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L&FP, 42a: The limit on Mathematical knowledge

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Here, a video series explores Godel’s incompleteness results:

The core point is that Hilbert’s scheme collapsed, nicely summarised. The Godel incompleteness results and the Turing machine halting challenge made Mathematics irreducibly complex. So, Mathematics, too, is a venture of knowledge as warranted, credibly true (so reliable) belief, which must be open to correction.

An exercise of rational, responsible faith, not utter certainty on the whole, once a sufficiently complex system is on the table. (Yes, first duties of reason obtain . . . here, there be dragons that love chick peas [Cicero . . .].)

The defeasible [= defeat-able] framework for understanding knowledge extends to Mathematics. A fortiori to Computer Science and Physics, then onward across the spectrum of disciplines and praxis.

We walk by faith, and not by sight. The question, then — given the Agrippa trilemma —

A summary of why we end up with foundations for our worldviews, whether or not we would phrase the matter that way}

. . . is the worldviews question: which core first plausibles, why. END

25 Replies to “L&FP, 42a: The limit on Mathematical knowledge

  1. 1
    kairosfocus says:

    The limit on Mathematical knowledge

  2. 2

    Irreducibly complex mathematics, what a horrible idea.

    Some mathematicians say they have found a way to avoid Godel’s incompleteness theorem. Something about deriving mathematics, including the mathematical operators, from zero.

    I saw a video of him lecturing to students to forget about a creator. But I guess that is no problem, because the entire creator category is subjective anyway, so would not be part of science.

    Again, what a totally horrible idea, that mathematics would be irreducibly complex.

  3. 3
    tjguy says:

    @2 Mr. Mohammad says the idea of irreducibly complex math is a horrible idea.

    My response is this. I doubt the truth/facts are determined by his emotional revulsion at the thought of irreducibly complex math.

    I can relate to not liking something and to not wanting something to be true, but in the end, our emotions do not determine truth.

    I’m not smart enough to argue this point either way so the above statement is neither pro or con towards the article – just an observation about his comment. It seems like he is letting his revulsion for something influence what he believes to be true. I guess we all do that at times because we all have biases.

  4. 4
    jerry says:

    My prediction is this too will deviate quickly from the OP into an indecipherable morass. The question is does this prediction have mathematical certainty?

    In the video the presenter uses the number zero to illustrate Godel’s ideas. Since zero does not exist, I assume if he chose an actual integer, it would end up the same. In other words there is nothing that depends on the initial number chosen?

    Did anyone else think the video of the show The Office have a guy that reminds one of the horned guy that entered the Capitol Building on January 6th? (Already setting up an off topic discussion).

  5. 5
    kairosfocus says:

    Jerry, zero exists as a key quantity. Indeed:

    {} –> 0
    {0} –> 1
    {0,1} –> 2
    . . .
    {0,1, 2 . . . } –> W

    etc.

    KF

  6. 6
    jerry says:

    Zero does not exist in the real world. It’s use is extremely useful but it doesn’t exist. Point to an example of zero. An empty set is an abstract concept not a real concept.

    https://www.youtube.com/watch?v=11EwyJ5fcBI

    Here we go again. Let’s not pursue this any further. It has been done before. It’s off topic.

    The question still stands, I assume any number could be used in Godel’s proof and doesn’t depend on using zero.

  7. 7
    kairosfocus says:

    Jerry, this is already tangential but it is reasonable that abstracta have reality that constrains states of affairs, including quantities in structured sets. 0 is as real as 1 and 1 + 3j, not to mention h and H. What is more, these are necessary, and that gives math its power. KF

  8. 8
    JVL says:

    Jerry: Zero does not exist in the real world. It’s use is extremely useful but it doesn’t exist. Point to an example of zero. An empty set is an abstract concept not a real concept.

    You’re leaving out man-made examples like temperature scales, longitude and latitude, economic growth, time stamps on recordings, time measurements in general, weights, acceleration, navigation bearings, shot gauges, etc.

    So, what are you saying doesn’t exist exactly?

  9. 9

    https://www.researchgate.net/publication/1956135_A_universal_alphabet_and_rewrite_system

    Peter Rowlands:
    “By rejecting the ‘loaded information’ that the integers represent, and basing our mathematics on an immediate zero totality, we believe that we are able to produce a mathematical structure which has the potential of avoiding the incompleteness indicated by Gödel’s theorem. ”

    The wisdom is, “Creatio ex nihilo”, and “ex nihilo, nihil fit”. And not, creatio ex complexity, whatever.

    One of Rowlands’ book is the only science book I ever read, half of it. It was getting too much new stuff to remember half way through, and there weren’t any overviews in the book. I am always thinking I will finish reading his book sometime, by making overview notations.

    As said, the man is apparently not a creationist, which in my book means he is a savage. So basically I have discovered this theory for creationism / real science.

    To not be a creationist, meaning the people who have a problem with acknowledging the human spirit on a properly subjective basis, including ofcourse my own spirit, are savages. That is justice. All the people who throw out subjectivity can take a hike.

    I have excellent subjective judgement. This rewrite theory is a great theory. I use subjective judgement all the time, works great! I guess there is also such a thing as subjectively liking the odds of a theory being factually true.

    This theory I found, while generally looking on the internet for the best theory in which choice is regarded as an actual reality of physics. Because creationism is just free will writ large, over the entire universe. And I already consider it proven that creationism is true, so then it was only a question of finding the best theory in which choice is regarded as a reality.

  10. 10
    kairosfocus says:

    MNY,

    it seems we are forever doomed to deal with tangents rather than focal issues, leading to having to refocus through the relevance of fundamental, first principles. Okay, I take a key opening salvo from your linked paper:

    In a recent paper,1 Deutsch et al state that, ‘Though the truths of logic and pure
    mathematics are objective and independent of any contingent facts or laws of nature, our knowledge of these truths depends entirely on our knowledge of the laws of physics.’
    According to these authors we have been forced by ‘recent progress in the theory of computation’, ‘to abandon the classical view that computation, and hence mathematical proof, are purely logical notions independent of that of computation as a physical process’. Mathematical structures, however autonomous, ‘are revealed to us only through the physical world’. We would go further and state that that mathematical structure which is most fundamental in understanding the physical world is also likely to be the structure which is most fundamental to understanding mathematics itself.

    The highlighted is my key point of concern. For, it is not just suggesting that mind is effectively computational but that our knowledge of laws of physics — which are mathematical in structure — controls our knowledge of Mathematics, effecting an inadvertent self-referential loop. I don’t think so.

    For, we are back to logic of being as applied to structure and quantity, across possible worlds, which was the hinge of my recent attack on the Wigner challenge. Let’s clip, starting with a supposition to be substantiated:

    t is possible that the reason for the effectiveness of mathematics is not because mathematics is in any way causative, but instead because mathematics studies the structure of logical possibility and constraint. When plugged into a possible world, mathematics gives us
    the tools to analyze the logically possible outcomes. There-fore, when a possible world that is expressed mathemati-cally sufficiently aligns with reality, mathematics becomes effective at expressing relationships and outcomes.

    That is, we have math as studying an aspect of the logic of being, across possible worlds, namely structure and quantity. So, if we can isolate aspects of structure and quantity that are necessary, world framework entities, they become common to all possible worlds, i.e. are trans-world universalisable and permanent. These include thought worlds . . . abstract, logic model worlds . . . and at least one world generally accepted as physically realised, our common world [we happen to deal with some who imagine differently, here at UD]. Thus, this core of mathematics is physically independent, or rather ontologically antecedent to physics and is thus non-circular on the physics or our knowledge of it.

    Accordingly, we see the relevance of the onward argument on distinct identity of a given possible world and what distinguishes from a near neighbour. And yes, we are forced to do an analysis, not rely on clever rhetorical talk points that play on emotions or inclinations toward candidate authorities. In the end it is facts and reasoning with assessment of worldview or model world level first plausibles or postulates that will be decisive.

    Clipping onward:

    . . . A key insight is that for any world W to be distinct from W’ [–> a close, near neighbour possible world] it requires some factor A in W that is absent in W’. We may then partition the factors of W as W = {A|~A}. After parti-tioning, we will have two distinct groups—the factor A and all of the factors which are not A. The null set corresponds to zero. Each particular set in the partition can be counted as the number one [–> note “simple” and “complex units], and the combination of both partitions (even in a single world where A is an empty set [–> yes, think on it: W = {{} = 0|~0]) is two. [–> this bridges to the paper] Thus, for any particular possible world W, the quantities 0, 1, 2 are necessarily present. Taking the von Neumann construction [–> {} –> 0, {0Z} –> 1, {0,1} –> 2 on], immediately we find N, thence (using addi-tive inverses) Z, so also (taking ratios [of integers]) Q and (summing convergent power series {~ decimal numbers}) R; where Z provides unit-stepped mileposts in R [the reals]. That is, a structured core of quantities will be present in any W, and we may regard mathematics as [–> substantial definition!] the study of the logic of structure and quantity. Extensions to the hyperreals R* follow by construction of some H that has as reciprocal h = 1/H closer to 0 than 1/n for any n in N. [We can go on to linked structures such as C and so forth, h of course is a gateway to Calculus, post Robinson, through non-standard analysis.]

    Therefore, relationships and linked operations across such quantities will also be present, or may be constructed as needed. Illustrating, after Abraham Robinson (Robinson, 1966), hyperreals allow calculus to be treated as extensions of algebra in R*.

    Thus, while bare distinct identity and coherence focused on quantities will not cause things by the inherent potential or action of such entities, they instead are logical constraints on being and are tied to what can or must be or cannot be or happens not to be. So, too, we may see that the abstract logic model worlds that we may construct then lead to key entities that if necessary are framework to any possible world; thus applicable to our common world. By contrast, if certain quantities and relationships are merely part of the contingencies of some W?? that is close enough to our own, they may provide adequate analogies for mod-elling.

    Here, we see a major province of mathematics that is independent of physical knowledge, but instead operates on the logic of beings with distinct identity. Ironically, the paper proceeds in the spirit of the Turing machine model and von Neumann’s construction from {}, to start with 0 and what is distinct from 0 albeit as yet undefined, i.e. with distinct identity and contrast:

    ” A key step in rewriting is the fact that there is an initial state. Here, we present a string representation of 0. We begin with the idea that only 0 is unique. Everything that is not 0 is undefined. In rewriting, we will start with an argument denying that we have a non-0 starting-point. We assume that we are not entitled to posit anything other than 0, and that we are force[d] to rewrite when we start from any other position. “

    In effect, we are back to von Neumann, they go on to set up a model to use 0 as start point to extend to other elements. However, we already see distinct identity, strings, symbols etc in a logic model world. Extensibility to all worlds begs for necessary being, as was already highlighted.

    That’s enough for the point that there is a universalisable core of Math, and further, to see the significance of the collapse of Hilbert’s programme.

    Thus, tangent becomes loop but not self referential circle.

    We really need to face the limitations on warrant for sufficiently complex mathematical systems, and we have to swallow weak form knowledge. Never mind, crow is delicious, see how well daddy makes it go down the hatch, Johnny. Jan, you too. (Curry covers a multitude of culinary sins!)

    And yes, see how he loves his carrots, peas, broccoli etc too.

    KF

  11. 11
    bornagain77 says:

    It is interesting to note that despite the fact that “Gödel halted the achievement of a unifying all-encompassing (mathematical) theory of everything in his theorem,,,”

    “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)

    ,,, It is interesting to note that despite that fact, most every theoretical physicist alive today still believes that they eventually will find a purely mathematical theory of everything.

    Despite Godel’s proof being around for 90 year’s now, It is still as if the word has not gotten out that Hilbert vision for mathematics, which is engraved on Hilbert’s tombstone, “We must know. We shall know.”, is still somehow tenable.

    David Hilbert
    Excerpt: The epitaph on his tombstone in Göttingen consists of the famous lines he spoke at the conclusion of his retirement address to the Society of German Scientists and Physicians on 8 September 1930. The words were given in response to the Latin maxim: “Ignoramus et ignorabimus” or “We do not know, we shall not know”:[25]

    Wir müssen wissen.
    Wir werden wissen.

    In English:

    We must know.
    We shall know.

    The day before Hilbert pronounced these phrases at the 1930 annual meeting of the Society of German Scientists and Physicians, Kurt Gödel—in a round table discussion during the Conference on Epistemology held jointly with the Society meetings—tentatively announced the first expression of his incompleteness theorem.[f] Gödel’s incompleteness theorems show that even elementary axiomatic systems such as Peano arithmetic are either self-contradicting or contain logical propositions that are impossible to prove or disprove.
    per wikipedia

    In short, despite the fact that it has been known, via Godel, for 90 years now that mathematics can’t possibly serve as the basis for its own foundation, (and that mathematics therefore has a contingent existence, not a necessary existence), most theoretical physicists alive today still act as if there exists some purely mathematical theorem that will be capable of explaining everything in the universe without reference to God whatsoever. i.e. a purely mathematical ‘theory of everything’.

    Theory of everything
    A theory of everything (TOE[1] or ToE), final theory, ultimate theory, or master theory is a hypothetical single, all-encompassing, coherent theoretical framework of physics that fully explains and links together all physical aspects of the universe.[2]:6 Finding a TOE is one of the major unsolved problems in physics.[3] String theory and M-theory have been proposed as theories of everything.
    – per wikipedia

    It is also interesting to note that the belief that mathematics itself, all by its lonesome, without any reference to God whatsoever, will be capable of explaining everything in the universe is in direct contradiction to what the Christian founders of modern science themselves believed.

    Namely, the Christian founders of modern science, (via Neoplatonic philosophy and Augustinian theology), believed that any mathematics that might describe this universe were based on the thoughts of God.

    KEEP IT SIMPLE by Edward Feser – April 2020
    Excerpt: Mathematics appears to describe a realm of entities with quasi-­divine attributes. The series of natural numbers is infinite. That one and one equal two and two and two equal four could not have been otherwise. Such mathematical truths never begin being true or cease being true; they hold eternally and immutably. The lines, planes, and figures studied by the geometer have a kind of perfection that the objects of our ­experience lack. Mathematical objects seem immaterial and known by pure reason rather than through the senses. Given the centrality of mathematics to scientific explanation, it seems in some way to be a cause of the natural world and its order.
    How can the mathematical realm be so apparently godlike? The traditional answer, originating in Neoplatonic philosophy and Augustinian theology, is that our knowledge of the mathematical realm is precisely knowledge, albeit inchoate, of the divine mind. Mathematical truths exhibit infinity, necessity, eternity, immutability, perfection, and immateriality because they are God’s thoughts, and they have such explanatory power in scientific theorizing because they are part of the blueprint implemented by God in creating the world. For some thinkers in this tradition, mathematics thus provides the starting point for an argument for the existence of God qua supreme intellect.
    There is also a very different answer, in which the mathematical realm is a rival to God rather than a path to him. According to this view, mathematical objects such as numbers and geometrical figures exist not only independently of the ­material world, but also independently of any mind, including the divine mind. They occupy a “third realm” of their own, the realm famously described in Plato’s Theory of Forms. God used this third realm as a blueprint when creating the physical world, but he did not create the realm itself and it exists outside of him. This position is usually called Platonism since it is commonly thought to have been ­Plato’s own view, as distinct from that of his Neoplatonic followers who relocated mathematical objects and other Forms into the divine mind. (I put to one side for present purposes the question of how historically accurate this standard narrative is.)
    https://www.firstthings.com/article/2020/04/keep-it-simple

    Perhaps the best example of just how integral this Augustinian view of mathematics, (i.e. “because they are God’s thoughts”), was for the rise of modern science is this following quote by Johannes Kepler. A quote which he made shortly after he discovered the laws of planetary motion in 1618,

    “O, Almighty God, I am thinking Thy thoughts after Thee!”
    – Johannes Kepler, “The Harmonies of the World.”, book five – 1619 – best known for his three laws of planetary motion,

    This view that mathematics exists “because they are God’s thoughts” and the Christian view that God created the universe and that the universe has not always existed, (as Aristotle had held), were, in fact, presuppositions that were necessary for modern science to take root in Medieval Christian Europe.

    In fact, the birth of modern science can be traced back to the quote-unquote ‘outlawing’ of Aristotle’s ‘deterministic and necessitarian views of creation.’

    The War against the War Between Science and Faith Revisited – July 2010
    Excerpt: … If science suffered only stillbirths in ancient cultures, how did it come to its unique viable birth? The beginning of science as a fully fledged enterprise took place in relation to two important definitions of the Magisterium of the Church. The first was the definition at the Fourth Lateran Council in the year 1215, that the universe was created out of nothing at the beginning of time. The second magisterial statement was at the local level, enunciated by Bishop Stephen Tempier of Paris who, on March 7, 1277, condemned 219 Aristotelian propositions, so outlawing the deterministic and necessitarian views of creation.
    These statements of the teaching authority of the Church expressed an atmosphere in which faith in God had penetrated the medieval culture and given rise to philosophical consequences. The cosmos was seen as contingent in its existence and thus dependent on a divine choice which called it into being; the universe is also contingent in its nature and so God was free to create this particular form of world among an infinity of other possibilities. Thus the cosmos cannot be a necessary form of existence; and so it has to be approached by a posteriori investigation. The universe is also rational and so a coherent discourse can be made about it. Indeed the contingency and rationality of the cosmos are like two pillars supporting the Christian vision of the cosmos.
    http://www.scifiwright.com/201.....revisited/

    Thus, the present view of theoretical physicists that mathematics, contrary to Godel, can serve as the basis of its own foundation, and that it has a necessary existence instead of a contingent existence that is based on ”God’s thoughts’, is, philosophically speaking, a major step backwards for modern theoretical physicists to take.

    God In Mathematics – 2016
    Jerry Bowyer – Interview with Vern Poythress
    Excerpt: The standard modern culture-war revolves around God vs. the mathematical sciences. Take your choice: Faith or physics. Then there are the voices of mutual toleration, which attempt to leave room for science among the faithful and for faith among the scientific. Poythress, though, taps into a different tradition entirely, one which is seldom heard in modern debate: That God and science are neither enemies, nor partners, but rather that God is the necessary foundation for mathematics and therefore of every science which uses it.
    The argument is that mathematical laws, in order to be properly relied upon, must have attributes which indicate an origin in God. They are true everywhere (omnipresent), true always (eternal), cannot be defied or defeated (omnipotent), and are rational and have language characteristics (which makes them personal). Omnipresent, omnipotent, eternal, personal… Sounds like God. Math is an expression of the mind of God. Sound strange? It isn’t. Modern natural science was created by people who said that they were trying to “think God’s thoughts after Him.”
    https://www.forbes.com/sites/jerrybowyer/2016/04/19/where-does-math-come-from-a-mathematiciantheologian-talks-about-the-limits-of-numbers/

    And although modern day theoretical physicists are seemingly loathe to ever allow a ‘Divine foot in the door’, it is interesting to note that the belief that any mathematics that might describe this universe are “God’s thoughts” has not yet completely died for modern theoretical physicists.

    In fact, Eugene Wigner, (who’s insights into quantum mechanics continue to drive breakthroughs in quantum mechanics; per A. Zeilinger), and Albert Einstein, who needs no introduction, are both on record as to regarding it as a miracle that math should even be applicable to the universe. Moreover, Wigner questioned Darwinism in the process of calling it a miracle, and Einstein even went so far as to chastise ‘professional atheists’ in his process of calling it a miracle.

    The Unreasonable Effectiveness of Mathematics in the Natural Sciences – Eugene Wigner – 1960
    Excerpt: ,,certainly it is hard to believe that our reasoning power was brought, by Darwin’s process of natural selection, to the perfection which it seems to possess.,,,
    It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them.,,,
    The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.
    http://www.dartmouth.edu/~matc.....igner.html

    On the Rational Order of the World: a Letter to Maurice Solovine – Albert Einstein – March 30, 1952
    Excerpt: “You find it strange that I consider the comprehensibility of the world (to the extent that we are authorized to speak of such a comprehensibility) as a miracle or as an eternal mystery. Well, a priori, one should expect a chaotic world, which cannot be grasped by the mind in any way .. the kind of order created by Newton’s theory of gravitation, for example, is wholly different. Even if a man proposes the axioms of the theory, the success of such a project presupposes a high degree of ordering of the objective world, and this could not be expected a priori. That is the ‘miracle’ which is constantly reinforced as our knowledge expands.
    There lies the weakness of positivists and professional atheists who are elated because they feel that they have not only successfully rid the world of gods but “bared the miracles.”
    -Albert Einstein
    http://inters.org/Einstein-Letter-Solovine

    As to the fact that the Christian founders of modern science believed the universe to be “contingent in its existence and thus dependent on a divine choice which called it into being; (and that) the universe is also contingent in its nature and so God was free to create this particular form of world among an infinity of other possibilities”,

    As to that fact, it is also interesting to note that “an infinite number of true mathematical theorems exist that cannot be proved from any finite system of axioms.”

    The Limits Of Reason – Gregory Chaitin – 2006
    Excerpt: Unlike Gödel’s approach, mine is based on measuring information and showing that some mathematical facts cannot be compressed into a theory because they are too complicated. This new approach suggests that what Gödel discovered was just the tip of the iceberg: an infinite number of true mathematical theorems exist that cannot be proved from any finite system of axioms.
    http://www.umcs.maine.edu/~chaitin/sciamer3.pdf

    In other words, an infinite number of true mathematical theorems could have described this universe but don’t.

    This puts the atheistic theoretical physicist in an awkward position.

    As Nobel Laureate Steven Weinberg, (an atheist), honestly admitted, “I don’t think one should underestimate the fix we are in. That in the end we will not be able to explain the world. That we will have some set of laws of nature (that) we will not be able to derive them on the grounds simply of mathematical consistency. Because we can already think of mathematically consistent laws that don’t describe the world as we know it. And we will always be left with a question ‘why are the laws of nature what they are rather than some other laws?’. And I don’t see any way out of that.”

    “I don’t think one should underestimate the fix we are in. That in the end we will not be able to explain the world. That we will have some set of laws of nature (that) we will not be able to derive them on the grounds simply of mathematical consistency. Because we can already think of mathematically consistent laws that don’t describe the world as we know it. And we will always be left with a question ‘why are the laws of nature what they are rather than some other laws?’. And I don’t see any way out of that.
    The fact that the constants of nature are suitable for life, which is clearly true, we observe,,,”
    (Weinberg then comments on the multiverse conjecture of atheists)
    “No one has constructed a theory in which that is true. I mean,, the (multiverse) theory would be speculative, but we don’t even have a theory in which that speculation is mathematically realized. But it is a possibility.”
    – Steven Weinberg – as stated to Richard Dawkins at the 8:15 minute mark of the following video
    – Leonard Susskind – Richard Dawkins and Steven Weinberg – 1 in 10^120 – Cosmological Constant points to intelligent design – video
    https://youtu.be/z4E_bT4ecgk?t=495

  12. 12
    bornagain77 says:

    Moreover, we, obviously, don’t have an infinite number of mathematical theorems that describe the universe, but we have only two. General Relativity and Quantum Mechanics respectfully.

    Yet these two theories simply refuse to be mathematically reconciled. In fact, there is found to be a ‘infinite mathematical divide’ that separates the two theories.

    Professor Jeremy Bernstein states the ‘infinite mathematical divide’ between the two theories as such, “there remains an irremediable difficulty. Every order reveals new types of infinities, and no finite number of renormalizations renders all the terms in the series finite.The theory is not renormalizable.”

    Quantum Leaps – Jeremy Bernstein – October 19, 2018
    Excerpt: Divergent series notwithstanding, quantum electrodynamics yielded results of remarkable accuracy. Consider the magnetic moment of the electron. This calculation, which has been calculated up to the fifth order in ?, agrees with experiment to ten parts in a billion. If one continued the calculation to higher and higher orders, at some point the series would begin to break down. There is no sign of that as yet. Why not carry out a similar program for gravitation? One can readily write down the Feynman graphs that represent the terms in the expansion. Yet there remains an irremediable difficulty. Every order reveals new types of infinities, and no finite number of renormalizations renders all the terms in the series finite.
    The theory is not renormalizable.
    https://inference-review.com/article/quantum-leaps
    Jeremy Bernstein is professor emeritus of physics at the Stevens Institute of Technology.

    And as theoretical physicist Sera Cremonini noted, “You would need to add infinitely many counterterms in a never-ending process. Renormalization would fail.,,,”

    Why Gravity Is Not Like the Other Forces
    We asked four physicists why gravity stands out among the forces of nature. We got four different answers.
    Excerpt: the quantum version of Einstein’s general relativity is “nonrenormalizable.”,,,
    In quantum theories, infinite terms appear when you try to calculate how very energetic particles scatter off each other and interact. In theories that are renormalizable — which include the theories describing all the forces of nature other than gravity — we can remove these infinities in a rigorous way by appropriately adding other quantities that effectively cancel them, so-called counterterms. This renormalization process leads to physically sensible answers that agree with experiments to a very high degree of accuracy.
    The problem with a quantum version of general relativity is that the calculations that would describe interactions of very energetic gravitons — the quantized units of gravity — would have infinitely many infinite terms. You would need to add infinitely many counterterms in a never-ending process. Renormalization would fail.,,,
    Sera Cremonini – theoretical physicist – Lehigh University
    https://www.quantamagazine.org/why-gravity-is-not-like-the-other-forces-20200615/

    Dr. William Dembski in this following comment, although he was not directly addressing the ‘infinite’ mathematical divide that exists between General Relativity and Quantum Mechanics, offers this insight into what the ‘unification’ of infinite God with finite man might look like mathematically:, Specifically he states, “The Cross is a path of humility in which the infinite God becomes finite and then contracts to zero, only to resurrect and thereby unite a finite humanity within a newfound infinity.”

    The End Of Christianity – Finding a Good God in an Evil World – Pg.31
    William Dembski PhDs. Mathematics and Theology
    Excerpt: “In mathematics there are two ways to go to infinity. One is to grow large without measure. The other is to form a fraction in which the denominator goes to zero. The Cross is a path of humility in which the infinite God becomes finite and then contracts to zero, only to resurrect and thereby unite a finite humanity within a newfound infinity.”
    http://www.designinference.com.....of_xty.pdf

    Philippians 2:8-9
    And being found in appearance as a man, He humbled Himself and became obedient to the point of death, even the death of the cross. Therefore God also has highly exalted Him and given Him the name which is above every name,

    Moreover, when we rightly allow the Agent Causality of God ‘back’ into physics, as the Christian founders of modern science originally envisioned, and when we rightly allow the Agent Causality of God back into physics as quantum mechanics itself now empirically demands with the closing of the free will loophole by Anton Zeilinger and company,

    Cosmic Bell Test Using Random Measurement Settings from High-Redshift Quasars – Anton Zeilinger – 14 June 2018
    Excerpt: This experiment pushes back to at least 7.8 Gyr ago the most recent time by which any local-realist influences could have exploited the “freedom-of-choice” loophole to engineer the observed Bell violation, excluding any such mechanism from 96% of the space-time volume of the past light cone of our experiment, extending from the big bang to today.
    – per journals

    ,,, then that very reasonable concession on our part then provides us with a very plausible resolution for the much sought after ‘theory of everything’ in that Christ’s resurrection from the dead provides an empirically backed reconciliation, via the Shroud of Turin, between quantum mechanics and general relativity into the much sought after ‘Theory of Everything”.

    In regards to gravity being dealt with in the Shroud of Turin, the following article states that ‘The bottom part of the cloth (containing the dorsal image) would have born all the weight of the man’s supine body, yet the dorsal image is not encoded with a greater amount of intensity than the frontal image.’

    Particle Radiation from the Body – July 2012 – M. Antonacci, A. C. Lind
    Excerpt: The Shroud’s frontal and dorsal body images are encoded with the same amount of intensity, independent of any pressure or weight from the body. The bottom part of the cloth (containing the dorsal image) would have born all the weight of the man’s supine body, yet the dorsal image is not encoded with a greater amount of intensity than the frontal image. Radiation coming from the body would not only explain this feature, but also the left/right and light/dark reversals found on the cloth’s frontal and dorsal body images.
    https://academicjournals.org/journal/SRE/article-full-text-pdf/CC774D029455

    And in the following video, Isabel Piczek states,,, ‘The muscles of the body are absolutely not crushed against the stone of the tomb. They are perfect. It means the body is hovering between the two sides of the shroud. What does that mean? It means there is absolutely no gravity.’

    “When you look at the image of the shroud, the two bodies next to each other, you feel that it is a flat image. But if you create, for instance, a three dimensional object, as I did, the real body, then you realize that there is a strange dividing element. An interface from which the image is projected up and the image is projected down. The muscles of the body are absolutely not crushed against the stone of the tomb. They are perfect. It means the body is hovering between the two sides of the shroud. What does that mean? It means there is absolutely no gravity. Other strange you discover is that the image is absolutely undistorted. Now if you imagine the clothe was wrinkled, tied, wrapped around the body, and all of the sudden you see a perfect image, which is impossible unless the shroud was made absolutely taut, rigidly taut.”
    Isabel Piczek –
    Turin shroud – (Particle Physicist explains the ‘event horizon’ on the Shroud of Turin) – video
    https://www.youtube.com/watch?v=27Ru3_TWuiY

    Kevin Moran, an optical engineer, describes the Shroud Image in this way, “The unique front-and-back only image can be best described as gravitationally collimated. The radiation that made the image acted perfectly parallel to gravity. There is no side image. The radiation is parallel to gravity,,,”

    Optically Terminated Image Pixels Observed on Frei 1978 Samples – Kevin E. Moran – 1999
    Discussion
    Pia’s negative photograph, from 1898, showed what looked to be a body that was glowing, but slightly submerged in a bath of cloudy water. This condition is more properly described as an image that is visible, at a distance, but by locally attenuated radiation. The unique front-and-back only image can be best described as gravitationally collimated. The radiation that made the image acted perfectly parallel to gravity. There is no side image. The radiation is parallel to gravity and, if moving at light speed, only lasted about 100 picoseconds. It is particulate in nature, colliding only with some of the fibers. It is not a continuum or spherical-front radiation that made the image, as visible or UV light. It is not the X-ray radiation that obeys the one over R squared law that we are so accustomed to in medicine. It is more unique,,,
    Theoretical model
    It is suggested that the image was formed when a high-energy particle struck the fiber and released radiation within the fiber at a speed greater that the local speed of light. Since the fiber acts as a light pipe, this energy moved out through the fiber until it encountered an optical discontinuity, then it slowed to the local speed of light and dispersed.
    Discussion
    The fact that the pixels don’t fluoresce suggests that the conversion to their now brittle dehydrated state occurred instantly and completely so no partial products remain to be activated by the ultraviolet light. This suggests a quantum event where a finite amount of energy transferred abruptly. The fact that there are images front and back suggests the radiating particles were released along the gravity vector. The radiation pressure may also help explain why the blood was “lifted cleanly” from the body as it transformed to a resurrected state.”
    https://www.shroud.com/pdfs/moran.pdf

    Moreover, besides gravity being dealt with on the Shroud of Turin, the Shroud also gives us evidence that Quantum Mechanics itself was dealt with. In the following paper, it was found that it was not possible to describe the image formation on the Shroud in classical terms but they found it necessary to describe the formation of the image on the Shroud in discrete quantum terms.

    The absorbed energy in the Shroud body image formation appears as contributed by discrete (quantum) values – Giovanni Fazio, Giuseppe Mandaglio – 2008
    Excerpt: This result means that the optical density distribution,, can not be attributed at the absorbed energy described in the framework of the classical physics model. It is, in fact, necessary to hypothesize a absorption by discrete values of the energy where the ‘quantum’ is equal to the one necessary to yellow one fibril.
    http://cab.unime.it/mus/541/1/c1a0802004.pdf

    Moreover, the following rather astonishing study on the Shroud, found that it would take 34 Trillion Watts of what is termed VUV (directional) radiation to form the image on the shroud.

    Astonishing discovery at Christ’s tomb supports Turin Shroud – NOV 26TH 2016
    Excerpt: The first attempts made to reproduce the face on the Shroud by radiation, used a CO2 laser which produced an image on a linen fabric that is similar at a macroscopic level. However, microscopic analysis showed a coloring that is too deep and many charred linen threads, features that are incompatible with the Shroud image. Instead, the results of ENEA “show that a short and intense burst of VUV directional radiation can color a linen cloth so as to reproduce many of the peculiar characteristics of the body image on the Shroud of Turin, including shades of color, the surface color of the fibrils of the outer linen fabric, and the absence of fluorescence”.
    ‘However, Enea scientists warn, “it should be noted that the total power of VUV radiations required to instantly color the surface of linen that corresponds to a human of average height, body surface area equal to = 2000 MW/cm2 17000 cm2 = 34 thousand billion watts makes it impractical today to reproduce the entire Shroud image using a single laser excimer, since this power cannot be produced by any VUV light source built to date (the most powerful available on the market come only to several billion watts)”.
    Comment
    The ENEA study of the Holy Shroud of Turin concluded that it would take 34 Thousand Billion (trillion) Watts of VUV radiation to make the image on the shroud. This output of electromagnetic energy remains beyond human technology.
    – per predator

    So thus in conclusion, when we rightly allow the Agent Causality of God back into physics as the Christian founders of modern science originally envisioned then a very plausible solution to the number one unsolved mystery in science today, of finding a reconciliation between General Relativity and Quantum Mechanics, readily pops out for us in that, as the Shroud of Turin gives witness to, both Gravity and Quantum Mechanics were dealt with in Christ’s resurrection from the dead.

    Verses and Video

    Matthew 28:18
    Then Jesus came to them and said, “All authority in heaven and on earth has been given to me,”

    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.

    Jesus Christ as the correct “Theory of Everything” – video
    https://youtu.be/Vpn2Vu8–eE

  13. 13
    kairosfocus says:

    BA77,

    good to hear from you again; trust you are well. Yes the vid shows how Hilbert’s programme collapsed. The consequences are laid out in OP.

    I note a clip from Feser:

    There is also a very different answer, in which the mathematical realm is a rival to God rather than a path to him. According to this view, mathematical objects such as numbers and geometrical figures exist not only independently of the ­material world, but also independently of any mind, including the divine mind. They occupy a “third realm” of their own, the realm famously described in Plato’s Theory of Forms. God used this third realm as a blueprint when creating the physical world, but he did not create the realm itself and it exists outside of him. This position is usually called Platonism since it is commonly thought to have been ­Plato’s own view, as distinct from that of his Neoplatonic followers who relocated mathematical objects and other Forms into the divine mind. (I put to one side for present purposes the question of how historically accurate this standard narrative is.)

    I think it is clear that the logic of distinct possible worlds embeds a necessary framework of structure and quantity tied to logic of being and distinct identity, much as rational thought relies on same. That gives universalisability of that core as we study it. As for independence of God, I think Plato had no concept of God as ultimate wellspring of reality so of worlds of thought or as actualised. Their creator in his vision, the demiurge, made an imperfect copy of the forms in a chaos that struggles to have some order.

    Ethical theism is utterly different. The inherently good utterly wise God is a necessary, maximally great being, the creator- sustainer- provider, source of order and of creatures with power of choice, thus ability to reflect the rational essence of the creator including ability to love, root of goodness and virtue. Logic of being is not independent of the creator, it is part of his wisdom and eternal contemplation. Including, logic of structure and quantity framework to any possible world.

    Where, without that necessary being root of reality, there would be no worlds pondered or actualised, so in the sense of anything existing rather than perpetual utter non being [it is hard to even conceive of utter negation of existence] there are no structures and quantities, relationships and principles independent of the creator and root of reality. That said, certain things starting with distinct identity so duality thence N, Z, Q, R, C, R* etc are inherent as abstract logical constraints on any world. Complete with structures, relationships, properties, principles. These too are necessary, world framework entities but have no power to be a root of worlds.

    KF

  14. 14
    jerry says:

    Still no one answered the question on whether Godel’s proof depends on there being a zero. I assume it doesn’t but the video makes a big deal of using zero.

    Aside: the video was sponsored by Brilliant.org. I just signed up for the 7 day free trial. It’s expensive to do so for a year. I am doing the logic course and many of their initial puzzles depend on analyzing self referencing truth/false statements.

    Aside2: I’m probably one of the few here who has seen the Shroud. Barrie Schwortz operates the oldest website on it. It hasn’t anything to do with OP.

    https://shroud.com/

  15. 15
    kairosfocus says:

    Jerry, Godel numbers depend on integers. Note, I highlight the von Neumann framework, which starts from {} –> 0, and observe the fact that distinct identity of a possible world W requires some difference A from a near neighbour W’. So we can structure W = {A|~A} where the partition, crucially is empty, i.e. it expresses 0. We also see two distinct units and thus duality; one of which may be compound. Roll up von Neumann and we have counting numbers from 0, thence by additive inverses Z, thence rationals Q and reals R. C and R* also come up. 0 is as real and as necessary as any other relevant quantitative entity. Note, it is the identity element of + for just one key functionality. Absent the emptiness of that partition, LNC and LEM collapse. KF

  16. 16

    Come on, Rowlands isn’t there making any explanation of how the mind works. It is just saying that natural objects, such as planets, compute things themselves.

    It was always an obvious nonsense that planets would “follow the laws of nature”, as like people following the laws in the lawbook of the land.

    So then the laws of nature are inherent in objects, and objects compute their next state themselves. Objects function logically.

    Another remark Rowlands makes elsewhere, is about “naive” understanding of physics. That is physics that starts out with seeing stuff like a rock or whatever, and then explains everything else in terms of what has already been seen. That’s ofcourse a prejudicial view based on the solids, because there are also liquids and gasses, and all kinds of other things.

    So really you have to build things from the ground up, and that is what he is doing. An ordering of mathematical structures in efficient steps from 0. That is the unprejudiced view of things. And then rocks would also fit in somewhere to that view.

    You, start with many unexplained factors, a complexity of them. Ofcourse, when your task is explaining things, it doesn’t help to bring in many unexplainable things. Aren’t you supposed to explain things in science? So even if what you say is true, you have to accept it as a loss, that you bring in many fundamental unexplained things, because your task is to explain things.

    You constantly make snide remarks about emotions, subjectivity. Which is a total outrage. To argue the superiority of fact over opinion, is totally absurd. Fact and opinion are obviously each valid in their own right, in their own domain. That is the main issue, for as far as worldviews is concerned.

    It is materialism vs creationism. It is people who are obsessed with facts vs people who acknowledge the validity of opinion and fact, with the 2 fundamental categories of creator and creation.

  17. 17
    kairosfocus says:

    MNY, the bridge between opinion and fact has two piers, credible truth and adequate warrant. We are all finite, fallible, morally struggling, too often ill-willed and sometimes stubborn or outright closed-minded or even selectively hyperskeptical. That is why we need to acknowledge and submit to the legitimate, inescapable authority of the self-evident Ciceronian first duties to truth, right reason, prudence [including warrant], sound conscience [cf. above on error-proneness], neighbour, so too fairness and justice etc. KF

  18. 18

    There is no bridge between opinion and fact, whatsoever. They each have a totally different underlying logic. Categorically different. And to make a mistake with the categories, by putting a matter of opinion in the fact category, or viceversa, is a very serious logic error.

    An opinion is formed by choice, and expresses what is that makes a choice.

    A fact is obtained by evidence of a creation, forcing to produce a 1 to 1 corresponding model of it it in the mind.

    See the difference, opinions are chosen in freedom, while facts are forced. Opinions are about what makes a choice, a creator, and facts are about chosen things, creations.

    You’re losing the entire point of creationism / intelligent design vs evolution.

    It’s about fighting for the proper domain of subjectivity, personal opinion, the spirit. The human spirit, as well as God the holy spirit. Straightforward and unequivocal acceptance of the validity of subjectivity, besides objectivtity. Each validated in their own right, with the 2 categories of creator and creation.

    You’re totally losing, and only my way is the winning way.

    You can just teach the difference between opinion and fact in school, with the creationist conceptual scheme, because teaching the difference between opinion and fact is already an established education goal.

    And with teaching the creationist conceptual scheme, materialism, atheism and socialism are destroyed.

    It is so obvious that the only reason why materialism and socialism are so bloody awful, is because they provide no room for emotions and personal opinion.

  19. 19
    kairosfocus says:

    MNY:

    AmHD:

    o·pin·ion n. 1. A belief or conclusion held with confidence but not substantiated by positive knowledge or proof: “The world is not run by thought, nor by imagination, but by opinion” (Elizabeth Drew). See Synonyms at view.

    be·lief n. 1. The mental act, condition, or habit of placing trust or confidence in another: My belief in you is as strong as ever.
    2. Mental acceptance of and conviction in the truth, actuality, or validity of something: His explanation of what happened defies belief.
    3. Something believed or accepted as true, especially a particular tenet or a body of tenets accepted by a group of persons.

    fact n. 1. Knowledge or information based on real occurrences: an account based on fact; a blur of fact and fancy.
    2. a. Something demonstrated to exist or known to have existed: Genetic engineering is now a fact. That Chaucer was a real person is an undisputed fact.
    b. A real occurrence; an event: had to prove the facts of the case.

    So, in epistemology it is in order to hold that “the bridge between opinion and fact has two piers, credible truth and adequate warrant”

    KF

  20. 20
    PaV says:

    Two simple thoughts:

    First, Godel’s theorem is treated at length in Roger Penrose’s, The Emporer’s New Mind. That’s a good starting point.

    Second, St. Thomas Aquinas says in the Summa Theologica that first principles of morality are intuited. From these first principles, all of morality flows. If you change the assumptions used in Euclidean Geometry, you can then formulate Non-Euclidean Geometry. Two completely different mathematical realities with shared first principles.

    I can’t help adding this:

    I’m a Neo-Platonist on this. Our intuitions borrow from the “world of ideas.” In a similar fashion, if you change morality’s first principles, the whole of this new form of morality is radically different from the prior system. Augustine says: “How can I find the idea of God within me, unless God first placed it there.” (Book 10 of the Confessions) [He means how can I grasp an infinite concept with my finite mind–think mathematics here] That is, Truth comes from God. He is the radical source of “objective” reality. Science has no way of helping us here. It loses all of its force.

  21. 21

    So you are basically saying, that the statement that the painting is beautiful, is of the same sort as the statement that the painting is on canvas, and of a man on a horse.

    That there is no categorical difference between statements of opinion and statements of fact. That all statements are tied up with the same underlying logic of warrant etc.

    While I am saying there is a categorical difference. That opinions are formed by choice and express what it is that makes a choice. And facts are obtained by evidence of a creation, forcing to produce a 1 to 1 corresponding model of it in the mind.

    1. Creator / chooses / spiritual / subjective / opinion
    2. Creation / chosen / material / objective / fact

    So basically you would be against teaching the difference between opinion and fact in school, because there is no fundamental difference, because they are all based on the same underlying logic.

    What is generally taught in school is that, opinion is what you feel, and fact is what you can prove.

    Which is consistent with what I say. You choose an opinion, based on feeling. The opinion is formed, by spontaneous expression of emotion with free will.

    And if it is demonstrated that the model corresponds, with what it is a model of, then the fact is proven. It is accurate.

    Ofcourse what is not taught in school is that when a statement of fact is found to be false, then it is still categorically a statement of fact, and does not turn into a statement of opinion.

  22. 22
    kairosfocus says:

    MNY, no. But as it turns out there are objective aesthetics principles that capture a lot of what is going on in our recognition of beauty and ability to appreciate same. KF

    PS: knowledge is structurally related to beliefs, and said beliefs often express opinions. Hence, the OP, knowledge is warranted, credibly true belief. Belief here is thought with assent towards accepting as true, accurate description or portrayal etc of states of affairs. I believe that Mr Abraham Lincoln was once president of the US and that there was a person who could be called King Arthur. Warrant is readily seen for the first, it is known. The second is far harder to pin down, I may know there may be a late Roman era figure who could be at root of legendary accretions, and can argue for some degree of reliabilioty but that such an individual was, is not knowledge.

  23. 23

    If the evidence is insufficient, then it’s called a guess, a hypothesis, or something. It is still the same basic structure as the concept of fact, it is still proposed as being a 1 to 1 corresponding model, it is just that the evidence is lacking. It’s still totally different from the logic of opinion.

    What you say doesn’t add up, obviously. Either fact and opinion are categorically different, or they are all the same based on the logic of warrant etc. Then they are only different in degree, and not categorically different.

    Basically you tolerate teaching the difference between opinion and fact in school. Out of the kindness of your heart you allow some ground for subjectivity. Please sir, please can I have an emotion? Oh alright then.

    It doesn’t actually follow from you stated beliefs.

    And you never go straightforwardly into subjectivity. With aesthethics, you only take the part that you say is objective, not explaining the part that is subjective.

  24. 24
    kairosfocus says:

    MNY, opinion is inevitable, freedom to responsibly express same is a right, but in a world where we as subjects are finite, fallible, error-prone, morally struggling and too often ill willed, we need to recognise the bridge between opinion and knowledge, warrant and credible truth. KF

  25. 25
    kairosfocus says:

    F/N: It is interesting that the focal issue for the OP, that Mathematics itself exhibits characteristics of weak form knowledge, seems to be something that cannot be readily, plausibly objected to. Such carries with it the force that we must recognise that reasonable, responsible belief is deeply embedded in our intellectual endeavours. We must all face the issue of our first plausibles, related worldviews and comparative difficulties analysis. KF

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