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At Aeon: Do philosophy and math still need each other?

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A photograph of the Greek letter pi, created as a large stone mosaic embedded in the ground.
pi in mosaic, Berlin/Holger Motzkau

One question is whether people are looking for logically certain answers any more:

Whereas mathematics seeks precise and certain answers, obtaining them in real life is often intractable or outright impossible. In such circumstances, what we really want are algorithms that return reasonable approximations to the right answers in an efficient and reliable manner. Real-world models also tend to rely on assumptions that are inherently uncertain and imprecise, and our software needs to handle such uncertainty and imprecision in robust ways.

Many of philosophy’s central objects of study – language, cognition, knowledge and inference – are soft in this sense. The structure of language is inherently amorphous. Concepts have fuzzy boundaries. Evidence for a scientific theory is rarely definitive but, rather, supports the hypotheses to varying degrees. If the appropriate scientific models in these domains require soft approaches rather than crisp mathematical descriptions, philosophy should take heed. We need to consider the possibility that, in the new millennium, the mathematical method is no longer fundamental to philosophy.

Ultimately, mathematics and the sciences can muddle along without academic philosophy, with insight, guidance and reflection coming from thoughtful practitioners. In contrast, philosophical thought doesn’t do anyone much good unless it is applied to something worth thinking about. But the philosophy of mathematics has served us well in the past, and can do so again. We should therefore pin our hopes on the next generation of philosophers, some of whom have begun to find their way back to the questions that really matter, experimenting with new methods of analysis and paying closer attention to mathematical practice. The subject still stands a chance, as long as we remember the reasons we care so much about it.Jeremy Avigad, “Principia
Is it possible that, in the new millennium, the mathematical method is no longer fundamental to philosophy?
” at Aeon

Hmmm. Maybe it is simpler than all this: One wonders, at times, when people say they have no use for philosophy or for mathematics, whether they really mean that they have no use for rationality or logic in their assessments of things.

See also: Prime numbers are not “nearly as scattershot” as previously thought

7 Replies to “At Aeon: Do philosophy and math still need each other?

  1. 1
    jdk says:

    Whereas mathematics seeks precise and certain answers, obtaining them in real life is often intractable or outright impossible. In such circumstances, what we really want are algorithms that return reasonable approximations to the right answers in an efficient and reliable manner. Real-world models also tend to rely on assumptions that are inherently uncertain and imprecise, and our software needs to handle such uncertainty and imprecision in robust ways.

    Many of philosophy’s central objects of study – language, cognition, knowledge and inference – are soft in this sense. The structure of language is inherently amorphous. Concepts have fuzzy boundaries. Evidence for a scientific theory is rarely definitive but, rather, supports the hypotheses to varying degrees.

    These are good statements, and I whole-heartedly agree.

    I didn’t read the whole article, though, so am not sure what it is saying about the role of philosophy.

  2. 2
    bornagain77 says:

    as to:

    “Real-world models also tend to rely on assumptions that are inherently uncertain and imprecise, and our software needs to handle such uncertainty and imprecision in robust ways.

    Many of philosophy’s central objects of study – language, cognition, knowledge and inference – are soft in this sense.”

    It is refreshing to see this honest admission that mathematics is extremely limited in its explanatory power when is comes to explaining reality, and humans in particular, and that a robust philosophy is also needed to make sense of it all.

    The smug, dismissive, attitude of many mathematicians towards philosophy can be summed up by this following quip,

    “Philosophy? We don’t need no stinkin philosophy!”

    Hawking infamously stated “philosophy is dead”

    “philosophy is dead. Philosophy has not kept up with modern developments in science, particularly physics. Scientists have become the bearers of the torch of discovery in our quest for knowledge.”
    – Hawking – The Grand Design
    https://www.age-of-the-sage.org/stephen_hawking/philosophy_is_dead.html

    Likewise Lawrence Krauss was equally scathing of philosophy. In fact Krauss called David Albert, a highly respected philosopher of physics at Columbia University, a “moronic philosopher” since Albert dared level a devastating philosophical critique against Krauss’s book ‘A Universe From Nothing”

    Lawrence Krauss: another physicist with an anti-philosophy complex – April 25, 2012
    Excerpt: Krauss understandably doesn’t like what Albert wrote. Still, I wonder if Krauss is justified in referring to Albert as a “moronic philosopher,” considering that the latter is not only a highly respected philosopher of physics at Columbia University, but also holds a PhD in theoretical physics. I didn’t think Rockefeller University (where Albert got his degree) gave out PhD’s to morons, but I could be wrong.
    http://rationallyspeaking.blog.....-with.html

    Tyson was similarly dismissive of philosophy

    Neil deGrasse Tyson and the value of philosophy
    Excerpt: Neil made his latest disparaging remarks about philosophy as a guest on the Nerdist podcast [4], following a statement by one of the hosts, who said that he majored in philosophy. Neil’s comeback was: “That can really mess you up.” The host then added: “I always felt like maybe there was a little too much question asking in philosophy [of science]?”
    https://scientiasalon.wordpress.com/2014/05/12/neil-degrasse-tyson-and-the-value-of-philosophy/

    As the preceding quotes highlight, often times many mathematicians tend to think that mathematics alone, minus a robust philosophy, can derive all truth about reality and about humans in particular.

    In fact, Steven Weinberg rejects the ‘instrumentalist approach’ in quantum mechanics precisely because it, “turns its back on a vision that became possible after Darwin, of a world governed by impersonal physical laws that control human behavior along with everything else.”

    The Trouble with Quantum Mechanics – Steven Weinberg – January 19, 2017
    Excerpt: The instrumentalist approach,, (the) wave function,, is merely an instrument that provides predictions of the probabilities of various outcomes when measurements are made.,,
    In the instrumentalist approach,,, humans are brought into the laws of nature at the most fundamental level. According to Eugene Wigner, a pioneer of quantum mechanics, “it was not possible to formulate the laws of quantum mechanics in a fully consistent way without reference to the consciousness.”11
    Thus the instrumentalist approach turns its back on a vision that became possible after Darwin, of a world governed by impersonal physical laws that control human behavior along with everything else. It is not that we object to thinking about humans. Rather, we want to understand the relation of humans to nature, not just assuming the character of this relation by incorporating it in what we suppose are nature’s fundamental laws, but rather by deduction from laws that make no explicit reference to humans. We may in the end have to give up this goal,,,
    Some physicists who adopt an instrumentalist approach argue that the probabilities we infer from the wave function are objective probabilities, independent of whether humans are making a measurement. I don’t find this tenable. In quantum mechanics these probabilities do not exist until people choose what to measure, such as the spin in one or another direction. Unlike the case of classical physics, a choice must be made,,,
    http://www.nybooks.com/article.....mechanics/

    As well, Max Tegmark went so far as to claim that humans are “nothing more than a consistent mathematical structure”

    In the following article, George Ellis remarks that “Tegmark has argued that every consistent mathematical structure exists in some disconnected universe. Tegmark also believes that nothing else exists beyond the consistent mathematical structures. Tegmark is himself nothing more than a consistent mathematical structure. This is a view that assigns to mathematical structures a degree of agency that they are not otherwise thought to possess.”

    Physics on Edge – George Ellis – August 2017
    Excerpt: Tegmark has argued that every consistent mathematical structure exists in some disconnected universe. Tegmark also believes that nothing else exists beyond the consistent mathematical structures. Tegmark is himself nothing more than a consistent mathematical structure. This is a view that assigns to mathematical structures a degree of agency that they are not otherwise thought to possess.
    http://inference-review.com/ar.....cs-on-edge

    Sheldon Glashow, professor of Mathematics and Physics at Boston University, in regards to Tegmarks belief that he nothing but a mathematical stucture, quips that “I may be a blockhead but I am certainly not a mathematical structure akin to a triangle.”

    A Hand-Waving Exact Science – Sheldon Glashow
    Excerpt: And our ToE is just one among an infinity of mathematical structures, each of them its own universe. If Tegmark is correct, there must exist a slightly different mathematical structure, whose equations are emblazoned on another T-shirt, wherein I am Tegmark’s psychiatrist rather than a physicist. I do not believe a word of it. Paraphrasing Danny, I may be a blockhead but I am certainly not a mathematical structure akin to a triangle.
    – Sheldon Glashow
    Sheldon Glashow is professor of Mathematics and Physics at Boston University and professor emeritus of Physics at Harvard University. He received the Nobel Prize in Physics in 1979.
    http://inference-review.com/ar.....ct-science

    Seeing as the article in the OP was touching upon the limits of what mathematics could ascertain about reality, and about humans in particular, I was surprised that no mention of Gödel was ever made in the article.

    If anything has ever been devastating to the thought that mathematics alone, minus any coherent philosophy, can provide all the answers to life it has been Gödel’s incompleteness theorem

    Mathematicians Bridge Finite-Infinite Divide – May 24, 2016
    Excerpt: Hilbert tasked mathematicians with proving that set theory and all of infinitistic mathematics is finitistically reducible, and therefore trustworthy. “We must know; we will know!” he said in a 1930 address in Königsberg — words later etched on his tomb.
    However, the Austrian-American mathematician Kurt Gödel showed in 1931 that, in fact, we won’t. In a shocking result, Gödel proved that no system of logical axioms (or starting assumptions) can ever prove its own consistency; to prove that a system of logic is consistent, you always need another axiom outside of the system. This means there is no ultimate set of axioms — no theory of everything — in mathematics. When looking for a set of axioms that yield all true mathematical statements and never contradict themselves, you always need another axiom. Gödel’s theorem meant that Hilbert’s program was doomed: The axioms of finitistic mathematics cannot even prove their own consistency, let alone the consistency of set theory and the mathematics of the infinite.
    https://www.quantamagazine.org/20160524-mathematicians-bridge-finite-infinite-divide/

    THE GOD OF THE MATHEMATICIANS – DAVID P. GOLDMAN – August 2010
    Excerpt: we cannot construct an ontology that makes God dispensable. Secularists can dismiss this as a mere exercise within predefined rules of the game of mathematical logic, but that is sour grapes, for it was the secular side that hoped to substitute logic for God in the first place. Gödel’s critique of the continuum hypothesis has the same implication as his incompleteness theorems: Mathematics never will create the sort of closed system that sorts reality into neat boxes.
    http://www.firstthings.com/art.....ematicians

    “Clearly then no scientific cosmology, which of necessity must be highly mathematical, can have its proof of consistency within itself as far as mathematics go. In absence of such consistency, all mathematical models, all theories of elementary particles, including the theory of quarks and gluons…fall inherently short of being that theory which shows in virtue of its a priori truth that the world can only be what it is and nothing else. This is true even if the theory happened to account for perfect accuracy for all phenomena of the physical world known at a particular time.”
    Stanley Jaki – Cosmos and Creator – 1980, pg. 49

    Even Hawking himself admitted that, “Gödel’s incompleteness theorem (1931), proves that there are limits to what can be ascertained by mathematics. Kurt Gödel, halted the achievement of a unifying all-encompassing theory of everything,,,”

    Gödel’s incompleteness theorem (1931), proves that there are limits to what can be ascertained by mathematics. Kurt Gödel, halted the achievement of a unifying all-encompassing theory of everything in his theorem that: “Anything you can draw a circle around cannot explain itself without referring to something outside the circle—something you have to assume but cannot prove”.
    – Stephen Hawking & Leonard Miodinow, The Grand Design (2010)

  3. 3
    bornagain77 says:

    As to explaining humans in particular, Gödel stated, “The formation in geological time of the human body by the laws of physics (or any other laws of similar nature), starting from a random distribution of elementary particles and the field is as unlikely as the separation of the atmosphere into its components. The complexity of the living things has to be present within the material [from which they are derived] or in the laws [governing their formation].”

    Conservation of information, evolution, etc – Sept. 30, 2014
    Excerpt: Kurt Gödel’s logical objection to Darwinian evolution:
    “The formation in geological time of the human body by the laws of physics (or any other laws of similar nature), starting from a random distribution of elementary particles and the field is as unlikely as the separation of the atmosphere into its components. The complexity of the living things has to be present within the material [from which they are derived] or in the laws [governing their formation].”
    Gödel – As quoted in H. Wang. “On `computabilism’ and physicalism: Some Problems.” in Nature’s Imagination, J. Cornwall, Ed, pp.161-189, Oxford University Press (1995).
    Gödel’s argument is that if evolution is unfolding from an initial state by mathematical laws of physics, it cannot generate any information not inherent from the start – and in his view, neither the primaeval environment nor the laws are information-rich enough.,,,
    More recently this led him (Dembski) to postulate a Law of Conservation of Information, or actually to consolidate the idea, first put forward by Nobel-prizewinner Peter Medawar in the 1980s. Medawar had shown, as others before him, that in mathematical and computational operations, no new information can be created, but new findings are always implicit in the original starting points – laws and axioms.,,,
    http://potiphar.jongarvey.co.u.....ution-etc/

    And indeed Gödel’s incompleteness theorem has now been extended to physics thus proving that mathematicians will never be able to discover, as Weinberg believed, “a world governed by impersonal physical laws that control human behavior along with everything else.”

    In the following article entitled ‘Quantum physics problem proved unsolvable: Gödel and Turing enter quantum physics’, which studied the derivation of macroscopic properties from a complete microscopic description, the researchers remark that even a perfect and complete description of the microscopic properties of a material is not enough to predict its macroscopic behaviour.,,, The researchers further commented that their findings challenge the reductionists’ point of view, as the insurmountable difficulty lies precisely in the derivation of macroscopic properties from a microscopic description.”

    Quantum physics problem proved unsolvable: Gödel and Turing enter quantum physics – December 9, 2015
    Excerpt: A mathematical problem underlying fundamental questions in particle and quantum physics is provably unsolvable,,,
    It is the first major problem in physics for which such a fundamental limitation could be proven. The findings are important because they show that even a perfect and complete description of the microscopic properties of a material is not enough to predict its macroscopic behaviour.,,,
    “We knew about the possibility of problems that are undecidable in principle since the works of Turing and Gödel in the 1930s,” added Co-author Professor Michael Wolf from Technical University of Munich. “So far, however, this only concerned the very abstract corners of theoretical computer science and mathematical logic. No one had seriously contemplated this as a possibility right in the heart of theoretical physics before. But our results change this picture. From a more philosophical perspective, they also challenge the reductionists’ point of view, as the insurmountable difficulty lies precisely in the derivation of macroscopic properties from a microscopic description.”
    http://phys.org/news/2015-12-q.....godel.html

    Darwinism vs Biological Form – video
    https://www.youtube.com/watch?v=JyNzNPgjM4w

    Thus it was surprising to see that no mention Gödel’s incompleteness theorem was made in the article in the OP since Gödel’s incompleteness theorem definitively proves that humans themselves and therefore “philosophy’s central objects of study – language, cognition, knowledge and inference” will forever be beyond a ‘complete’ mathematical explanation.

    Of supplemental note to mathematics needing a coherent overarching philosophy, mathematics itself points to a Theistic, even Christian, view of reality, not an atheistic and/or materialistic view of reality (as is apparently presupposed by many atheistic mathematicians).

    Mathematics itself exists in a transcendent, beyond space and time, realm which is not reducible any possible material explanation. This transcendent mathematical realm has been referred to as a Platonic mathematical world.

    Platonic mathematical world – image
    https://image.slidesharecdn.com/quantuminformation2-120301000431-phpapp01/95/quantum-information-14-728.jpg?cb=1330561190

    Naturalism and Self-Refutation – Michael Egnor – January 31, 2018
    Excerpt: Mathematics is certainly something we do. Is mathematics “included in the space-time continuum [with] basic elements … described by physics”? It seems a stretch. What is the physics behind the Pythagorean theorem? After all, no actual triangle is perfect, and thus no actual triangle in nature has sides such that the Pythagorean theorem holds. There is no real triangle in which the sum of the squares of the sides exactly equals the square of the hypotenuse. That holds true for all of geometry. Geometry is about concepts, not about anything in the natural world or about anything that can be described by physics. What is the “physics” of the fact that the area of a circle is pi multiplied by the square of the radius? And of course what is natural and physical about imaginary numbers, infinite series, irrational numbers, and the mathematics of more than three spatial dimensions? Mathematics is entirely about concepts, which have no precise instantiation in nature,,
    https://evolutionnews.org/2018/01/naturalism-and-self-refutation/

    Simply put, Mathematics itself, contrary to the materialistic presuppositions of Darwinists, does not need the physical world in order to exist. And yet Darwinists, although they deny that anything beyond nature exists, need this transcendent world of mathematics in order for their theory to be considered scientific in the first place. The predicament that Darwinists find themselves in regards to denying the reality of this transcendent, immaterial, world of mathematics, and yet needing validation from this transcendent, immaterial, world of mathematics in order to be considered scientific, should be the very definition of self-refuting.

    What Does It Mean to Say That Science & Religion Conflict? – M. Anthony Mills – April 16, 2018
    Excerpt: In fact, more problematic for the materialist than the non-existence of persons is the existence of mathematics. Why? Although a committed materialist might be perfectly willing to accept that you do not really exist, he will have a harder time accepting that numbers do not exist. The trouble is that numbers — along with other mathematical entities such as classes, sets, and functions — are indispensable for modern science. And yet — here’s the rub — these “abstract objects” are not material. Thus, one cannot take science as the only sure guide to reality and at the same time discount disbelief in all immaterial realities.
    https://www.realclearreligion.org/articles/2018/04/16/what_does_it_mean_to_say_that_science_and_religion_conflict.html

    As David Berlinski states in the following article,“There is no argument against religion that is not also an argument against mathematics. Mathematicians are capable of grasping a world of objects that lies beyond space and time…. The number four, after all, did not come into existence at a particular time, and it is not going to go out of existence at another time. It is neither here nor there. Nonetheless we are in some sense able to grasp the number by a faculty of our minds. Mathematical intuition is utterly mysterious. So for that matter is the fact that mathematical objects such as a Lie Group or a differentiable manifold have the power to interact with elementary particles or accelerating forces. But these are precisely the claims that theologians have always made as well – that human beings are capable by an exercise of their devotional abilities to come to some understanding of the deity; and the deity, although beyond space and time, is capable of interacting with material objects.”

    An Interview with David Berlinski – Jonathan Witt
    Berlinski: There is no argument against religion that is not also an argument against mathematics. Mathematicians are capable of grasping a world of objects that lies beyond space and time….
    Interviewer:… Come again(?) …
    Berlinski: No need to come again: I got to where I was going the first time. The number four, after all, did not come into existence at a particular time, and it is not going to go out of existence at another time. It is neither here nor there. Nonetheless we are in some sense able to grasp the number by a faculty of our minds. Mathematical intuition is utterly mysterious. So for that matter is the fact that mathematical objects such as a Lie Group or a differentiable manifold have the power to interact with elementary particles or accelerating forces. But these are precisely the claims that theologians have always made as well – that human beings are capable by an exercise of their devotional abilities to come to some understanding of the deity; and the deity, although beyond space and time, is capable of interacting with material objects.
    http://tofspot.blogspot.com/20.....-here.html

  4. 4
    bornagain77 says:

    To further support the notion that Theism, and Christian Theism in particular, should be the philosophical basis for an overall philosophical structure for mathematics, it should be noted that atheists have no compelling evidence for all the various parallel universe and/or multiverse scenarios that they have put forth.

    Multiverse Mania vs Reality – video
    https://www.youtube.com/watch?v=nQJV4fH6kMo

    And whereas, atheists have no compelling evidence for all these various parallel universe and/or multiverse scenarios that they have put forth, Christians, on the other hand, can appeal directly to the higher dimensional mathematics behind Quantum Mechanics, Special Relativity and General Relativity to support their belief that God upholds this universe in its continual existence, as well as to support their belief in a heavenly dimension and in a hellish dimension.

    The first part of the following video goes over that evidence,

    Quantum Mechanics, Special Relativity, General Relativity and Christianity – video
    https://www.youtube.com/watch?v=h4QDy1Soolo

    Moreover, the main problem with trying to unify General Relativity and Quantum Mechanics into a purely ‘mathematical’ theory of everything is that mathematicians have basically completely tossed agent causality, (and more specifically, have tossed the Agent Causality of God), by the wayside.

    Yet with the closing of the ‘free will loop-hole, quantum mechanics itself now demands that the agent causality of people, and more specifically the Agent Causality of God, be let back into the picture of modern physics.

    Quantum mechanics: Pushing the “free-will loophole” back to 7.8 billion years ago – September 14, 2018
    https://uncommondescent.com/intelligent-design/pushing-the-free-will-loophole-back-to-7-8-billion-years-ago/

    Moreover, if we rightly let the Agent causality of God ‘back’ into the picture of modern physics, (as Quantum Mechanics itself now demands and as the Christian founders of modern physics originally envisioned , Sir Isaac Newton, James Clerk Maxwell, Michael Faraday, and Max Planck, to name a few), then an empirically backed reconciliation between Quantum Mechanics and General Relativity, i.e. the ‘Theory of Everything’, readily pops out for us in Christ’s resurrection from the dead.

    Copernican Principle, Agent Causality, and Jesus Christ as the “Theory of Everything”
    https://www.youtube.com/watch?v=NziDraiPiOw

    Thus, unlike all the various extra dimensions, parallel universe and/or multiverse scenarios of atheists for which they have no evidence, the Christian, on the other hand, can appeal directly to the higher dimensional mathematics behind Quantum Mechanics, Special Relativity and General Relativity, to support his belief in heaven, hell, as well as in the providence of God. And can also appeal, (as was mentioned towards the end of the last video), to the Shroud of Turin to empirically support his contention that Jesus Christ successfully dealt with both quantum mechanics and general relativity, i.e. gravity, in his resurrection from the dead to provide us with a true, and philosophically robust, ‘Theory of Everything’

    Verse and video:

    Colossians 1:15-20
    The Son is the image of the invisible God, the firstborn over all creation. For in him all things were created: things in heaven and on earth, visible and invisible, whether thrones or powers or rulers or authorities; all things have been created through him and for him. He is before all things, and in him all things hold together. And he is the head of the body, the church; he is the beginning and the firstborn from among the dead, so that in everything he might have the supremacy. For God was pleased to have all his fullness dwell in him, and through him to reconcile to himself all things, whether things on earth or things in heaven, by making peace through his blood, shed on the cross.

    Shroud of Turin: From discovery of Photographic Negative, to 3D Information, to Hologram
    https://youtu.be/F-TL4QOCiis

  5. 5
    polistra says:

    Actually we could use an Auditing Philosopher to check simple things like epistemology and ontology.

    Most of our “news” and “politics” now is just atrocious epistemology. Long chains of said-that and knows-that and didn’t say that she didn’t know about his failure to condemn her lack of sufficient certainty ……. with no actual fact or event at the base of the chain.

    If it isn’t bad epistemology, it’s ontology disguised as facts. Persia is “attacking us” because Persia exists. Russia is “interfering” because Russia exists. Neither country is actually DOING anything to us.

  6. 6

    because it’s embarrassing to the rationalizing human mind. What’s required for certain answers? A true premise and valid logical form. But where does one get a true premise that needs no further proof? Where does one get a true premise without divine revelation? This question is embarrassing to the person who’s working hard to argue against God. And this is especially true when confronted with those of us who are following Christ and experiencing His moment-by-moment revelation in the form of leading, teaching, and correcting.

    God is well able to impart knowledge with certainty. In fact, He reveals Himself to every person, and failing to acknowledge Him takes extreme effort and desire to avoid Him.

  7. 7
    Fasteddious says:

    Some years ago, engineers promoted “fuzzy logic” for controlling everyday processes based on noisy or uncertain measurements and inputs. This was to be applied to various real-life processes, including washing machines and other household appliances, if I remember correctly. I suspect that it merely amounted to clever use of hysteresis and some real-world experience. In any case, it seems to have faded away and I haven’t heard much about it lately. Rather, such groups are now taken by “deep learning” as their buzz word for the latest and greatest in smart tech.
    Thus, if there is uncertainty and vagueness in mathematics and philosophy, there must surely be theories and processes to deal with it; Bayesian logic and analysis, for example.

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