Abstract Explanations in Science


by Scientia SalonMathematical_equations

This is a paper by Christopher Pincock, a philosopher at Ohio State University, tackling the interesting issue of whether, and in what sense, mathematical explanations are different from causal / empirical ones. Here is the abstract:

This article focuses on a case that expert practitioners count as an explanation: a mathematical account of Plateau’s laws for soap films. I argue that this example falls into a class of explanations that I call abstract explanations. Abstract explanations involve an appeal to a more abstract entity than the state of affairs being explained. I show that the abstract entity need not be causally relevant to the explanandum for its features to be explanatorily relevant. However, it remains unclear how to unify abstract and causal explanations as instances of a single sort of thing. I conclude by examining the implications of the claim that explanations require objective dependence relations. If this claim is accepted, then there are several kinds of objective dependence relations.

And in what follows are some of the highlights from the paper, which we thought were particularly interesting while reading it. We have alerted Prof. Pincock of this, so hopefully he will chime in during the discussion.

In the philosophy of science, discussions of explanation tend to assume that most explanations are causal explanations. Disagreements arise concerning what a cause is and how the explanatorily relevant causes are to be identified.

This methodology more or less guarantees a certain kind of satisfying result: there turns out to be only one kind of explanation. However, the worry remains that this unified category of explanation is more an artifact of the tools used than an underlying unity.

The case study for this article is Plateau’s laws for soap-film surfaces and bubbles. …

The power of Plateau’s laws is that they work for any such soap-film system. The frame can be any shape. In fact, there might not be a frame at all. An unconstrained soap film may form a bubble that encloses a given volume. Such a system almost trivially meets Plateau’s laws as it has only one smooth surface. A more interesting case involves more than one bubble. These systems obey Plateau’s laws as well

Taylor’s solution was to develop a new mathematical theory of surfaces that incorporated these complexities. … Prior to Taylor’s ground-breaking work in the 1970s, mathematicians were forced to resort to experiments with soap films to discern the structure of this or that surface. … With these definitions, the key result is that ‘any such configuration of surfaces must of mathematical necessity conform exactly to the three geometric principles stated at the beginning’ (Almgren and Taylor [1976], p. 86), namely, Plateau’s three laws.

We think we have an explanation when we have found a (1) classification of systems using (2) a more abstract entity that is (3) appropriately linked to the phenomenon being explained. Whenever an explanation has these three features I will say that we have an abstract explanation. … An easy way to see that causal dependence is not involved would be by emphasizing the highly mathematical character of the Taylor case. As nobody thinks that causal relations obtain in pure mathematics, we have a non-causal explanation. … Woodward is at pains to allow for explanations via causal generalizations that are not laws. These generalizations may lack the necessity or universal scope that many require laws to have, but yet they can still explain. For Woodward (writing with Hitchcock), these generalizations explain because they say how an object would change under this or that intervention

An attractive aspect of Woodward’s approach to causal explanation is that he can trace the value of explanations to the description of causal dependence relations. We can generalize Woodward’s approach by taking the essential features of abstract explanation to correspond to an abstract dependence relation. That is, we can say that a soap-film surface obeying Plateau’s laws depends on its being an instance of an almost minimal set. As with causal dependence, there can be subtle questions about exactly how we should interpret talk of abstract dependence … Strevens allows important explanatory contributions from more abstract entities like mathematical objects. But he ultimately requires these entities to represent underlying causal processes: ‘The ability of mathematics to represent relations of causal dependence—wherever it comes from—is what qualifies it as an explanatory tool’

Jackson and Pettit distinguish between a causally efficacious property and a causally relevant property. A causally efficacious property is one whose bearer thereby gains certain causal powers, for example to bring about a certain effect in a given situation. Clearly, one sort of causal explanation would explain that effect by noting the presence of the appropri- ate causally efficacious property. However, Jackson and Pettit insist that not all causally relevant properties are causally efficacious.

Lange is clear that the value of these explanations is that they do something that causal explanations cannot do: ‘these [distinctively mathematical] explanations work not by describing the world’s network of causal relations in particular, but rather by describing the framework inhabited by any possible causal relation’ … Lange discusses the explanation for why a person cannot carry out a certain crossing over the bridges of Konigsberg. The purely mathematical part of the explanation pertains to abstract topological structure (Lange [2013], p. 489). But the rest involves ‘various contingent facts presupposed by the why question that the explanandum answers, such as that the arrangement of bridges and islands is fixed’

According to Salmon’s classic discussion, conceptions of explanation can be divided into epistemic, modal, and ontic approaches … Few defend an epistemic conception today, primarily because the link between explanation and prediction seems too confining. Salmon himself advocates an ontic conception that ties explanations to features of the world: ‘The ontic conception sees explanations as exhibitions of the ways in which what is to be explained fits into natural patterns or regularities’ (Salmon [1998], p. 320). The more specific ontic approach that Salmon defends is of course a causal approach. … On a modal approach ‘scientific explanations do their jobs by showing that what did happen had to happen’


74 thoughts on “Abstract Explanations in Science

  1. Patrice Ayme: “It seems that an agreement is actually surfacing that axioms are ‘causative’.”

    I think so!

    From the Curry-Howard correspondence, we have “the direct relationship between computer programs and mathematical proofs.” And programs are causative: in the practical sense (like where the object programs are produced by molecular assemblers of synthetic biology), and in the theoretical sense (“Causality For Free! Parametricity Implies Causality for Functional Reactive Programs”).


  2. Coel wrote:

    “It is surely for nature to tell us (science) how things are.”


    Nature does not come pre-conceptualized; categorized; classified; etc. Perception is an active, not a passive process, and all observation is theory-laden.

    This view of our relationship to nature, via science is simply incorrect — at least, if you mean it literally.


  3. Hi Aravis,

    > This view of our relationship to nature, via science is simply incorrect — at least, if you mean it literally.

    Are you perhaps a little over-keen to poke holes in Coel’s argument? I don’t see much conflict between what Coel is trying to say and what you have just said.

    I mean, it seems to me that there is a fact of the matter about how nature is (how things are), and this is all Coel was saying. I think what you said is correct, but to me it doesn’t really undermine Coel’s point so much as point out that there are different ways of describing or thinking about the way things are.

    To restate Coel’s point in another way, if given experimental results and observations there is no satisfactory account or theory of “the way things are” which is compatible with our intuitive notions regarding causation, then we ought to be open to accepting that our intuitions are wrong. Put that way, is the point really so objectionable?


  4. (Partial answer to Philip, with whom I agree; and Robin who seems to me unaware of the extent of Buridan’s revolution; much of Buridan is still in untranslated Medieval Latin, that may explain it, after centuries of Catholic war against him.)

    Know How To Dream… To Bring Up New Axiomatics

    Human beings communicate digitally (words and their letters or ideograms), and through programs (aka languages, including logic and mathematics). All of this used conventions, truths I call axioms, to simplify… the language (this is not traditional, as many of these axioms have had names for 25 centuries).

    So for example, I view the modus ponens (if P implies Q and P happens, then Q) as an axiom (instead of just a “logical form” or “rule of inference”).

    The reason to call basic “logic forms” “axioms” is that they are more fragile than they look. One can do with, or without them. All sorts of non-classical logics do without the “excluded third law” (for example fuzzy set theory).

    With such a semantic, one realizes that all great advances in understanding have to do with setting up more appropriate axioms.

    For example, in the Fourteenth Century, the intellectual movement launched by Buridan, included Oresme and the Oxford Calculators. They discovered inertia, momentum (“impetus”), graphs, the law of falling bodies, the heliocentric system (undistinguishable from the geocentric system, said Buridan, but we may as well stick to the latter, as it is in Scripture, said Buridan, wryly).


    These breakthroughs were major, and consisted in a number of new axioms (now often attributed to Galileo, Descartes, Newton). The axioms had a tremendous psychological effect. At the time, Buridan, adviser to no less than four Kings, head of the University of Paris, was untouchable.

    The philosopher cum mathematician, physicist and politician, died in 1360. In 1473, the pope and king Louis XI conspired to try to stop the blossoming Renaissance.

    More than a century after his death, Buridan’s works, his new axioms, were made unlawful to read. (However Buridan was mandatory reading in Cracow, and Abbot Copernic re-published the work, when he was sure to have taken refuge on his death bed already).

    The mind, the brain, is quite fuzzy (in the sense of fuzzy set theory; the dreaming part). Axioms, and axons enable to code it digitally. So mathematization, and programmation are intrinsic human mental activities.


  5. DM wrote:

    “I mean, it seems to me that there is a fact of the matter about how nature is (how things are).”


    I think this is highly problematic, if stated without substantial qualification. That’s why the realism/constructivism/anti-realism debate in metaphysics is so hard — and so important.

    Nelson Goodman, for example, would disagree with this statement entirely. I am not saying he is correct — or incorrect — but the arguments given in Ways of Worldmaking are worth considering.

    I don’t know about any “over-keenness” on my part. I have barely commented on Scientia at all, lately, because the topics are not of interest to me, and regardless, I have no expertise with respect to them. This is the first comment that has caught my attention, and so, I decided to say something.

    Trust me, I have plenty of other worthwile things to do, beyond arguing with Coel about metaphysics.


  6. Hi Joe,

    I go along with this until you start saying modus ponens is tested. I can not imagine any empirical data that would make me reject modus ponens.

    Well, your brain and imagination evolved to model a world in which modus ponens holds, so we wouldn’t expect you to be good at imagining it not holding.

    But, we can ask, if it didn’t hold, what we would expect to see? The answer might be “incoherent, chaotic, nothingness”. In which case the fact that we see order and structure in the world (and the fact that it is sufficiently regular to support complex beings like ourselves) is the empirical source for modus ponens.

    Just try substituting not-modus-ponens into the web of ideas and ask, do our predictions of solar eclipses then fall apart? If the answer is yes, then the web that includes modus ponens is empirically preferred (gives better explanatory and predictive power) and so is adopted.

    Hi Aravis,

    This view of our relationship to nature, via science is simply incorrect …

    The “job” of science is to adjust our web-of-ideas to best match the natural world (or, even more basically, to model our stream of experiences). Yes, all observation is theory laden, which just means that all observation is interpreted through the web-of-ideas, but that doesn’t alter the basic point of what science is trying to do.

    What we should not do is insist on prior expectations of how the world is, nor indeed about what counts as science. Thus we should not insist on “all events have a cause”. That is something we investigate by adjusting the web-of-ideas to nature. If we get a better model of, say, an electron emitting a photon, by relaxing the idea “all events have a cause”, then that’s what we do.

    Similarly, we should not insist on “either A or not-A”. If nature tells us that an electron can be both spin-up and spin-down, then we accept “both A and not-A” in some circumstances. Thus even basic logic is something that we derive, empirically, from nature.

    Liked by 4 people

  7. Coel is wearing his empiricist hat, and says that causality should be rejected if quantum electrodynamics requires it, and that the law of the excluded middle should be rejected if an electron can be shown to have 2 different properties at the same time. No such things have ever been shown, of course. QED is more explicitly causal that Newtonian mechanics ever was, and while an electron superposition shows the possibility of different outcomes, the electron has never been shown to be in 2 different places at the same time, or anything like that.

    Those who go back to Hume or Russell for ideas on causality are going to have a hard time, as neither one has a firm grasp on the subject. Ruesell’s essay is so spectacularly wrong in everything he says that it is comical. You will not see any scientist citing that essay in a favorable way. Saying that Russell’s view is naive would be an understatement. Pre-Aristotle would be more accurate.


  8. Coel:

    Your reply isn’t really responsive to what I wrote. You said that “it’s up to nature tell us (science) how things are.” I rejected that as untrue, as it represents a naive view of perception and observation. Implicit in that rejection is either something of a constructivism — a la Kant — or more radically, an anti-realism, a la Goodman. You accepted the point regarding perception and observation, but ignored its implications, which is nothing more than footstamping. Oh, and causality is implicit in what is *meant* by an explanation, in science, so you’re going to have to redefine ‘explanation’ too.

    Schlafly wrote:

    “Those who go back to Hume or Russell for ideas on causality are going to have a hard time, as neither one has a firm grasp on the subject. Rusell’s essay is so spectacularly wrong in everything he says that it is comical. ”


    I would say that besides the topics on Scientia becoming increasingly technical and from my perspective, abstruse, it is precisely your attitude that has caused me to almost completely stop commenting on this forum. And I certainly would never submit anything, given the sort of dismissive commenting that you and others seem to habitually engage in. You are welcome to your tunnel-visioned conversation — I will have no part in it.

    Liked by 2 people

  9. Coel,

    I interpreted Aravis’ phrase “all observation is theory laden” in a different way to you. Your statement that it “just means that all observation is interpreted through the web-of-ideas” seems to refer to the *results* of observation rather than the *act* of observation itself which seems to imply that you believe the results are nature’s raw, uncontaminated data. If the *act* of observation is theory-laden, which is the sense I got from the phrase, then the data are not as “raw” as you might think.
    We should not of course allow any prior expectations of results to influence how we carry out an experiment, but what questions we can ask of nature, and how and when we should ask them are all influenced by prior expectations of what constitutes a proper experiment. In other words, the web-of-ideas determines the kind of nature we think we are interrogating.

    Liked by 1 person

  10. dadooq: Yes, you have interpreted my correctly.

    With the aim of being productive, both Coel and DM could benefit from the following article, which speaks to the issue that I am raising. It is a very well-known piece in the history of analytic philosophy and has had a great influence on the course the discussion has taken in the analytic tradition.

    Hilary Putnam, “Why There Isn’t a Ready-Made World,” Synthese, Vol. 51, No. 2 (1982).


  11. I think I would have to agree with Aravis (assuming I understand his position correctly), that all attempts to explain or understand nature will always involve some degree of abstraction. While Aravis provides some references from the analytical tradition another perspective that compliments this view goes way back to the book of Zhuangzi (ChuangiTzu).

    The second of the ‘inner chapters’ titled ‘Equalizing Assessments of Things’ contains the bulk of his thought in this regard. Zhaungzi felt that every human distinction was an abstraction from an underlying natural wholeness. He felt that whenever some assertion was made part of the whole was brought to light while another aspect became hidden in darkness. Thus he gave the advice:

    ‘Allow both Alternatives to Proceed’

    It wasn’t just that we tend to focus on the assertion while losing track of it’s denial. Zhaungzi felt that even if we consider both ‘A’ and ‘not A’, if we only think of them as exclusively separate we lose awareness of how they might interact. For Zhuangzi the way things interact due to circumstance would be lost to any fixed theory. He also however recognized how the downside of this concept could lead one into absolute relativity and an inability to make make judgements.

    ‘Therefore, the Sage brings all into harmony through assertion and denial but rests it upon the balance of heaven: this is called “Walking a Double Path.’ This is accomplished by ‘Allowing Both Alternatives to Proceed’ & ‘Remaining at Rest at the Center of the Spontaneous Potter’s Wheel of Nature’ .

    For Zhanungzi there are many interpretations, but I think it is fair to say he conceptualzed ‘Heaven’ as the spontaneous and undesigned unfolding of nature.

    Sorry if this may see a bit off topic, but I thought this perspective might interest some who are unfamiliar with Zhuangzi. Even if all causal explanatory theories involve some degree of abstraction, I suppose there are many degrees of abstraction and the posted paper could be useful in classifying them.


  12. The whole thrust of the article seems to me to beside the point.

    A good mathematical model of soap films is described by Plateau’s laws. First of all, though, they only apply as a limiting stable situation. They don’t describe the dynamics of soap films. As a description of what’s observed, they’re a fine summary, but there are more fundamental principles that explain them. In fact, the laws themselves don’t explain anything.

    Mathematicians have sought out interesting minimal surfaces for a long time. Few were found in the 19th century, but more in the late 20th century with the help of computers. Physical soap bubbles have not been particularly helpful in their discovery.


  13. Dan,

    I assume your question is due to not having followed the debate and since my argument has bled over from prior threads, I will try to recap it;

    Our perception consists of a sequence of impressions. As Coel stated above, “our stream of experiences.”

    From this sequential process of perception, we derive the concept of time as just such a universal vector, from the starting point of the Big Bang, to the entropic fadeout of dispersed energy, on which the point of the present is a dimensionless concept that is either moving universally(Newton), subjectively(non-simultaneous), or is a subjective illusion(block time).

    For physics it presents a conundrum to go from a determined past, to a indeterminate future. The classical tendency is toward determinism, that since the “laws of nature” produce a finite outcome to every event, then this logically applies to all future events as surely as it does to all past events and so probability is an illusion based on our limited knowledge.

    Yet quantum theory has problems with testing this out and the link between input and output is fuzzy, so the “laws of nature” seem statistical, if not yielding multiple outcomes.

    My observation is this sequencing effect is a function of our subjective view of a dynamic context and that it is the process by which future becomes past, within this physical state we refer to as the present. Tomorrow becomes yesterday. The measure of duration being used as time in experiments is not external to the “point of the present,” but is the state of what is present, as the events form and dissolve.

    So while we experience A to B to C and assume A must be causative of B, etc, in this alternate view causation is entirely a function of energy transfer, not sequence. For instance, a batter hitting a ball is cause of it flying away, because there is energy transfer, but yesterday is not causative of today, because the energy transfer is light shining on a spinning planet creating this effect of days, which proceed from future to past, as they arise and recede. Only the energy is “conserved,” as the present state.

    So my argument has been that if we were to recognize this “stream of experiences” as non-fundamental and all we are really measuring is frequency, than we might better understand causation as a physical process and not just a complex analytic one.

    Then it is causation, that which physically occurs, as diverse input arrives and the lightcone of input is only completed by the event, that determines the outcome, as present becomes past.

    Obviously there are a multitude of other factors to consider, such as that making time an effect of action would have to re-open the question of space and whether it is three dimensional, or whether that is just a mapping device we apply to it, as with the dimension of narrative being used to define time. Consider that a multitude of such devices can be applied to the same space, effectively making it “multi-worlds.” Then the particle can be spinning up in one frame and down in the opposite frame.

    For example, ask the Israelis and Palestinians about using different coordinates, backed by different narratives, to describe the same space, without some objective, God’s eye view, to moderate it.

    My last post and against the word limit……..


  14. The OP and the comments are way too technical for me, but having thought a little about causality on a general level, I’ll say that I don’t see any problem with abstract facts in causal explanations. As I see it, causality generally concerns relations of truth-dependence; truths depend on other truths, and the cause of a given truth is some other truth on which it depends and which is relevant in the context.

    What might in part seem odd about use of abstract facts in causal explanations is that such truths are necessary truths and so the counterfactual dimension of causal consideration doesn’t apply. That is, usually causes are such that if there were false, the effect would not obtain, so in cases where the cause is such that it cannot be false, it might not seem like a cause.

    But if cause is just a matter of truth dependence, I see no reason truths must be contingent to be causes.


  15. Coel,

    The probabilistic dice-throwing is in itself entirely adequate to explain the 2nd law

    I think we have reached a point where it is obvious that we disagree on this, and all arguments for and against have already been said (in short, I don’t believe the above statement until I see a rigorous proof, while you believe the statement, hoping that a rigorous proof could eventually be formulated). So I see no point in discussing this any further, and I suggest we drop the subject. 🙂

    While I fully agree about the role of non-deterministic non-causality, I think you’re going way too far in throwing it out entirely and making it purely an artifact.

    This is because I understand causality as contingent on determinism, and I don’t see how determinism and nondeterminism can both be valid in Nature. Despite high precision of deterministic predictions of, say, solar eclipses and such, determinism is still (ultimately) only an approximation (granted, in some cases an exceedingly good one, but still), so the observed causality must be too.


    if we change the axioms concerning Platonic solids we are no longer talking about Platonic solids

    Well, we shouldn’t really plug axioms into the definition of a Platonic solid. Otherwise, the situation would be no different than saying “Look, these five polyhedra I drew on the paper are by definition Platonic solids, and no other polyhedron is Platonic solid, also by definition.”. Then it seems a bit trivial that there are only five of them, because it is not a *fact* in any stronger sense than by fiat of the definition. I could define them differently (so that there are only two of them), and my definition will be merely inequivalent to yours, but not false or wrong in any meaningful way.

    A formal system makes a distinction between axioms and definitions, precisely because of these kinds of pitfalls.


    A naive view implies, as I said, an unexamined view. A philosophical interest would imply an examined view. So you appear to be saying that they are obsessively examining their unexamined view, which is a contradiction.

    No, that’s not what I said. I said that they were obsessive about *causality*, while naive about *determinism* (by thinking about Nature in terms of, say, Newtonian mechanics). So they are extensively examining the issue of causality, while taking for granted a wrong assumption that Nature is deterministic. They are examining one thing, while ignoring the other. No contradiction there.

    That is my honest impression of this whole story — if one accepts that Nature is nondeterministic, any talk about causality pretty soon becomes a non sequitur. Continued discussions about causality therefore seem to imply ignorance of the nondeterministic nature of the real world. This was my original question, and I’d be really grateful if someone would explain to me how does it make sense to keep talking about causality (as something fundamental) in a probabilistic-only Nature.


  16. Joe,

    I go along with this util you [Coel] start saying modus ponens is tested. I can not imagine any empirical data that would make me reject modus ponens.

    Could lack of imagination be the reason? 😉 But seriously, in some approaches to trivalent propositional logic (see here for some details), i.e. if we move from Boolean true/false values for propositions to three valued set true/false/unknown, modus ponens (among other things) ceases to be a tautology. You can verify that explicitly using the trivalent truth tables for and/or/not/implies operations given in the wiki page. For example, in Kleene’s propositional calculus it is known that there can be no tautologies at all.

    So the question of testability of modus ponens is really the question of why we prefer Boolean logic over, say, Kleene’s logic. And this preference is based ultimately in our experience — we use Boolean logic because it is more useful when dealing with the real world, as opposed to Kleene’s logic which is more useful when dealing with our own knowledge or computer databases. When you make an SQL query, the computer may respond in a way that is inconsistent with rules like modus ponens. And this is entirely ok, because absence of information (answer “I don’t know”) cannot be interpreted as either true or false statement.

    Logic is not something fixed in stone. It is all about being a language useful for a particular purpose. And this is always contingent on the empirical, as Coel often stresses.


    causality is implicit in what is *meant* by an explanation, in science

    Not in physics, it isn’t! 🙂 A physical phenomenon can be *described* by a mathematical model. Such a model is called phenomenological, and sits somewhere in the reductionism chain (ermm, network) of various models. The model at the bottom of the chain (called a fundamental theory) also deals with *describing* certain phenomena. If the phenomenological theory can be fully reduced to the fundamental theory, the physical phenomenon *described* by the phenomenological theory is said to be *explained* in terms of the fundamental theory. This is the definition of “explanation” in physics.

    Note that, in the above, the distinction between description and explanation is somewhat arbitrary, and usually contingent on the historical context of which theory (pheno or fundamental) was formulated first. Beyond history, the distinction can make sense if the phenomenon being described/explained is not expressible directly in terms of phenomena described by the fundamental theory, but only indirectly (through a chain of reductionistic definitions of vocabulary of higher-level theories).

    As an example, the relation between the pressure, temperature and volume of an ideal gas is *described* by thermodynamics, and *explained* (ultimately) by Standard Model of elementary particles. While formally correct, it would be somewhat misleading that this relation is merely *described* by SM — because the formulation of SM does not deal explicitly with concepts of pressure, temperature and volume, but is built to describe only some more primitive concepts like particles. That’s why we say SM “explains” the equation of state, while thermodynamics “describes” it.

    Therefore, the distinction between description and explanation is made solely for purposes of convenience in classification of phenomena and theories.

    Finally, note that none of the above has anything whatsoever to do with causality. I see no reason why it should.

    This was my fifth post. I hope this discussion can be continued in some future thread. 🙂


  17. Realist conceptions of scientific explanation are based around the concept of causality. To the extent that philosophers of science have embraced conceptions of explanation that are not grounded in causality, they have been anti-realists (like Bas van Frassen), and Coel ain’t no anti-realist. This is stuff that can be found in basic encyclopedia entries on the subject, at Stanford and the IEP.

    I know, I know…”But that’s not what *scientists* mean by ‘explanation’.” Sigh. Every time I reach this point, I remember why I dropped out of the conversation the last time. It seems, unfortunately, that we cannot talk productively on these subjects, because we don’t agree on what the basic terms mean. So why bother? The conversations always end in the same place — arguing about the meanings of words.

    Fifth comment for me too, and it’s just as well.


  18. “Similarly, we should not insist on “either A or not-A”. If nature tells us that an electron can be both spin-up and spin-down, then we accept “both A and not-A” in some circumstances. Thus even basic logic is something that we derive, empirically, from nature.”

    IMO you are way too fast to embrace being illogical worldview.

    It is impossible for me to even imagine any immediate sense perception that would tell me to reject modus ponens or that A and not A are true at the same time in the same way.

    To come to those conclusions we would need to not just directly see something but also make inferences from what we saw. Those inferences would likely use the very logical axioms (like modus ponens) which you are suggesting we should deny. I suppose that wouldn’t be a problem since you might believe in logical axioms are true and not true at the same time. But it is about this point where everything becomes jibberish.

    I like science and find it interesting. But its statements like this that make me roll my eyes.

    Sure we don’t know allot of what happens at the quantum level. But jumping to embrace illogical views is not likely to be the answer. If it is we have much bigger problems than understanding quantum reality.

    “But, we can ask, if [modus ponens] didn’t hold, what we would expect to see? The answer might be “incoherent, chaotic, nothingness”. In which case the fact that we see order and structure in the world (and the fact that it is sufficiently regular to support complex beings like ourselves) is the empirical source for modus ponens.”

    It’s interesting that you claim to rely on your perceptions more than basic logical truths. Yet in your quote you seem to use the logical axiom of modus tollens to justify your view that perceptions come first.
    It would not be “nothingness” if logical truths didn’t hold but rather “anythingness.”

    Consider if you think your direct perception of “T” forces you to believe “A” and “not A.”
    Since A is true you can say A.
    Since A is true you can also say A *or* anything else.
    A or “not T”
    Not A
    Therefore Not T.

    Sure there is logic involved but there would be logic involved in your accepting that T lead to A and not A.

    I have been convinced of dreams that were incoherent – yet they seemed coherent in the dream. Clearly my mind can get confused about what it thinks are consistent memories – at least when I am sleeping. But to believe that about reality is a different matter. To put what I think my perceptions are above logical axioms seems unwarranted.

    We think we perceive the world a certain way. And we think it corresponds to reality. But there is no way we can verify this. This is one reason I tend to put my faith in logic more than perceptions. I can clearly imagine how my beliefs about the external world are mistaken, but I find it impossible to think how basic axioms of logic are mistaken.


  19. Hi dadooq,

    I interpreted Aravis’ phrase “all observation is theory laden” in a different way to you. Your statement that it “just means that all observation is interpreted through the web-of-ideas” seems to refer to the *results* of observation rather than the *act* of observation itself …

    My previous answer was intended to agree that *all* aspects of the observation are only known about through the web of ideas, this includes the act of observation itself. So I fully agree on that, there is no “raw” stage that is knowable independently of the web.

    But that doesn’t change the rest of my comments. It is still the case that the job of science is to adjust the web to match reality, and that we shouldn’t insist on pre-conceived ideas such as “events need causes”.

    Hi Marko,

    This is because I understand causality as contingent on determinism, and I don’t see how determinism and nondeterminism can both be valid in Nature.

    Why not? Why can’t nature be an entwined mixture of both? Isn’t that how observations are telling us nature actually is? No totally non-deterministic model (= the state at time T+1 is totally unrelated to the state at time T) would work, and yet no fully-deterministic model (the state at T+1 is completely specified by the state at T) works either.

    Hi Joe,

    It’s interesting that you claim to rely on your perceptions more than basic logical truths.

    How do we know that something is a “basic logical truth”? Indeed what does “truth” in that phrase even mean? I’d assert that the only sensible answer is that it models reality, which is then an empirical matter. The alternative iis declaring it by fiat, or claiming we “just know” it.

    I would agree that adopting “both A and not-A” *generally* would lead to incoherent “anythingness”. It is clearly not the world we live in, which is why we reject it. But, it may well be that in *some* situations “both A and not-A” is a better description of nature, and an electron being “both spin-up and not spin-up” (= spin-down) is one of those.

    So, overall, I’m asserting that even “basic logical truths” are ultimately empirical, and that nature tells us the extent to which they hold.

    Hi Aravis,

    To the extent that philosophers of science have embraced conceptions of explanation that are not grounded in causality, they have been anti-realists …

    But Marko and I are talking about science, whereas you are talking about something called “philosophy of science”, which is not necessarily about science.

    It is not correct to insist that “causality is implicit in what is *meant* by an explanation, in science”. Science had that debate around the time quantum mechanics was invented, continuing with EPR and Bell’s theorems. It is for nature to tell us whether or not all all events are caused, and thus whether or not all explanations need to be causal. It is not for us to impose our notions of what sorts of explanations we’ll accept. If we model reality better by relaxing ideas about causality, then that’s what scientists do.

    The “web” idea tells us that we can’t throw the whole lot up in the air at once, but we certainly can re-consider any part of that web, including notions of causality and basic logic.

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  20. I have seen definitions of causation as transfer of energy between material things (whether field or extended object). It seems to fail when we describe the physical evolution of a computer carrying out a particular computation – eg what is it that determines whether the computational process halts? I suppose the proximal cause is the nonlinear mapping of the state of a set of neurones onto the initial state of the machine. I definitely think that the idea of causation has to include mechanisms that are stochastic – it seems ridiculous to say that manipulations that lead to decoherence of a quantum state aren’t causing an effect, or applying an electric or magnetic field to alter the state functions of an atom. At the level of the models of causation I am familiar with, we always have a source of randomness (usually but presumably not always classical in nature) eg segregation of chromosomes in meiosis.


  21. Marko
    I didn’t see your comment before I posted my response to Coel. I think my response to Coel also touches on some of your comments as well.

    I suppose it is a lack of imagination that I do not think A and not A can be true at the same time in the same way. I admit I can not imagine how that is the case. Can you?

    Sure I can see how I might 1) know A, or 2) Know “not A”, or 3) not know A or “not A”. But I don’t think my knowing or lack of knowing makes A or “not A” any less real. Do you think the moons of Neptune were less real before we knew of them?

    I don’t deny that we can use a different rational methodologies to draw conclusions when we have unknowns. But I don’t think that means logic does not apply to reality. I understand truth to be that which corresponds to reality. And truth is what I seek. Now you might say truth is different than that. But if truth is disconnected from reality then I start to lose interest in “truth.” I will ask that we coin a different word that will be the word we use when our beliefs accord with reality. It will be that new worded thing I am interested in pursuing.


  22. It is great to see this vigorous discussion of my article. I have not been able to study all the comments, but maybe a few points of clarification would be helpful. 1) Plateau’s laws were arrived at through careful observation and measurement. The explanation that I discuss is the explanation of why those laws are true. In particular, it remained puzzling why the surfaces should meet in only two ways, with angles of 120 degrees and close to 109 degrees. 2) There are many theories of causal explanation that insist that in an explanation the mathematics is merely serving to represent underlying causes. So much of my paper is trying to show that this is _not_ the role of the mathematics in this case. The mathematics is explaining, but not by representing underlying causes of the laws holding. 3) It remains to be seen how to make sense of this case. I only argue for the conditional claim: if all explanations involve dependence, then this case involves a special sort of abstract dependence between platonic entities and concrete phenomena. This is a strange and puzzling conclusion, and I do not want to insist that this is the only way to approach this case. (By the way, readers interested in “mathematical explanation” quite generally might be interested in a companion article where I discuss a case of explanation within pure mathematics: http://quod.lib.umich.edu/p/phimp/3521354.0015.003/1/)

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  23. Disagreeable Me,

    You said:

    “There is some sense in which the object I am thinking of is not simply identical with the state of my particular brain.
    If mathematical objects exist, they must do so independently of mathematicians.”

    This sounds like the Platonism I have become fascinated with because of its independent existence.

    “Platonism is not in any way mystical or mysterious. It is just an attitude with respect to what we should deem to exist, and so can be regarded as a particular way of defining the concept of “existence”. When I say “there are 5 platonic solids”, I am not saying quite the same kind of thing as “There are 2 surviving members of the Beatles”. The latter is a statement about the physical state of our world, the former is not, nor is it a statement about the physical state of some other world. Mathematical objects don’t exist in a world, they just exist, outside of space and time (for certain values of “exist”, anyway!).

    First you said “what we should deem to exist” and then said “they just exist”. It seems like Platonism, at least in its strong form, is not saying that we just deem (imagine) them existing but that in some sense, as you said at first, they actually exist. If Platonism is “just an attitude” then it is about the form of the mind not an ontological Form. If the Forms exist, then it seems very mystical or mysterious.

    As John Von Neumann said, “You never understand mathematics you just get used to it.” Maybe you have gotten used to Platonism and it doesn’t seem mysterious to you any more, but for me, it is as mystical as a deity that is beyond space and time.

    Chris Pincock

    You said:

    “I only argue for the conditional claim: if all explanations involve dependence, then this case involves a special sort of abstract dependence between platonic entities and concrete phenomena.”

    Sounds like the old Cartesian dualism problem of interaction (dependence). How can we understand a dependence relation between the abstract and concrete?


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