Progress in Mathematics and Logic — IV

paradoxes[for a brief explanation of this ongoing series, as well as a full table of contents, go here]

A panoply of logics

The striking thing about contemporary logic is that it is plural. Indeed, Logics (Nolt 1996) was the title of the book I used as a graduate student at the University of Tennessee, and that in itself was a surprise for me, since I had naively always thought of logic as a single, monolithic discipline. (But why, really? We have different ways of doing geometry and mathematics, and certainly a plethora of natural sciences!). The following brief look at modern logic should be enough to convince readers that the field is both vibrant and progressive, in the sense discussed above.

Consider, for instance, modal logic (Garson 2009), which deals with the behavior of sentences that include modal qualifiers, such as “necessarily,” “possibly,” and so on. It is actually a huge field within logic, as it includes deontic (“obligatory,” “permitted,” etc.: Hilpinen 1971), temporal (“always,” “never”), and doxastic (“believes that”) logics. Arguably the most familiar type of modal logic is the so-called K-logic (named after influential 20th century philosopher Saul Kripke), which is a compound logical system made of propositional logic, a necessitation rule (using the operator “it is necessary that”) and a distribution axiom. A stronger system, called M, is built by adding the axiom that whatever is necessary is the case, and this in turn yields an entire family of modal logics with different characteristics. Starting again with K-logic, one can also begin to build a deontic logic by adding an axiom that states that if x is obligatory, then x is permissible. Similarly, one can build temporal logics (with the details depending on one’s assumptions about the structure of time) and other types of modal logics. Indeed, Garson (2009) presents an elegant “map” of the relationships among a number of modal logics, demonstrating how they can all be built from K by adding different axioms. Furthermore, Lemmon and Scott (1977) have shown that there is a general parameter (G) from the specific values of which one can derive many (though not all) the axioms of modal logic. Garson’s connectivity map  and Lemmon and Scott’s G parameter are both, well, highly logical, and aesthetically very pleasing, at the least if you happen to have a developed aesthetic sensibility about logical matters. (Notice, once again, that these are all nice examples of non-teleonomic progress propelled by internally generated problems and consisting in exploring additional possibilities in a broad conceptual space that is evoked once one adds axioms or assumptions.)

A major issue with modal logic is that — unlike classical logic (Benthem 1983) — it is not possible to use truth tables to check the validity of an argument, for the simple reason that nobody has been able to come up with truth tables for expressions of the type “it is necessary that” and the like. The accepted solution to this problem (Garson 2009) is the use of possible worlds semantics (Copeland 2002), where truth values are assigned for each possible world in a given set W. Which implies that the same proposition may be true in world W1, say, but false in world W2. Of course one then has to specify whether W2, in this example, is correctly related to W1 and why. This is the sort of problem that has kept modal logicians occupied for some time, as you might imagine. Modal logic has a number of direct philosophical applications, as in the case of deontic logic (McNamara 2010), which deals with crucial notions in moral reasoning, such as permissible / impermissible, obligatory / optional, and so forth. Deontic logic has roots that go all the way back to the 14th century, although it became a formal branch of symbolic logic in the 20th century. As mentioned, deontic logic can be described in terms of Kripke-style possible worlds semantics, which allows formalized reasoning in metaethics (Fisher 2011).

Of particular interest, but for different reasons, are also the next two entries in our little catalog: many-valued and “fuzzy” logics. The term many-valued logic actually refers to a group of non-classical logics that does not restrict truth values to two possibilities (true or false; see Gottwald 2009). There are several types of many-valued logics, including perhaps most famously Łukasiewicz’s and Gödel’s. Some of these admit of a finite number of truth values, others of an infinite one. The Dunn/Belnap’s 4-valued system, for instance, has applications in both computer science and relevance logic. Multi-valued logic also presents aspects that reflect back on philosophical problems, as in the area of concepts of truth (Blackburn and Simmons 1999), or in the treatment of certain paradoxes (Beall 2003), like the heap and bald man ones. More practical applications are found in areas such as linguistics, hardware design and artificial intelligence (e.g., for the development of expert systems), as well as in mathematics. Fuzzy logic is nested within many-valued logics (Zadeh 1988; Hajek 2010), consisting of an approach that makes possible to analyze vagueness in both natural language (about degrees of beauty, age, etc.) and mathematics. The basic idea is that acceptable truth values range across the real interval [0,1], rather than lying only at the extremes (0 or 1) of that interval, as in classical logic. Fuzzy logic deals better than two-valued logic with the sort of problems raised by Sorites paradox, since these problems are generated in situations in which small/large, or many/few quantifiers are used, rather than simple binary choices. Therefore, fuzzy logic admits of things being almost true, rather than true, and it is for this reason that some authors have proposed that fuzzy logic can be thought of as a logic of vague notions.

A whole different way of thinking is afforded by so-called intuitionistic logic (Moschovakis 2010), which treats logic as a part of mathematics (as opposed to being foundational to it, as in the old Russell-Whitehead approach that we have discussed above), and — unlike mathematical Platonism (Linnebo 2011) — sees mathematical objects as mind-dependent constructs. Important steps in the development of intuitionistic logic were Gödel’s (who, ironically, was a mathematical Platonist!) proof (in 1933) that it is as consistent as classical logic, and Kripke’s formulation (in 1965) of a version of possible-worlds semantics that makes intuitionistic logic both complete and correct. Essentially, though, intuitionistic logic is Aristotelian logic without the (much contested) law of the excluded middle, which was developed for finite sets but was then extended without argument to the case of infinities.

Finally, a couple of words on what some consider the cutting edge, and some a wrong turn, in contemporary logic scholarship: paraconsistent and relevance logics. Paraconsistent logic is designed to deal with what in the context of classical logic are regarded as paradoxes (B. Brown 2002; Priest 2009). A paraconsistent logic is defined as one whose logical relations are not “explosive,” [12] with the classical candidate for treatment being the liar paradox: “This sentence is not true.” The paraconsistent approach is tightly connected to a general view known as dialethism (Priest et al. 2004), which is the idea that — contra popular wisdom — there are true contradictions, as oxymoronic as the phrase may sound. Paraconsistent logic, perhaps surprisingly, is not just of theoretical interest, as it turns out to have applications in automated reasoning: since computer databases will inevitably include inconsistent information (for instance because of error inputs), paraconsistent logic can be deployed to avoid wrong answers based on hidden contradictions. Similarly, paraconsistent logic can be deployed in the (not infrequent) cases in which people hold to inconsistent sets of beliefs, sometimes even rationally so (in the sense of instrumental rationality). More controversially, proponents of paraconsistent logic argue that it may allow us to bypass the constraints on arithmetic imposed by Gödel’s theorems. How does this work? One approach is known as “adaptive logic,” which begins with the idea that consistency is the norm, unless proven otherwise, or alternatively that consistency should be the first approach to a given sentence, with the alternative (inconsistency) being left as a last resort. That is to say, classical logic should be respected whenever possible. Paraconsistent logics can be generated using many-valued logic, as shown by Asenjo (1966), and the simplest way to do this is to allow a third truth value (besides true and false), referred to as “indeterminate” (i.e., neither true nor false).

Yet another approach within the increasingly large family of paraconsistent logics it to adopt a form of relevance logic (Mares 2012), whereby one stipulates that the conclusion of a given instance of reasoning must be relevant to the premises, which is one way to block possible logical explosions. According to relevance logicians what generates apparent paradoxes is that the antecedent is irrelevant to the consequent, as in: “The moon is made of green cheese. Therefore, either it is raining in Ecuador now or it is not,” which is a valid inference in classical logic (I know, it takes a minute to get used to this). The typical objection to relevance logic is that logic is supposed to be about the form, not the content, of reasoning — as we have seen when briefly examining the history of both Western and Eastern logic — and that by invoking the notion of “relevance” (which is surprisingly hard to cash out, incidentally) one is shifting the focus at least in part to content. Mares (1997), however, suggests that a better way to think about relevance logic comes with the realization that a given world X (within the context of Kripe-style many worlds) contains informational links, such as laws of nature, causal principles, etc. It is these causal links, then, that are deployed by relevance logicians in order to flesh out the notion of relevance (in a given world or set of possible worlds) — perhaps reminiscent of Dharmakīrti’s invoking of causal relations to assure the truth of a first premiss, as we have discussed. Interestingly, some approaches to relevance logic (e.g., Priest 2008) can be deployed to describe the difference between a logic that applies to our world vs a logic that applies to a science fictional world (where, for instance, the laws of nature might be different), but relevance logic has a number of more practical applications too, including in mathematics, where it is used in attempts at developing mathematical approaches that are not set theoretical, as well as in the derivation of deontic logics, and in computer science (development of linear logic).

So, what do we get from the above historical and contemporary overviews of logic, with respect to how the field makes progress? The answer is, I maintain, significantly different from what we saw for the natural sciences in Chapter 4, but not too dissimilar from the one that emerged in the first part of this chapter in the case of mathematics. Logicians explore more and more areas of the conceptual space of their own discipline. Beginning, for instance, with classical two-valued logic it was only a matter of time before people started to consider three-valued and then multi-valued, and finally infinitely-valued (such as fuzzy) types of logical systems. Naturally enough, this progress was far from linear, with some historians of logic identifying three moments of vigorous activity during the history of Western logic, with the much maligned Middle Ages not devoid of interesting developments in the field. Other signs of progress can readily be seen in the expansion of the concerns of logicians, beginning with Aristotelian syllogisms or similar constructs (as in the parallel developments of the Stoics) and eventually exploding in the variety of contemporary approaches, including deontic, temporal, doxastic logics and the like. Even dialethism and the accompanying paraconsistent and relevance logics can be taken as further explorations of a broad territory that began to be mapped by the ancient Greeks: once you chew for a while (in this case, a very long while!) on the fact that classical logic can yield explosions and paradoxes, you might try somewhat radical alternatives, like biting the bullet and treating some paradoxes as “true,” or pushing for the need for a three-valued approach when considering paradoxes. As in the case of mathematics, there have been plenty of practical (i.e., empirical) applications of logic, which in themselves would justify the notion that the field has progressed. But in both mathematics and logic I don’t think the empirical aspect is quite as crucial as in the natural sciences. In science there is a good argument to be made that if theory loses contact with the empirical world (Baggott 2013) then it is essentially not science any longer. But in mathematics and logic that contact is entirely optional as far as any assessment of whether those fields have been making progress by the light of their own internal standards and how well they have tackled the problems that their own practitioners set out to resolve.

What about philosophy, then? It is finally to that field that I turn next, examining a number of examples of what I think clearly constitutes progress in philosophical inquiry. As we shall see, however, the terrain is more complex and perilous. More complex because while philosophy is very much a type of conceptual activity, sharing in this with mathematics and logic, it also depends on input from the empirical world, both in terms of commonsense and of scientific knowledge. More perilous because a relatively easy case could be made that a significant portion of philosophical meanderings aren’t really progressive, and some even smack of mystical nonsense. Unfortunately, this sort of pseudo-philosophy does appear in the philosophical literature, side by side with the serious stuff, and occasionally is even the result of the writings of the same authors! Nothing like that appears to be happening either in science or in mathematics and logic, which I believe is a major reason why philosophy keeps struggling to be taken seriously in the modern academic world.

Notes

[12] A logical explosion is a situation where everything follows from a contradiction, which is possible within the standard setup of classical logic.

References

Asenjo, F.G. (1966) A Calculus of Antinomies. Notre Dame Journal of Formal Logic 7:103-105.

Baggott, J. (2013) Farewell to Reality: How Modern Physics Has Betrayed the Search for Scientific Truth. Pegasus.

Beall, J.C. (ed.) (2003) Liars and Heaps. Clarendon Press.

Benthem, J.F. van (1983) Modal Logic and Classical Logic Bibliopolis.

Blackburn, S. and Simmons, K. (eds.) (1999) Truth. Oxford University Press.

Brown, B. (2002) On Paraconsistency. In: D. Jacquette (ed.), A Companion to Philosophical Logic, Blackwell, pp. 628-650.

Copeland, B.J. (2002) The genesis of possible worlds semantics. Journal of Philosophical Logic 31:99-137.

Fisher, A. (2011) Metaethics: An Introduction. Acumen Publishing.

Garson, J. (2009) Modal Logic. Stanford Encyclopedia of Philosophy (accessed on 31 May 2013).

Gottwald, S. (2009) Many-valued logic. Stanford Encyclopedia of Philosophy (accessed on 22 August 2013).

Hajek, P. (2010) Fuzzy logic. Stanford Encyclopedia of Philosophy (accessed on 30 March 2011).

Hilpinen, R. (1971) Deontic Logic: Introductory and Systematic Readings D. Reidel.

Lemmon, E. and Scott, D. (1977) An Introduction to Modal Logic. Blackwell.

Linnebo, Ø. (2011) Platonism in the philosophy of mathematics. Stanford Encyclopedia of Philosophy  (accessed on 11 October 2012).

Mares, E.D. (1997) Relevant Logic and the Theory of Information. Synthese 109:345–360.

Mares, E. (2012) Relevance logic. Stanford Encyclopedia of Philosophy (accessed on 28 August 2013).

McNamara, P. (2010) Deontic logic. Stanford Encyclopedia of Philosophy (accessed on 22 August 2013).

Moschovakis, J. (2010) Intuitionistic logic. Stanford Encyclopedia of Philosophy (accessed on 22 August 2013).

Nolt, J. (1996) Logics. Cengage Learning.

Priest, G. (2008) An Introduction to Non-Classical Logic: From If to Is. University of Cambridge Press.

Priest, G. (2009) Paraconsistent logic. Stanford Encyclopedia of Philosophy (accessed on 14 September 2012).

Priest, G., Beall, J.C. and Armour-Garb, B. (eds.) (2004) The Law of Non-Contradiction. Oxford University Press.

Zadeh, L.A. (1988) Fuzzy logic. Computer 21:83-93.

135 thoughts on “Progress in Mathematics and Logic — IV

  1. brodix

    I don’t see how a view that is implicitly, if not explicitly, absolutist, is reconciled with one that is reciprocal. Other than they are both trying to explain morality
    It seems a bit like saying black is similar to white, because they are both colors.

    Like

  2. synred

    http://philosophyfaculty.ucsd.edu/faculty/wuthrich/

    So does anybody know this guy? He’s a philosopher working on the problem of QM-Gravity unification.

    I stumbled across him because I was searching for references to my uncle Hartland Snyder’s quantized space-time paper and this one popped up:

    Click to access WuthrichChristian2005PhilSci_QuantizeOrNot.pdf

    in which he seems to be raising significant issues missed by physicist. If he turns out to be correct that could real progress; even if he’s wrong it might help clarify the issues.

    That’s progress. But who gets the credit – physics or philosophy?

    Like

  3. garthdaisy

    “Neuroscientists and psychologists are forever telling something we already know about ourselves in tones which suggest that we ought to be surprised at it..”

    In tones? That’s hilarious. So much anger towards science on this blog. “Stop telling philosophers about human nature! We already know! We knew before you, science!” The people who participated in the studies sure didn’t know. They thought they were reasoning separate from their desires but the studies showed they were wrong and so was Kant. He thought you can separate your logic from your desires but you Kant.

    As for hope, are you sure you did’t mean faith?

    Liked by 3 people

  4. garthdaisy

    And perhaps you didn’t notice the person I’m debating with here doesn’t even seem to believe in the existence of “so called human nature” as he puts it. Perhaps that’s who those “tones” you speak of are intended for.

    Like

  5. brodix

    Arthur,

    That is an interesting paper. I’ve wondered something that may be similar. Of gravity as a vacuum effect from energy being quantized, as a quantum would seem to occupy less space than the field from which it coalesced. Though that would seem to imply space as an equilibrium or vacuum, that is distinct from the quantum field.

    I’d better not speculate any further.

    Like

  6. synred

    Uber Munch for the Will to Power Lunch

    –joke missing from typo! Can’t Cut&Paste from an image — anymore!

    Like

  7. Daniel Kaufman

    garthdaisy:

    My skepticism about there being a “human nature” is due to a more fundamental anti-essentialism. You may disagree with it, but it’s hardly some lampoonable position, as you seem to want to paint it. Indeed, it’s actually quite common.

    Maybe you should learn a little bit more about these things, before you come out so stridently.

    Like

  8. synred

    I also am baffled by what you mean by ‘no human nature’

    However, it’s way off topic (I think), so perhaps we should leave it for another time or another forum.

    A funny thing happened on the way to the forum:

    Like

  9. garthdaisy

    Daniel,

    “You may disagree with it, but it’s hardly some lampoonable position”

    Said the man who claims he just had to weigh in on my Kant comment because it was “a howler.” (that many other credible philosophers agree with)

    Lampooning is what philosophers live for. If you can’t take it, don’t dish it out.

    Like

  10. Daniel Kaufman

    synred wrote:

    Lighten up guys! It’s only a blog ;_)

    ——————————————

    How much lighter can I get? I have withdrawn from the argument.

    Like

  11. synred

    Does ‘deflationary theory of truth’ represent progress or BS?

    http://plato.stanford.edu/entries/truth-deflationary/

    According to the deflationary theory of truth, to assert that a statement is true is just to assert the statement itself. For example, to say that ‘snow is white’ is true, or that it is true that snow is white, is equivalent to saying simply that snow is white, and this, according to the deflationary theory, is all that can be said significantly about the truth of ‘snow is white’

    –this makes no sense. To say ‘Snow is black’ is certainly to assert that the statement ‘Snow is black’ is true.

    –But snow is not black!

    –How would you verify the highlighted (‘is all that…’) claim?

    To start of ‘deflation’ seems trivial:

    Of course ‘Snow is white’ means (denotes) the same things as ‘The truth is snow is white’. The latter is just bad, redundant writing my 12th grade English teacher taught us about, akin to ‘I think snow is white’ as likely you would not assert ‘Snow is white’ if you didn’t think it was white (even if you’re wrong).

    How you get to ‘all that can be said significantly…’ is baffling. The assertion ‘Snow is black’ is false. The assertion ‘The truth is snow is black’ is redundant and false.

    The statement “The snow is yellow, best not eat it’ could be true and might be significant to you if you just ran out of water during the Iditarod.


    ‘Significantly’ to whom?

    “The truth is saying simply that snow is white, is, according to the deflationary theory, all that can be said significantly about the truth of ‘snow is white’. “ is true, i.e., that is what they say, but what the hell does it mean?

    http://iditarod.com/race/2016/standings/

    Frege:
    It is worthy of notice that the sentence ‘I smell the scent of violets’ has the same content as the sentence ‘it is true that I smell the scent of violets’. So it seems, then, that nothing is added to the thought by my ascribing to it the property of truth. (Frege 1918)

    -Duh?

    And still it might be untrue that you ‘smell the scent of the violets’. You might say ‘It is untrue that I smell the scent of the violets.’ A poorly phrased admission of lying. I always lie.

    What people mean when they assert ‘The truth is X’ is that they’re not lying when they say X. When a used car salesman or a politician say ‘The truth is…’ watch out. They likely are lying and that may well be significant to you.

    So to me, a physicist, this kind convoluted reasoning/logic gives philosophy a bad rep. It seems like a pseudo-profound gloss on things that are not that complicated.
    I assert ‘Snow is white’. My assertion could be true or false.

    ‘The truth is snow is white’ is just awkward way asserting the assertion is true–bad English (and subtle physics/neurology). Literally redundant, but we know what it means.

    Like

  12. Mark Szlazak

    Since i cannot post back where Duhem-Quine first came up, I am kind of wondering if critical rationalism/Popper threatens so much of philosophy that philosophers resort to all kinds of straw-man and misdirection tactics to avoid a tragic denouement. Discrediting Bayesianism would be perhaps to unpleasant since it is used now as the current point of progress but it has a history of coming and going. Maybe much has to be pushed back to the time of Popper and what his followers have done since, instead of what is popular now.

    Like

  13. couvent2104

    Mark,

    In what sense Duhem wasn’t practicing “critical rationalism”?
    You may feel he was wrong, but what I know about him gives me the impression that he was a rational, critical thinker.
    I also have the impression – but I’m a mere physicist – that physicists apply Duhem. It’s no accident that there were two different experimental groups (ATLAS and CMS) working on the discovery of the Higgs. Duhem didn’t believe in “crucial experiments”, and physicists – Whig historiography of physics notwhitstanding – do not believe in them either. An experiment is only deemed to be “crucial” if it is repeated and confirmed, preferably with other experimental techniques and approaches. If only ATLAS would have made the crucial detection of the Higgs, there would have been doubts. Why? Because physicists know that a cutting-edge experiment also is a test of the cutting-edge equipment used and the idea that it does what’s it supposed to do.

    You may find this trivial – and from a very detached, I would almost say philosophical viewpoint it perhaps is – but personally I don’t find the ingenuity that went into ATLAS and CMS trivial.

    If you only wanted to point out that Duhem-Quine is trivial in certain circumstances, I have no problem agreeing with that. If I’m timing a cake in the oven, and if I’m saying to myself “It’s an hour now” I don’t care about people shouting “Duhem-Quine” when I’m reading the timer.

    If have more difficulties with the Quine-part of Duhem-Quine. It’s general enough to become quite meaningless (an to undermine itself).

    Liked by 1 person

  14. Massimo Post author

    garth, Dan,

    as synred says, please lighten up a bit, it’s just a blog.

    (That reminds me, synred, perhaps a bit fewer YouTube videos?)

    garth, I don’t find generic accusations to “you philosophers” to be particularly useful, or accurate.

    Philosophers aren’t the only ones to react to a lot of euro-babble coming from contemporary neuroscience:

    http://www.amazon.com/Brainwashed-Seductive-Appeal-Mindless-Neuroscience/dp/

    which doesn’t mean neuroscience is bunk, it just means that there is a lot of hype, and one of the job of philosophers is to critically look at hype.

    As for human nature, Dan’s position is perfectly sensible, though I myself disagree with it. I’m not an essentialist, but there are non-essentialist ways to recover the concept of human nature. Still, that concept is by no means accepted universally not only in philosophy, but even in biology.

    At any rate, please turn down the heat a couple of notches (talking to Dan also, here): truth springs from argument amongst friends, not from hurling insults or sarcastic remarks to each other. Thanks.

    Liked by 2 people

  15. garthdaisy

    Massimo,

    I agree. Noticing that there is such a thing as human nature does not require essentialism. I’m neither an essentialist nor an anti-essentialist. I know it when I see it. And when a good study sees it.

    Apologies for the “you philosophers” comment I can’t even believe I said that as I consider myself a philosopher. But it may have been in reaction to an anti science vibe I get from certain philosophers. You yourself have admitted using the term “scientifically minded” as a bit of a pointed stick and you tend to deride science and scientists a lot. In the philosophy/science wars I’m the one who just wants mom and dad to stop fighting. I’m mad at the scientists who say philosophy is dead or useless or doesn’t make progress, and I’m mad at the philosophers who get territorial on subjects like human nature and reject good science on the subject or point to bad science reporting on the subject as an example of science on that subject.

    Stop fighting, mom and dad, you’re both valid and useful.

    Liked by 1 person

  16. Robin Herbert

    Hi garthdaisy,

    In tones? That’s hilarious. So much anger towards science on this blog

    Anger? What anger? Gently mocking science is being angry at science.

    And you don’t think writing can have a tone? You thought my post had a tone.

    They thought they were reasoning separate from their desires but the studies showed they were wrong and so was Kant. He thought you can separate your logic from your desires but you Kant.

    So tell me – could the people who conducted and reported upon those studies separate their reasoning from their desires? If not then surely we must reject what they say.

    But if they can, do they think that they have special reasoning abilities that we mere mortals don’t?

    No, the fact that they are presenting a reasoned argument indicates that they believe that it is possible to separate reasoning from desire.

    Like

  17. garthdaisy

    Robin,

    “So tell me – could the people who conducted and reported upon those studies separate their reasoning from their desires?”

    No. Their studies showed that. They are aware that nothing they do is perfectly reasoned without influence from their intuitions.

    “If not then surely we must reject what they say.”

    No, we just have to factor that information into our interpretation as they themselves do, and that interpretation itself will also be influenced and we have to factor that in. They are just reporting findings not making prescriptions. You can look at their data and decide for yourself how you’d like to take it.

    “But if they can, do they think that they have special reasoning abilities that we mere mortals don’t?”

    No. They can’t, and so they don’t.

    “the fact that they are presenting a reasoned argument indicates that they believe that it is possible to separate reasoning from desire.”

    No, it doesn’t mean that at all. They are making as reasoned an argument possible knowing from their own studies that it can only ever be so reasoned. Just because a study shows you can not be perfectly reasoned is no reason to stop trying top be as reasoned as possible.

    Like

  18. synred

    One physicist later wrote that, had Maurice Goldhaber not existed, “I am not sure that the helicity of the neutrino would ever have been measured.”

    Crease, Robert. The Prism and the Pendulum: The Ten Most Beautiful Experiments in Science (p. 208). Random House Publishing Group. Kindle Edition.

    https://en.wikipedia.org/wiki/Maurice_Goldhaber

    Up there with Beethoven? :_)

    Like

  19. synred

    http://www.robertpcrease.com/

    Hi Massimo, Do you know this guy (the author of our current Martin Perl book club book).? He is relatively near you (Stonybrook).

    He is a philosopher. The book seems a lot more level headed than many books we’ve read by scientist. Not so much hype and bad analogies..

    Like

  20. garthdaisy

    ““I am not sure that the helicity of the neutrino would ever have been measured.”

    “Ever” is just such a long time. Perhaps that’s why he was not sure.

    Like

  21. Robin Herbert

    Hi garthdaisy,

    “No, it doesn’t mean that at all. ”

    Really, they believe they can make a reasoned argument without being able to separate reason and desire at all?

    Will you grant me at least that either the people who conducted and reported on those studies managed to somewhat differentiate reason and desire, then their reports would be completely useless?

    Will you grant me that the rest of us might also be able to somewhat separate reason and desire to the extent that they do?

    “Just because a study shows you can not be perfectly reasoned is no reason to stop trying top be as reasoned as possible.”

    I am not sure that anyone is claiming that we can engage in perfect reasoning. Indeed, philosophers have been pointing out the imperfection of our reasoning processes for millennia. It was a very big theme in ancient Greek philosophy.

    If all they are pointing out is that we cannot reason perfectly then I am afraid that really does appear to be telling us what is perfectly obvious.

    Like

Comments are closed.