k
    - I agree. Theism and evolution are both examples of unified theories. Theism is a case where mind and matter are said to come from mind; evolution is a case where mind and matter are said to come from matter. In both cases one side is given a primacy, even if the explananda are only thought to be virtually or implicitly present in the explanans. Of course there are also more robust dualistic theories than the brute fact theory noted.
  • Wayfarer
    22.5k
    From the abstract of the paper:

    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

    This is very much a question about the ontological status of mathematical laws—whether they are conceptual tools imposed by observers to describe the world, or whether they reveal some deeper, inherent structure of the universe, suggesting an a priori necessity. I think that the idea that the world might exhibit a kind of mathematical necessity independent of human observation is resisted because it seems to suggest that something beyond the physical and observable, namely abstract mathematical truths, might be causally efficacious. Empiricism doesn't like that.

    “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

    Isn't mathematical physics a distillation of quantifiable values mapped against observable data? Insofar as mathematics is used to quantify and then model some object of analysis, then mathematical logic is applicable to those models. The mathematical structures qua predictive principles accurately capture and model the relevant attributes of the objects in question, such that they will conform with those predictions. It doesn't mean that the outcome is constrained by the model, but that the model accurately reflects the real attributes of the objects and relationships in question. 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. But because those relationships are faithfully captured by the mathematics then mathematical logic can be applied to it's analysis, and further, often unexpected, entailments can be discovered (as discussed in Eugene Wigner's famous paper The Unreasonable Effectiveness of Maths in the Natural Sciences.)
  • unenlightened
    9.2k
    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
    A1. 23 divides by 3 exactly into 7 & 2/3.
    A2. You have introduced 'whole' and 'unbroken'.

    If you have 23 objects you have already mathematicised them by counting: they are pre-labeled, as it were, and the act of division is a relabelling, labels which we can call in this case ,' a, b, and c,' instead of numbers again (that would be confusing). Then we have objects:

    1a, 2b, 3c, 4a, 5b, ... and so on. And because the labels are always applied in the same sequence and we always stop the sequence in the same place, the result is always the same.

    First you learn the label names in sequence, and then you apply the labels to themselves and that is the abstraction that is arithmetic. And the meaning of the name is its position in the sequence, and nothing else.

    And when philosophers and mathematicians have thoroughly forgotten their childhood, they wonder that the world should mysteriously agree with their abstractions, as though they were abstracted from nowhere at all.
  • RussellA
    1.8k
    I don't just see a bunch of atoms grouped together- I see a type of object.schopenhauer1

    Yes, as you say, "I see" a notepad.

    In the world are many objects, where each object is a sheet of paper, but it is the "I" that sees them as a single object, a notepad.

    It is the "I" that sees a relation between many different objects in the world. It is not the world that is relating a particular set of objects together.
  • RussellA
    1.8k
    But that doesn't make Q2 a linguistic problem, since we've stipulated what an "object" will be in this question.J

    The moment we've stipulated what an "object" will be, Q2 becomes a linguistic problem, because there are many different ways an "object" can be stipulated.
    ===============================================================================
    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

    Q2 is defining an object as being whole and unbroken.

    Therefore, 24 objects can be evenly divided into three collections each of 8 objects

    Also, 23 objects can be evenly divided into three collections each of things.

    By Q2's definition of "object", 23 objects cannot be evenly divided into three collections of objects.

    However, other definitions of "object" are possible.

    One of the Merriam Webster's definitions of "object" is "something material that may be perceived by the senses".

    Using this definition, as of an object is something material that may be perceived by the senses, we can say that of an object is also an object. In that event, 23 objects can be evenly divided into three collections of objects.

    Ambiguity arrives through deciding what exactly is the definition of an "object".

    Knowing whether 23 objects can be evenly be divided into three collections depends on the definition of "object". This is a linguistic problem that has to be resolved even before we consider mathematical facts about the world.
    ===============================================================================
    Q4: Why can’t my cat be on my lap and in Paris at the same time? (constraint: I live in Maryland)J

    Presumably, "my cat", being an average cat, has a length of 30cm, height of 20cm and width of 15cm, meaning spatially extended.

    In other words, "my cat" does exist in more than one location at the same time.

    Perhaps not as extreme as Paris and Maryland, but spatially extended nevertheless.

    Though perhaps your cat unfortunately died, the brain sent to Paris for medical research and the body buried in Maryland.

    In that event, one could rightly say that your cat is both in Paris and Maryland at the same time.

    However, this depends on what exactly does "my cat" mean, raising the question as to the meaning of the terms "my" and "cat". This takes us back again into having to solve the linguistic problem before being able to solve the ontological problem.
  • J
    621
    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

    I think this is Jha et al.'s thesis, pretty much. It's the world's (causal) regularities that permit math to function as part of an explanation. A different world, if there could be such, would reveal different regularities, but the role of math would be unchanged.
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