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Stirring the pot: on the apparent mathematical ordering of reality, and linked worldview/ philosophical/ theological issues . . .

This morning, in the Gonzalez video post comment exchange, I saw where Mung raised a question about how Young Earth Creationists address the Old Cosmos, Old Earth implications of the view raised.

I thought it useful to respond briefly, but then the wider connexions surfaced.

I would like to stir the pot a bit [--> pl. note the new category], by headlining some sketched out thoughts for consideration, on the mathematical ordering of reality, and related worldviews level philosophical and even theological issues. Indeed, somewhere along the line, the whole project of the validity of a natural theology (and Biblical references to same) crops up as connected to the concerns.

Kindly, consider the below scoop-out from my response to Mung as very much preliminary notes on thoughts in progress, and do help me clarify and correct my thinking:

___________________

>> I hear you, though there are of course at least two YEC responses: (i) the for-argument, where one shows that even if X were so, then “architect-ing” of the cosmos is still credible, and (ii) something like the Humphreys argument of an old cosmos, frozen time, young earth model.

A third is indeed to reject — per scripture-interpretation grounds — the design approach as a variant on old earth theistic evolutionary thought, which is I think (not 100% sure) near to the “[Semi-?]official view” of one or two of the major YEC groups.

We may well argue one way or another on whether such are valid approaches, but they are options open to YEC advocates.

What I find interesting about the design argument — setting aside the YEC issues — is how robust it is against age, multiverse etc speculations, it is almost as if someone wanted to make sure that the cosmos would “speak” to us, whether looked at with an appreciative eye or analysed scientifically and/or mathematically and/or philosophically.

Right now, what is exercising my mind in the background, is the concept of mathematical ordering of the cosmos as a sign of rational mind behind it.

You will recall where I recently started with the empty set {} and then constructed the natural numbers by successively equating {} = 0, {0} = 1, {0, 1} = 2, etc. This of course already brings to bear infinity. We can then define operations and mappings so we have addition, subtraction, equality etc. Then, I moved to the reals by using a Baire-like construction that exploits the properties of decimal fractions [WLOG] by defining ordered points in the interval [0, 1] through a countably infinite tree with ten branches at each forking node in succession 0.abc . . . Such allows us to fill up the continuum between 0 and 1, which is all we need to extend to any continuum.

After that, I had used the i operator concept (where i*i*x = -1*x  so we see sqrt -1 playing an important role) to get us to space (and BTW, to angles and rotation in time too, implicitly using the series expansion definitions of e^x, cos x and sin x, where x can in turn be generated as w*t, w angular velocity and t, time). Once we have an interval [0, 1], where it can also be shown that there is some continuous function on a proposed space S that will map points in [0,1] to its points, S is continuous too and is pathwise connected. The space of points (x, i*x) is obviously such as r*e^i*q can span it, r being the magnitude of the vector where r is in [0,1]. In addition, it is possible to iteratively define a Peano space filling curve that in effect allows a moving point — here comes kinematics, the study of motion without regard to forces — to fill it by continuously touring all points in the space. By identifying an orthogonal set of unit vectors i, j, k, and by bringing on board vectors and matrices as usually defined, we are at 3-d space and we are also at kinematics in time long since. For dynamics, we only need to move from points and images to inertial properties and forces across space and time thus momentum and energy. Along the way, calculus enters and gives tools to analyse — or to define — dynamics in the continuum. By this time, we are in familiar territory and can keep going to all of maths and physics in space-time domains.

This would fit in with the effectiveness of mathematics in physical science, as if the world is ordered mathematically in a unified way, then it would follow the logical consequences of those underlying principles and consequent dynamics.

That is, we have here a frame that is at least suggestive of an ordering frame for physical reality: logical, mathematical mind. (And BTW, I find a unifying view of mathematics and moving onwards to physical dynamics helps fill my urge to find coherent unity. I therefore find it satisfying and motivating of explorations. Much moreso than the actual isolated and rather utilitarian way in which I was actually exposed to such things way back. In effect I am reviewing and back-filling, connecting dots etc. I wonder if that failure of unifying vision is a part of the problem we have with Math? And with science? Etc?)

That brings me back to a scripturally based theological point (yes!), as we may see some interesting and empirically testable assertions in Jn 1, Col 1 and Heb 1, i.e. points where three of those heavily packed brief phil statements I recently talked about crop up — I cite AMP:

Jn 1:1 In the beginning [before all time] was the Word ([a --> NB: Word is LOGOS, meaning, communication, rationality himself . . . ] Christ), and the Word was with God, and the Word was God [b]Himself.

2 He was present originally with God.

3 All things were made and came into existence through Him; and without Him was not even one thing made that has come into being. [--> Notice the implied contrast of contingent vs necessary being, and so creation is the zone of the contingent that has a beginning. By contrast, mathematical reality as above is mental and logical, and insofar as it denotes necessary truth and implications of such, is beginningless, held eternally in the mind of God]

4 In Him was Life, and the Life was the Light of men.

5 And the Light shines on in the darkness, for the darkness has never overpowered it [put it out or absorbed it or appropriated it, and is unreceptive to it]. [--> this brings out the moral ordering of the world and the true balance of power]

Col 1:15 [Now] He is the [o]exact likeness of the unseen God [the visible representation of the invisible]; He is the Firstborn of all creation.

16 For it was in Him that all things were created, in heaven and on earth, things seen and things unseen, whether thrones, dominions, rulers, or authorities; all things were created and exist through Him [by His service, intervention] and in and for Him.

17 And He Himself existed before all things, and in Him all things consist (cohere, are held together).

Heb 1:1 In many separate revelations [[a]each of which set forth a portion of the Truth] and in different ways God spoke of old to [our] forefathers in and by the prophets,

2 [But] in [b]the last of these days He has spoken to us in [the person of a] Son [--> Word/Logos again], Whom He appointed Heir and lawful Owner of all things, also by and through Whom He created the worlds and the reaches of space and the ages of time [He made, produced, built, operated, and arranged them in order].

3 He is the sole expression of the glory of God [the Light-being, the [c]out-raying or radiance of the divine], and He is the perfect imprint and very image of [God’s] nature, upholding and maintaining and guiding and propelling the universe by His mighty word of power. [--> Definition of natural law in the perspective of "thinking God's (creative and sustaining, ordering) thoughts after him"!] When He had by offering Himself accomplished our cleansing of sins and riddance of guilt, He sat down at the right hand of the divine Majesty on high,

4 [Taking a place and rank by which] He Himself became as much superior to angels as the glorious Name (title) which He has inherited is different from and more excellent than theirs.

These contexts are of course historically quite important in Christian theology and in civilisations shaped by such. Indeed, we see here outlined the philosophical theology that shaped the mindset that propelled the scientific revolution by giving confidence that there was an intelligible, coherently rational natural order made and sustained by God, c 1200 – 1700 and which in accordance with our creation mandate, we were commissioned to explore, discover and use for good. Unfortunately, too often, we have abused that order on the one hand to do evil, and on the other, have too often turned the very order into an improper substitute for the creator who stands behind it.

It is against that backdrop that we run into a key pair of texts:

Ps 19:1 The heavens declare the glory of God; and the firmament shows and proclaims His handiwork.

2 Day after day pours forth speech, and night after night shows forth knowledge.

3 There is no speech nor spoken word [from the stars]; their voice is not heard.

4 Yet their voice [in evidence] goes out through all the earth, their sayings to the end of the world . . .

Rom 1:19 For that which is known about God is evident to them and made plain in their inner consciousness, because God [Himself] has shown it to them.

20 For ever since the creation of the world His invisible nature and attributes, that is, His eternal power and divinity, have been made intelligible and clearly discernible in and through the things that have been made (His handiworks). So [men] are without excuse [altogether without any defense or justification],

21 Because when they knew and recognized Him as God, they did not honor and glorify Him as God or give Him thanks. But instead they became futile and [c]godless in their thinking [with vain imaginings, foolish reasoning, and stupid speculations] and their senseless minds were darkened.

22 Claiming to be wise, they became fools [professing to be smart, they made simpletons of themselves].

23 And by them the glory and majesty and excellence of the immortal God were exchanged for and represented by images, resembling mortal man and birds and beasts and reptiles.

24 Therefore God gave them up in the lusts of their [own] hearts to sexual impurity, to the dishonoring of their bodies among themselves [abandoning them to the degrading power of sin],

25 Because they exchanged the truth of God for a lie and worshiped and served the creature rather than the Creator, Who is blessed forever! Amen (so be it).

Now, I find in these an echo of the line of thought I have been exploring, and also some pretty bold, empirically testable claims and implications. We had better believe — on evident track record — that if the evidence had come up that the cosmos is a chaos instead, this would have been cast in the teeth of theists, loud and long.

Instead, we find an ordered, mathematically coherent system of reality of amazing beauty. The reaction? Stridency, or insistence, to dismiss the idea that even possibly, there lies behind reality an ordering highly mathematical-logical mind.

Ooops, I raised another issue there, didn’t I.

God, plainly, is a serious candidate to be a necessary being, and the NB that would be foundational to reality — indeed, inter alia the eternal mind in which eternal necessary truths reside eternally. Especially mathematical ones, of course.

But that brings up a challenge, that there is a logic of contingency vs necessity of being. Contingent beings [CB's] possibly may not exist, i.e there are possible worlds in which they would not be. Thus, they depend on external enabling causal factors [XEF's], think of these as “on/off switches” that must be on or the CB cannot begin or be sustained; e.g. the four factors for a fire highlighted through the fire tetrahedron: fuel, heat, oxidiser, chain reaction.

A genuine NB, on the other hand, has no XEF’s, so it would have no beginning and no possibility of ending.

It would be eternal.

So, if we consider a candidate NB (CNB), the issue is whether it is genuine or not: that is, it either exists in all possible worlds (GNB), or it is IMPOSSIBLE (IB), it can exist in no possible world. For example there is no possible world (PW) in which 3 + 2 = 5 will fail, not even a world empty of material objects. (That is guaranteed by the force of the cascade from the empty set up, mental constructs can create a mathematical universe!)

Likewise, there is no possible world in which 2 + 3 = 6, or the like; such is an IB.

This brings us to a significant challenge that faces those who would deny or dismiss the reality of God.

As God is indeed a serious CNB, we have the choice: GNB or IB. Or as has been put elsewhere, if a candidate necessary being is possible, it will be actual. That is, if it can feasibly exist in at least one possible world, it is not impossible and by virtue of the logic, it will not only be in one possible world but all possible worlds, including the actual one.

So, to deny or dismiss God is to imply — one may not be aware of this, of course — that one holds God to be IMPOSSIBLE.

That is a serious logical-metaphysical commitment indeed.

And, on track record of the objectors, one that many such are very reluctant to take up. Indeed, in part the rhetorical tack of projecting an imagined unique burden of proof on theism, and assuming that in default of absolute deductively valid and sound proof they can rest comfortable on the default assumption that God does not exist, is patently fallacious.

I say this, as invariably, we have worldviews that trace to core beliefs as a cluster, about reality, ourselves in it etc. On the “every tub must stand on its own bottom” principle my Grandpa was so fond of, each worldview faces the same challenge of being reasonable and coherent, matching well to and explaining the world as we experience it. So, playing at rhetorical default games in a worldviews foundation context — and I here assert that every worldview has a foundation of some sort (as can be seen from the implications of the abstract chain of warrant A –> B –> C –> . . . entered into once we ask in succession, why accept A, B, C . . . once we also accept that infinite regress is impossible for the finite and fallible such as we are) — is a fallacy.

{Let me insert an illustration:

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

 

I think some serious re-thinking on worldviews is in order.>>

___________________

Am I on to something here, or am I barking up the wrong tree(s)?

Why or why not? END

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41 Responses to Stirring the pot: on the apparent mathematical ordering of reality, and linked worldview/ philosophical/ theological issues . . .

  1. Oops, correction no 1: r in z = e^iq should range [0, sqrt_2]. Such gives a circle of radius sqrt_2, that then enfolds the unit box of points [x, i*x] = z, such that 1 >/= |z|. Pardon me if I occasionally slip up and use the physicist/engineering j for i to mean sqrt_-1. KF

  2. No 2: To make “tiling” (and onward, “brick-ing”) easy, restrict |x| LT/= 1. We can also define x_1 and x_2, where for convenience x_2 = y and x_1 = x. Of course on extension x_3 = z, where z is not the complex no in this case.

  3. What is it about the word, ‘Yes’, atheists don’t understand, I wonder?

    What is it about the total cogency of empirical physical proofs, confirmed by mathematics, that they don’t understand?

    There really is a strange ambivalence, uncertainty even in their minds, even of Nobel prize-winners.

    I saw a YouTube video-clip of Murray-Gell, and he was chuckling fit to bust that, when he had told Einstein someone had found a fault in one of his relativity theories, he had simply replied: ‘It’ll go away…’

    He had done the math, and evidently had it checked. Indeed, he sometimes had better mathematicians to do the computations. So what part of the cogency of properly-performed mathematics is subject to uncertainty, and if it is, why, in what way?

  4. For all their bombast re the Promissory Note, they clearly do not feel able to really repose much faith in mathematics and science. It’s as if they’re ‘winging it’, and hoping for the best.

  5. I find the natural theology – in the- Bible aspect of the above is also significant. Any thoughts? KF

  6. F/N: For instance, in Rom 1, it speaks of God showing to people the deliverances of observation and reflection on the world and the inner man (if we add in Rom 2, this seems to in part speak to core morality stamped in conscience), i.e. natural reason informed by the world is viewed as speaking with a revelatory voice. One that can be resisted, but only with culpability.

    How would such speak at worldviews analysis level, to issues of design evident in observable, testable signs in nature, and to the voice of morality, as well as the reflection on “the unreasonable effectiveness of Mathematics [and by extension, Logic]” in understanding our physical world, in light of the onward circumstances of natural law, so deeply stamped that even chance processes typically obey stable statistical laws?

    In that context, did Craig have a point in his recent debate, where he highlighted the mathematical order of the world as one of the signs pointing to its origin in a supremely rational mind?

    And, so forth?

    Onwards, what does this say to those who would say, more or less, that design theory is stealth Creationist theology and theocracy?

    To those, who would dismiss the idea of objective, observable and intelligible evidence pointing to design of our world and of our own selves?

    And so forth?

    KF

  7. Axel, There is also the challenge of grounding the reliability of the perceiving, reasoning, knowing mind, on evo mat premises. Indeed, i am prepared to argue that on such premises, science, math, reasoning and knowledge of the world all become self-referentially incoherent. KF

  8. Sorry, I would just like to clarify a couple of mathematical points.

    What do you mean by “fill up the continuum” as in

    Such allows us to fill up the continuum between 0 and 1, which is all we need to extend to any continuum.

    Could you give me a reference to your usage?

    Also I’m a bit confused by this statement:

    I had used the i operator concept (where i*i*x = -1 so we see sqrt -1 playing an important role) to get us to space (and BTW, to angles and rotation in time too, implicitly using the series expansion definitions of e^x, cos x and sin x, where x can in turn be generated as w*t, w angular velocity and t, time).

    Obviously i times i = -1 but what is x? And what kind of series expansion are you using? And how is x being generated as w*t?

  9. Jerad,

    Maybe I am being too compressed?

    In the first case, I am using a tree of countably infinite depth with each successive node taking the relevant stage in the place value number 0.abc . . . , which summarises the series: 0 + a* 1/10 + b* 1/100 + c* 1/1000 + . . .

    I am sure you will recognise how such a tree will fill out the interval [0, 1]. Where of course 0.50000 . . . and 0.4999 . . . will converge to the same value. In short, I am spanning the continuum. a = 0, 1, 2, 3 . . . 9 marks the appropriate tenths, ab will give the hundredths between two tenths, abc the thousandths between two hundredths, and so on to an ever finer spotting of points. Obviously, this can go on to a countable infinity of places, so that every point in the continuum has been tagged “in principle.”

    In short, I am filling in the unit interval on the number line, using a series technique to cover the points similar to Baire.

    I assume you know that this can be extended in various ways to any relevant continuum.

    I am next using the old trick where i* i* [x] rotates the vector pi rads a/c, to yield – x. This gives us the “natural interpretation that i*x is an orthogonal axis, the y axis for convenience. The Cartesian and Argand planes are closely related, and complex numbers and (x,y) coords alike are vectors.

    Since e^iwt = cos wt + i*sin wt, we have a natural definition for angles and rotation. (I am ducking going for Greek letters.)

    The expansions in question for these relations are the Taylor series ones, which if I am not mistaken, were used by Newton.

    x = wt is saying, angular value — strictly, displacement — a/c from x axis (the polar axis) is angular velocity times time. I have avoided using theta here and omega.

    This is of course also the context for the Euler relationship 0 = 1 + e^ i*pi. Just work out for pi rads.

    In going on to 3-d space, I have used the ijk unit vectors as a basis.

    Kinematics is motion without reference to force etc and to inertia. Dynamics introduces such.

    At that point, we are in sufficiently familiar territory to say, and so on.

    (My point was to get a unifying core math framework for physical investigations. that is why I wanted to start from {} and get natural numbers, then Reals, then Complex and vectors, including rotations, vectors, phasors and angles etc. Then to span to space and time, kinematics and dynamics. And for my purposes I need not worry on onward bridges to the quantum and relativistic worlds. Build up from mental concepts to a virtual world then flesh out with kinematics and dynamics.)

    I hope that is less compressed.

    KF

  10. The first section in green doesn’t make much sense – neither compressed nor uncompressed: how is i an operator? i*i*x = -1 should be i*i*x = -x, but why do you introduce that way? Mixing Peano curves and kinematics is very baffling, too!

  11. F/N: The point of that exploration is to look at the ways that he logic of mathematics can set up a coherent framework for creating a physics, i.e. a simple model of “thinking God’s thoughts after him.” This, in answer to the unreasonable effectiveness of mathematics in science. Where I am going is that from the empty set we go to the natural numbers by a familiar process. The next issue is how do we bridge to the continuum? Taking the hint of the place value notation decimal number system (and its near relatives) we use an approach of Baire, and consider a countably infinite depth tree that in effect fills the interval [0,1] by successively decimating 1/10 fractions of 1/10 fractions etc. Then the injection of the complex number concept and use of the Taylor series expansions of exponentials, leads to a plane and to rotation and angle. (Indeed, these days angle is defined on more or less this sort of approach.) The extension to further roots of unity following Hamilton, leads to the ijk vectors and to 3-D space. Motion of points leads to kinematics, and injection of inertia and forces to dynamics. Whereupon, we are in familiar territory. So, we have an exploration of how Mathematical concepts can be a unifying construct for a physical world, and can be embedded in the reality from the ground up. Is this a helpful summary of what I have been up to? KF

  12. F/N: BA has linked an interesting discussion by Craig, here:

    Whether one is a realist or an anti-realist about mathematical objects, I think that the theist enjoys a considerable advantage over the naturalist in explaining the uncanny success of mathematics.

    Take realism first. As philosopher of mathematics Mary Leng points out, for the non-theistic realist, the fact that physical reality behaves in line with the dictates of acausal mathematical entities existing beyond space and time is “a happy coincidence” (Mathematics and Reality [Oxford: Oxford University Press, 2010], p. 239). Think about it: If, per impossibile, all the abstract objects in the mathematical realm were to disappear overnight, there would be no effect on the physical world. This is simply to reiterate that abstract objects are causally inert. The idea that realism somehow accounts for the applicability of mathematics “is actually very counterintuitive,” muses Mark Balaguer, a philosopher of mathematics. “The idea here is that in order to believe that the physical world has the nature that empirical science assigns to it, I have to believe that there are causally inert mathematical objects, existing outside of spacetime,” an idea which is inherently implausible (Platonism and Anti-Platonism in Mathematics [New York: Oxford University Press, 1998], p. 136).

    By contrast, the theistic realist can argue that God has fashioned the world on the structure of the mathematical objects. This is essentially what Plato believed. The world has mathematical structure as a result.

    Now consider anti-realism of a non-theistic sort. Leng says that on anti-realism relations said to obtain among mathematical objects just mirror the relations obtaining among things in the world, so that there is no happy coincidence. Well and good, but what remains wanting on secular anti-realism is an explanation why the physical world exhibits so complex and stunning a mathematical structure in the first place. Balaguer admits that he has no explanation why, on anti-realism, mathematics is applicable to the physical world or why it is indispensable in empirical science. He just observes that neither can the realist answer such “why” questions.

    By contrast, the theistic anti-realist has a ready explanation of the applicability of mathematics to the physical world: God has created it according to a certain blueprint He had in mind. There are any number of blueprints He might have chosen. Philosopher of mathematics Penelope Maddy asks,

    can the Arealist account for the application of mathematics without regarding it as true? . . . contemporary pure mathematics works in application by providing the empirical scientist with a wide range of abstract tools; the scientist uses these as models—of a cannon ball’s path or the electromagnetic field or curved spacetime—which he takes to resemble the physical phenomena in some rough ways, to depart from it in others. . . . The applied mathematician labors to understand the idealizations, simplifications and approximations involved in these deployments of his abstract structures; he strives as best he can to show how and why a given model resembles the world closely enough for the particular purposes at hand. In all this, the scientist never asserts the existence of the abstract model; he simply holds that the world is like the model is some respects, not in others. For this, the model need only be well-described, just as one might illuminate a given social situation by comparing it to an imaginary or mythological one, marking the similarities and dissimilarities (Defending the Axioms: On the Philosophical Foundations of Set Theory [Oxford: Oxford University Press, 2011], pp. 89-90).

    On theistic anti-realism the world exhibits the mathematical structure it does because God has chosen to create it according to the abstract model He had in mind. This was the view of the Jewish philosopher Philo of Alexandria, who maintained that God created the physical world on the mental model in His mind.

    More food for thought.

    KF

  13. In short, I am filling in the unit interval on the number line, using a series technique to cover the points similar to Baire.

    I assume you know that this can be extended in various ways to any relevant continuum.

    I can see how you’ve created a countable set which has the same cardinality of the natural numbers. But, according to the continuum hypothesis, the cardinality of the real numbers is greater than the cardinality of the natural numbers. So there is a question of whether or not you’ve covered the whole interval.

    I am next using the old trick where i* i* [x] rotates the vector pi rads a/c, to yield – x. This gives us the “natural interpretation that i*x is an orthogonal axis, the y axis for convenience.

    I don’t know how ‘natural’ it is to use the basic imaginary number to create the complex plane but I know what you mean.

    I understand what you’re getting at . . . just wanted to know better what you meant by ‘continuum’ and whether or not you had indeed ‘filled’ the interval [0,1].

  14. Jerad:

    Thanks for thoughts.

    An infinitely deep digital string will pretty well cover any number that can be expressed as a decimal fraction (even of infinite length), by exhaustion. I am not indulging in a formal proof, which usually requires a major headache not suited to a blog or anything less than a Math Journal. Consider this as a model that brings out the possibility of infinitely deep sums of decimal fractions that exhaust the possibilities in the interval, rather than a proof.

    And, Absent our non standard analysis infinitesimals, I think that that should in principle cover any point on the interval.

    As to the imaginary numbers and ops on these, my thought is the rotation vector approach is pretty well intuitive. At least, this worked with my students.

    KF

  15. PS: Continuum: for any two distinct points in an interval, say p, q, another point belonging to same may be found or in some cases constructed.

  16. PPS: I think — remember, this thread is an exploration — the decimal fraction tree is uncountable, per a simple thought; it will not require as much analysis as I would have originally thought.

    What happens is that the scope of the tree as one reaches the transfinite aleph-null becomes of the cardinality of 2^aleph-null.

    Recall, at each level of nodes n, we have 10^n branches: root, 10^0 -> 1 value [0]; 1st branch,10^1 –> 10, 2nd, 10^2 –> 100, etc. So, as we go transfinite in the counting chain, we are looking at (5 * 2) raised to a transfinite, countable power of cardinality aleph null.

    It is reasonable to infer from this that the cardinality is that of the power set of aleph null, i.e. on a reasonable set of axioms, the continuum number.

    The decimal fractions summed to transfinite depth DO seem to exhaust the continuum in [0,1] as intuition would tell us.

    Or, have I missed something?

    KF

  17. As you say, it’s not the place to argue over Cantor’s Continuum Hypothesis. And of course you can get as close as you want to any particular value using a countable tree.

    But, it is also true the the cardinality of the Real numbers is greater than the cardinality of the integers (or any countable set). You can always construct a ‘value’ that is not on any given list of countable numbers. So, you can’t get to the reals from a list or tree of rational numbers. it doesn’t just cross a line somewhere.

    And there are sets whose cardinality is greater than that of the real numbers. The set of all functions that map the reals to the reals for example.

    This kind of mathematics is slippery stuff and doesn’t necessarily have much to do with the usefulness of applying math models to the real world.

    Anyway, carry on.

  18. Yes, this is slippery, but the key point is that the cardinality is that which we expect of a continuum; where also the non-finite nature of the tree becomes material to its cardinality, i.e this is not a discretisation, it is converging on any and all points in the interval [0,1]. That is where its cardinality turns into a power set cardinality on that of the natural numbers. And yes, we can see a succession of transfinite numbers of ever increasing cardinality. Weird as that at first sounds. (Wait till you see the hyper-reals!) KF

  19. F/N: To see the cardinality point, instead of using base 10, slip over to binary base. At each step in 0.abc . . . we now have 2^n branch points/ nodes, and so as we proceed to the end we are at 2^aleph_null nodes [where, simply changing counting base cannot change the cardinality of the points at exhaustion], which on reasonable grounds is the cardinality of the continuum. KF

  20. Dieb:

    Thanks for the catch on -x not – 1.

    The trick of viewing i*[] as an operator that rotates pi/2 rad a/c, is more often seen in a technology context. Then, i*i* [x] –> – x yields a geometric and algebraic meaning. The vector-phasor interpretation of complex no’s and co-ords in the plane should be patent.

    Peano curves span a space by taking a convoluted tour. That is significant in injecting motion as a step beyond rotation. It also brings up how a space is a continuum.

    Kinematics is the — usually less esoteric — study of motion without reference to forces. So we can introduce moving points/particles and assess mathematically. (Historically that happened, Galileo was before Newton. And BTW, ever reflected on how we speak of point masses [so, infinite density], a physical impossibility?)

    Then, inject inertia, force and action of force across space [work] and time [momentum].

    The point is to sketch a bridge from purely mental constructs such as {} to a frame of dynamics, via mathematics. That embeds math in physics, or explores how it could be done. Of course, there are a lot of offshoots here, such as the complex frequency domain analysis of systems that is never very far from my thoughts.

    But the core issue is an exploration of the unifying role and power of Math in physical analysis. Thus, a question of the reflection of highly mathematical mind behind physical reality.

    As in; going back to some of the cosmological points raised by Plato — of course, without needing to take on the theory of a world of ideal forms.

    KF

  21. The trick of viewing i*[] as an operator that rotates pi/2 rad a/c, is more often seen in a technology context

    What’s a?, what’s c? Why rad? Multiplication with i rotates the complex plane by 90° or pi/2, what’s the rest all about? Yes, you can see the multiplication with a constant as a linear operator on the complex plane. But why all that? That doesn’t become clear – you are just mentioning mathematical buzzwords:

    Peano curves span a space by taking a convoluted tour. That is significant in injecting motion as a step beyond rotation. It also brings up how a space is a continuum.

    How is the Peano curve significant in “injecting motion”? Because of the phrase “taking a convoluted tour”? But it is a course no one can take, as it is a little bit long even between arbitrarily near points in the plane…. And what kind of continuum are you talking about? Obviously a set-theoretical – but what has this to do with motion, again?

  22. Dieb:

    I’m kinda busy just now, pardon intermittency.

    Radians are the natural measure of angle, and the natural rotation sense is anticlockwise.

    The issue of introducing rotations and angles by conceptual and algebraic definition is that we want a conception not dependent on an external separate world from the model. That is also why numbers are introduced by playing with a collection of sets starting with the empty one — utterly abstract. The point of the exercise is abstract antecedent and foundational to concrete.

    In speaking fairly loosely about the Peano curve, I am speaking not only of the ultimate result but he chain of steps, from the first few forms, which bring us to a touring point.

    Next, The continuum starts with the [0,1] interval as I want to move from naturals to reals then complex/ plane then 3-d world as a virtual entity.

    A touring point in the continuum introduces kinematics, and thence we may inject force, inertia and dynamics to get to a model physical world. Instantiation as a computer simulation or in the context of the root discussion, creation of a world, would follow from that modelling and would embed that math in the order of the world.

    KF

  23. Forgive me, I have never seen “a/c” used as an abbreviation for “anticlockwise” – ACW, CCW, yes, but “a/c” no: using your own abbr. doesn’t make it easier for the rest of us to understand you!

    From the rest of your answer I get the impression that you introduce concepts like the Peano-Curve without necessity, just to sound more formidable.

  24. F/N: Notice, the above “pot stirring exercise” is an initial exploration of the sort of issues that I think would arise from the question of mathematics as a sign of mind behind the cosmos, and wider concerns linked to philosophy and theology, especially natural theology. As such it is looking at design theory issues and also at linked phil and theology concerns, in a context of canonical statements of the Christian tradition. Observe how, in so examining, the issues of science and the grounding of physics in a mind that is mathematical, are not to be found in a circular argument pivoting on debates on interpretation of scriptural texts and traditions in theology. But, notice how the implications of the canonical statements connect to findings that are tied to the science and math. In this case, the classic note on the uncanny utility and power of mathematics in physics. It looks like a fruitful avenue of exploration, is the issue as to how a mind could conceive of a physical world mathematically then in instantiating it, go on to build it in light of the mathematical framework of ideas. Comments, suggestions and thoughts are welcome. KF

  25. Dieb:

    Actually, what obtains is that I am very much concerned to look at the bridge to the continuum from the naturals. (And, this I guess points to my own views on how math naturally extends so that reals and complex numbers are not to be dismissed as arbitrary constructs.)

    Accordingly, I have multiple reasons for the Peano curve in particular. First, at low, initial steps, as an exploration of points moving, then as we push the to the limit, the issue of a genuine continuum being demonstrated is there.

    Similarly, I introduced the rotation approach to address moving to 2 D’s algebraically.

    I do think it is conceptually important to see that complex numbers are vectors and that such are also subjected to rotation and to angular measures. In going on to the ijk vectors, I am extending the approach to 3 ds.

    The notion of motion brings in kinematics, and of course in that somewhere lurks the issue of temporally convergent infinite series in answer to Zeno, e.g. on the overtake issue. Dynamics follows. Calculus lurks, with issues on limits, continuity, smoothness of curves and changes, and what infinitesimals can be conceived to be. All of this points to the significance of infinity in math and to the transfinite numbers with ever so many set issues in the background.

    And given inter alia the underlying associated paradoxes of naive set theory, building up the foundational naturals from the empty set is important, and extending to the continuum is a challenge in 1-D. The Baire-like countably infinite tree offers a way to make the bridge and on inspection we can see that it does yield the right sort of continuum cardinality, at least on reasonable grounds. (Inter alia, it does mean that sufficiently extended decimal representations — which are series — will converge to any real number; a point we often overlook. It also means that if we carry the series far enough, we will stably be within a given acceptable error limit.)

    And, underlying all, is the issue that a space-time continuum can be conceived abstractly and adequately, using appropriate math that builds on the key abstraction, {}. From this, we can build up a series of further abstractions in a framework model that hen is a basis for instantiation of a physical — or for that matter a virtual — world. So, we have a model for how math can be built into a physical cosmos from the ground up, and it puts the candidate of a unifying, grounding mind for that reality squarely on the table.

    I do apologise for not using a more conventional abbreviation, but I had thought that in context it should have been reasonably clear, in light of R*e^iwt = R*cis wt which of course introduces rotational kinematics, and thinking on it, it allows sweeping space in 2-ds by continuously varying R. I suspect a time dependent version on the 3-D form of the Pythagoras relationship may allow a sweep of 3-d space, Notice, how much of the just outlined depends on the issue of convergent infinite series analysis also.

    While I am at it, the same relationship of course yields that 0 = 1 + e^ i*pi, which I take to be a point of unexpected convergence and coherence that underscores the unity and naturalness of reals and complexes as well as certain key functions.

    I find the exploration of an underlying unity in Math as a ground for physics is worth exploring in itself, and that it is worth tossing up a few unexpected issues or odd sidelights to see what they can cross-illuminate.

    KF

  26. kf, speaking of bringing in motion, read Berlinski’s The King of Infinite Space: Euclid and His Elements as soon as you can. :)

  27. Speaking of Mathematics:

    Meaning in Mathematics

  28. Mung;

    yes, I am aware that in some of the higher proofs in Elements, and elsewhere in Greek Geometry, motion was used to derive a proof, though it was often seen as a lower standard of proof.

    For instance, this was a part of generating quadratures, which of course are a primitive precursor to modern integration.

    The quadrature of the circle is for instance the square equal in area to it, which requires integration to find area. Calculus was first placed on a rigorous limits basis in C19. Subsequently Robinson’s non standard analysis of the 1960′s has extended the reals to hyper real numbers and in effect the inverse of extremely large numbers, the infinitesimals. This puts the intuitive insights of Liebnitz and Newton on a firm footing.

    An effect of all this is that there are infinite series that converge in finite time, due to the implications of things being in motion. This resolves Zeno’s Paradox;es], and is relevant to the continuum and to series convergence. This is an example of L’Hopital’s rule that solves expressions of form 0/0 and the like where we have a convergence. Here, as we move to the limit of Achilles overtaking the tortoise, what happens is that the finer and finer increments in distance are happening in finer and finer increments of time, and the overtake point is the result in the limit.

    KF

    PS: Polkinghorne et all look to be well worth reading on the abstract realities of Math.

  29. F/N: Jim Loy has an interesting discussion:

    Achilles, being confident of victory, gives the tortoise a head start. Zeno supposedly proves that Achilles can never overtake the tortoise. Here, I paraphrase Zeno’s argument:

    Before Achilles can overtake the tortoise, he must first run to point A, where the tortoise started. But then the tortoise has crawled to point B. Now Achilles must run to point B. But the tortoise has gone to point C, etc. Achilles is stuck in a situation in which he gets closer and closer to the tortoise, but never catches him.

    What Zeno is doing here, and in one of his other paradoxes, is to divide Achilles’ journey into an infinite number of pieces. This is certainly permissible, as any line segment can be divided into an infinite number of points or line segments. This, in effect, divides Achilles’ run into an infinite number of tasks. He must pass point A, then B, then C, etc. And what Zeno is arguing is that you can’t do an infinite number of tasks in a finite amount of time. Why not?

    Zeno says that you can divide a line into an infinite number of pieces. And then he says that you cannot divide a time interval into an infinite number of pieces. This is inconsistent.

    There is no paradox here. Zeno was just showing (pretending?) some ignorance of the nature of time. A time interval is just another line segment (when you graph it), that you can divide up in any way you want . . . .

    I think that Zeno, and Euclid, and Archimedes all had a firm grasp of infinity. 90% of our knowledge of infinity is from these three people. We did not have to wait for Newton and Cantor to explain it to us. They merely clarified some of the details. Zeno may have been puzzled, somewhat. But, I think he had infinity mostly figured out. Euclid (in defining pi) and Archimedes (in estimating pi) used geometric objects with an infinite number of sides, as a limit (without using the term “limit”), many centuries before Newton and Leibniz. Paradox (self-contradiction) is an important way in which Euclid and Archimedes disproved things. They showed no doubts about the legitimacy of dividing something into an infinite number of pieces.

    A finite length can be divided up into an infinite number of pieces, all of zero length. You can imagine that, can’t you. Just divide a length into halves, then fourths, then eighths, etc. But, in the Zeno story above, we find that none of the pieces is of zero length. They are all, infinitely many of them, longer than zero length. That may be counter-intuitive. But, it obviously is no paradox, as the mathematics is simple and clear.

    I am not so sure of the 90%, but he has a point, they were dealing with limit processes, but did not have the advantage of the next 2,000+ years of Math to develop it into a formal and rigorous system.

    They laid the foundations for it!

    Similarly, the Arrow Paradox:

    If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.[11] – as recounted by Aristotle, Physics VI:9, 239b5

    My own answer to this is that in the freeze frame snapshot, the arrow is indeed in a location at any given instant, but per conservation of momentum — a dynamical concept that is one of the great pivots in physical reasoning — it has a momentum at he same instant, that is, it is still moving.

    Just, we here have in concept an infinitely fast freeze frame, an infinitesimal blink of time.

    In the spatial and temporal continuums (pardon the bad Latin form used for clarity) — cf the reason for my taking time to work from natural numbers to continuum via a Baire-like tree reduction to a countable infinity of places in a decimal fraction — what is happening is that successive instants and places vary smoothly in light of convergent series and L’Hopital type reductions where in effect we ask which thing heading for 0 gets there faster, that dominates and can give a zero, finite or infinite result depending. (This is of course rooted in a calculus view.)

    KF

  30. PS: One significance of the Peano Curve tour of the space is that it reduced the space to a continuous, space filling line [resemblance to the reduction of the line to a succession of points through an infinitely deep tree is not coincidental]. That is, we show that the cardinality of the space’s number of points is the same as that of a line. It can be further shown that the number of points in [0,1] is the same as that of the whole real line.

    Here is Wiki on a space-filling curve:

    Intuitively, a continuous curve in 2 or 3 (or higher) dimensions can be thought of as the path of a continuously moving point. To eliminate the inherent vagueness of this notion Jordan in 1887 introduced the following rigorous definition, which has since been adopted as the precise description of the notion of a continuous curve:

    A curve (with endpoints) is a continuous function whose domain is the [--> closed] unit interval [0,?1].

    In the most general form, the range of such a function may lie in an arbitrary topological space, but in the most commonly studied cases, the range will lie in a Euclidean space such as the 2-dimensional plane (a planar curve) or the 3-dimensional space (space curve).

    Sometimes, the curve is identified with the range or image of the function (the set of all possible values of the function), instead of the function itself. It is also possible to define curves without endpoints to be a continuous function on the real line (or on the open unit interval (0, 1)).

    Notice, the link to kinematics as highlighted, and of course the very Newtonian conception of function as flow or motion.

    I should note that my favourite approach to conveying he core principle of calculus uses TIME not space as the independent variable, and looks at rates and accumulations of flow.

    Typically, I use a cylindrical bucket under a pipe being filled up at a rate that may be constant or variable. From this we can look at rate and volume vs time relationships and intuitively see the fundamental theorem that boils down to the inverse nature of the two operations, differentiation [extract rate] and integration [extract accumulation].

    In the next step, I use an impulse of flow, where rate surges and falls in a bell curve, to show a sigmoid accumulation of change. (This leads in the extreme to how a unit impulse yields a unit step response, which is BTW the the time domain correlate of the Laplace complex frequency domain transfer function. In turn, the jw — complex frequency — part of that Laplace Transform, is the frequency response of the system. Thence Fourier, for those who need or want it. As well, given that –where s is now the Laplace variable sigma + j*omega, j being the engineers’ form of sqrt – 1 — s*G(s) –> d/dt(g(t) and {1/s}*G(s) –> Integral[g(t)*dt] we have a basis for both analysing ordinary differential equation models and for understanding the high pass/low pass frequency response characteristics of systems. The heavy rubber sheet model with poles sticking up and zeros nailed down allows us to visualise the structure and dynamical implications of a given transfer response. This can be extended to discrete time with the related z-transform, where 1/z corresponds to a unit delay element. here, I am just pointing out where this sort of stuff can go. I just note that partial differential equation models help us analyse fields.])

    That surge and effect curve is powerful in explaining impact of a crisis or a surge behaviour, e.g. the wave of adoptions in marketing that explains the creation of a market through pioneers, early adopters, the majority, and laggards, and of course we also have refusers. This sort of logistic curve is the same that explains epidemics, movements, growth surges, etc. (E.g: THE POLARISATION GAME BEING PLAYED BY EVER SO MANY OBJECTORS TO DESIGN THEORY IS INTENDED TO CONSTRICT ADOPTION.)

    The question of slopes of curves as limits of chords as they approach tangents, is a good way to grasp the limit concept for a slope, and similarly to make successively finer stepwise reconstructions of the area under a curve shows how integration is a limit sum.

    And so forth.

    KF

  31. F/N 2: The discussions of theology of Math here and here may be of significant interest, noting the second person is a double PhD, on both areas. So may be this on phil of math. And, yes, Virginia, there are such things as theology and philosophy of math. Where also, the Logos Theology as discussed here [cf. p 12 on] — and as adverted to in the OP (I note Gregory’s absence when all of this comes up . . . ) — may have something to say to us that we need to hear. KF

  32. F/N: Polythress highlights:

    It may surprise the reader to learn that not everyone agrees that ’2 + 2 = 4′ is true. But, on second thought, it must be apparent that no radical monist can remain satisfied with ’2 + 2 = 4.’ If with Parmenides2 one thinks that all is one, if with Vedantic Hinduism3 he thinks that all plurality is illusion, ’2 + 2 = 4′ is an illusory statement. On the most ultimate level of being, 1 + 1 = 1.4

    What does this imply? Even the simplest arithmetical truths can be sustained only in a worldview which acknowledges an ultimate metaphysical plurality in the world—whether Trinitarian, polytheistic, or chance-produced plurality. At the same time, the simplest arithmetical truths also presuppose ultimate metaphysical unity for the world—at least sufficient unity to guard the continued existence of “sames.” Two apples remain apples while I am counting them; the symbol ’2′ is in some sense the same symbol at different times, standing for the same number.

    So, at the very beginning of arithmetic, we are already plunged into the metaphysical problem of unity and plurality, of the one and the many. As Van Til and Rushdoony have pointed out, this problem finds its solution only in the doctrine of the ontological Trinity.5 For the moment, we shall not dwell on the thorny metaphysical arguments, but note only that without some real unity and plurality, ’2 + 2 = 4′ falls into limbo. The “agreement” over mathematical truth is achieved partly by the process, described elegantly by Thomas Kuhn and Michael Polanyi, of excluding from the scientific community people of differing convictions.6 Radical monists,for example, are not invited to contribute to mathematical symposia . . . .

    Consider the statements

    A: Somewhere in the decimal expansion of pi there occurs a sequence of seven consecutive 7′s.

    B: There are infinitely many primes p such that p + 2 is prime.

    In 1975 [appar., date of composition], no man knows whether either A or B is true. Nor is there any known procedure by which, in a finite amount of time, we could be assured of, obtaining a definite yes-or-no answer. For the intuitionists, this means that A and B should not be considered as either true or false.10 It makes no sense to talk about truth or falsehood so long as we have no way of checking. On the other hand, the Christian, on the basis of I John 3:20 (“God is greater than our hearts, and he knows everything”), Psalm 147:5, and other passages, is likely to feel that at least God knows perfectly definitely whether A or B is true. Our own limitations set no limits to His knowledge (§24) (cf. Isa. 55:8-9; Ps. 139:6, 12, 17-18).

  33. Notice, the above “pot stirring exercise” is an initial exploration of the sort of issues that I think would arise from the question of mathematics as a sign of mind behind the cosmos, and wider concerns linked to philosophy and theology, especially natural theology.

    But who is asking such questions? Only those who have assumed there is some designer, not those who are working in mathematics for its own sake. I too find Euler’s equation satisfying but I would never assume that it had anything to say about a possible mind behind the methods.

    I have never head any mathematician or physicist I know personally make the arguments you assert. But I hear design advocates make such arguments all the time. The existence or non-existance of an infinite number of prime pairs, the truth or non-truth of the Axiom of Choice, Goldbach’s conjecture, etc . . . these things are not proof of design. They are simply questions about the structure(s) of human created mathematics. When Andrew Wiles proved Fermat’s Last Theorem I rather doubt he felt he was finding an indication of a mind behind the math. He didn’t pray for an answer, he worked hard for 7 or 8 years. And after he was done he hadn’t changed or revised our view of mathematics. He had just illuminated one small corner of a vast and complicated landscape.

  34. Jerad:

    Pardon, but that is a bit of a fast one, dismissal by asserting/projecting presumably question-begging assumptions.

    The basic issue is the classical observation on the “unreasonable effectiveness of mathematics” in the sciences.

    That dates to what is it, Wigner in the early 60′s?

    Let’s clip Wigner:

    . . . mathematical concepts turn up in entirely unexpected con-nections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections. Secondly, just because of this circumstance, and because we do not understand the reasons of their usefulness, we cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate. We are in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in suc-cession, always hit on the right key on the ?rst or second trial. He became skeptical concerning the uniqueness of the coordination between keys and doors.

    Most of what will be said on these questions will not be new; it has probably occurred to most scientists in one form or another. My principal aim is to illuminate it from several sides. The ?rst point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories.

    In short, there is a pattern, a powerful one we routinely depend on, it is uncanny and it just does not fit the dominant materialist trend among many scientists.

    Maybe, we need to heed the issue, then? And, ask what is another way of looking at it?

    Long before Wigner, Plato in The Laws, Bk X — c 360 BC — made a cosmological inference on the ordering of and evident purpose in the cosmos.

    The highly mathematical ordering of the cosmos and surprising results like the Euler relationship that out of the blue brings together results from a world of math in one expression, are pointing somewhere.

    You, evidently committed to an ultimately incoherent view of the world, may not want to ask some pretty serious questions about such, but a lot of us do.

    And in my case, I am asking, how do we get to such an ordering that starts ex nihilo and ends with math deeply embedded in reality. That is the root of my exploration.

    Start with a collection that is empty, and create natural numbers using successive extensions. Move from there to the continuum, thence space, thence motion and dynamics, including the onward links into the complex frequency domain.

    Voila, math in a powerfully unifying role in science.

    I find that powerfully stirring and that it makes me think about and understand mathematical results in ways I had not brought together before. Which is well worth reflecting on for its own sake.

    For instance, just the question of getting to the continuum and onward to dimensionality and the notion of a “ball of string” filling of space that shows the continuum cardinality of space, are very useful for thinking about maths in understanding our world. On track record, just the bringing together of various results and perspectives in a fresh way is apt to bring up insights.

    That begs to be thought through, under whatever labels for such study one may want. And it begs to be integrated into an overall worldview.

    And, so much for your “laziness” or ‘science stopping” strawman.

    (Just look above at the track record in this thread. Who is actually thinking about math and its implications, and who is content to snipe and dismiss? Do you not see that that pattern tells the astute onlooker a lot?)

    Where also, given the very brute “thereness” of math AND its sheer mentality, you are not going to be able to play rhetorical games with the substantial matters. Namely, we undeniably have a powerful unifying phenomenon, where mathematics and mathematical results are a powerful part of the unification of the world. But math is allegedly a mental exercise, an exploration of a world that — so say ever so many — has no causal powers. Figments, that somehow bang, shine the x-ray spectacles on the phenomena.

    Or, is it?

    The almost resentfully jaundiced reaction above suggests that that is a serious question, worth thinking through — but obviously uncomfortable to some.

    Let’s put the lurking question — and that is a question not a question-begging assumption — Could the mathematical order of the cosmos be a powerful clue pointing to a mind as its source?

    Why, or why not?

    What superior explanation do you have on tap and why do you deem it superior?

    KF

  35. PS: I find your demands for “proof of design” and evident assumption that in absence of such, an inference thereto is to be dismissed, fails some basic tests. No physical result and no fact based result can meet the demand for deductive proof on universally acceptable axioms. None. So, to demand such implies either a surrender of the whole of inductive reasoning and science, or else it is an exercise in selective hyperskepticism.

  36. Mung:

    This, from IEP is a good bit of further pot stirring:

    The applicability of mathematics can lie anywhere on a spectrum from the completely trivial to the utterly mysterious. At the one extreme, mathematics is used outside of mathematics in cases which range from everyday calculations like the attempt to balance one’s checkbook through the most demanding abstract modeling of subatomic particles. The techniques underlying these applications are perfectly clear to those who have mastered them, and there seems to be little for the philosopher to say about such cases. At the other extreme, scientists and philosophers have often mentioned the remarkable power that mathematics provides to the scientist, especially in the formulation of new scientific theories. Most famously, Wigner claimed that “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.” And according to Kant, “In any special doctrine of nature there can be only as much proper science as there is mathematics therein.” Many agree that the problem of understanding the significant tie between mathematics and modern science is an interesting and significant challenge for the philosopher of mathematics . . . .

    Eugene Wigner (1902-1995) was a ground-breaking physicist who also engaged in some important philosophical reflections on the role of mathematics in physics. In his paper “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” in (Wigner 1960), he emphasizes “unreasonable effectiveness,” but it is not always clear what aspects of applicability he is concerned with. In a crucial stage of his discussion he distinguishes the role of mathematics in reasoning of the sort discussed above from the use of mathematics to formulate successful scientific theories: “The laws of nature must already be formulated in the language of mathematics to be an object for the use of applied mathematics” (Wigner 1960, p. 6). This procedure is surprisingly successful for Wigner because the resulting laws are incredibly accurate and the development of mathematics is largely independent of the demands of science. As he describes it, “Most advanced mathematical concepts … were so devised that they are apt subjects on which the mathematician can demonstrate his ingenuity and sense of formal beauty” (Wigner 1960, p. 3). When these abstract mathematical concepts are used in the formulation of a scientific law, then, there is the hope that there is some kind of match between the mathematician’s aesthetic sense and the workings of the physical world. One example where this hope was vindicated is in the discovery of what Wigner calls “elementary quantum mechanics” (Wigner 1960, p. 9). Some of the laws of this theory were formulated after some physicists “proposed to replace by matrices the position and momentum variables of the equations of classical mechanics” (Wigner 1960, p. 9). This innovation proved very successful, even for physical applications beyond those that inspired the original mathematical reformulation. Wigner mentions “the calculation of the lowest energy level of helium … [which] agree with the experimental data within the accuracy of the observations, which is one part in ten millions” and concludes that “Surely in this case we ‘got something out’ of the equations that we did not put in” (Wigner 1960, p. 9) . . .

  37. 38
    Kantian Naturalist

    Adorno, “Introduction” to The Positivism Dispute in German Sociology

    Carnap, one of the most radical positivists, once characterized as a stroke of good luck the fact that the laws of logic and of pure mathematics apply to reality. A mode of thought, whose entire pathos lies in its enlightened state, refers at this central point to an irrational — mythical — concept, such as that of the stroke of luck, simply in order to avoid an insight which, in fact, shakes the positivistic position; namely, that the supposed lucky circumstance is not really one at all but rather the product of the ideal of objectivity based on the domination of nature or, as Habermas puts it, the ‘pragmatistic’ ideal of objectivity. The rationality of reality, registered with relief by Carnap, is simply the mirroring of subjective ratio. The epistemological metacritique denies the validity of the Kantian claim to the subjective a priori but affirms Kant’s view to the extent that his epistemology, intent on establishing validity, describes the genesis of scientistic reason in a highly adequate manner. What to him, as a remarkable consequence of scientistic reification, seems to be the strength of subjective form which constitutes reality is, in truth, the summa of the historical process in which subjectivity —- liberating itself from nature and thus objectivating itself -— emerged as the total master of nature, forgot the relationship of domination and, thus blinded, re?interpreted this relationship as the creation of that ruled by the ruler. Genesis and validity must certainly be critically distinguished in the individual cognitive acts and disciplines. But in the realm of so?called constitutional problems they are inseparably united, no matter how much this may be repugnant to discursive logic. Since scientistic truth desires to be the whole truth it is not the whole truth. It is governed by the same ratio which would never have been formed other than through science. It is capable of criticism of its own concept and in sociology can characterize in concrete terms what escapes science —- society.

  38. KN:

    An interesting comment, though one I think flawed in interesting ways.

    To take just one pattern, within its domain of applicability — above molecules and lower than about 10% of the speed of light, Newtonian Dynamics is demonstrably, shockingly accurate. Accurate in ways that are often counter-intuitive and astonishing. (Think about the analysis on how tilting a bicycle will cause it to turn, through inertia-torque interactions in light of conservation laws, or the like.)

    Similarly, physical optics has a track record of astonishing precision, not just accident of subjective judgements and agreements. Quantum results, where the quantum is a required analysis, are credibly the best attested current theory.

    All of these are highly quantitative, to the point where routinely, mathematical analysis is used in prediction, observations and theory building, with the implicit reliance on its apt and exact description. One striking result from 200 years ago, was the inference from Young’s double-slit interference experiment that if the wave view were true, a small dot of light should lie in the centre of the shadow of a small sphere shone upon by a point source. This was seen as a demonstration that Young was wrong. Until, someone did the exercise, and behold, there was the little dot.

    Wigner is right to highlight the effectiveness of Mathematics in physical sciences — notice, I have little or no interest here in the notions and issues involved in sociology or anthropology etc — as an astonishing feature of our world.

    That is what needs to be explored and examined on the principles of inference to best explanation.

    I repeat, the notion that there is an ugly, unbridgeable gulch between the subjective world of the phenomenon and the noumenal world of things in themselves, is self referential and self stultifying. Yes, we have reason to believe that our perceptions filter our observations, and yes, to err is human. But, to go on from that to the notion of knowing of such an unbridgeable gulch, is to imply a claim to know what is being denied: the objective and knowable reality of the external world.

    Far better is to accept that we may and do err, so that our empirically grounded knowledge claims of that world are provisional, though in many cases — for good and sufficient reason — they are morally certain.

    As for first principles of right reason, there is good reason to understand that they are so, on pain of self-referential absurdity. The very attempt to deny, scant and dismiss such implies their reality; starting with the assumed stability of identity and the principle that what we say we mean, not also the opposite, etc.

    However, all of this is tangential.

    What we are really dealing with here, is foundational mathematical concepts that underpin reality. If we can identify and count distinct things, and see that two collections may or may not have the same number, per one-to-one matching or the like, we accept the reality of natural numbers. What I have done is to use revised Zermelo Frankel Set Theory — an adjustment post the discovery of antinomies in naive set theory — to construct the set of naturals starting with the empty set.

    In exploration, I then went on to exploring the way we can move from such to the reals, the continuum. Where the “ball of string filling space” principle [via the Peano space filling curve] then allows us to extend a 1-d continuum to 3-de space, by way of the power of complex numbers and vectors as mathematical structures.

    Motion is introduced as a primitive, in effect a marked point that takes up successive positions in time. Add in the matter of inertia and dynamics, and we are now in familiar territory.

    That is, I am pointing to a unification and integration of mathematics into the core of reality, explaining why the physical world is so intensely mathematical in its behaviour. However one wishes to explain this, that integration is not a matter of dubious assumption or guess work. It is a massively evident objective fact, to the point that its denial would raise questions of rhetorical game playing and points scoring, or else profound ignorance and/or disturbance.

    But at the same time, mathematics itself is NOT an empirical discipline, it is a conceptual-logical one. Where, its constructs have no causal power in themselves, they are utterly and patently inert. But, these constructs are deeply embedded in the cause-effect bonds and patterns of reality as we may easily experience or observe.

    If anything ever cries out for a good and adequate explanation on a comparative difficulties basis, this does.

    Notice, I am alluding here to the fundamental philosophical method of comparative difficulties. This is philosophy, not science. But that only means, we are at foundations and hard questions, hard because there are no easy answers.

    I contend that the best explanation of mental concepts being embedded in the heart of reality, is that reality is the product of supremely logical and mathematical mind. Mind, that used mathematics in designing and building our world. So that in mathematics and physics especially, as we explore and discover pivotal concepts and patterns that reflect quantity, logic and structure, we are finding the complex patterns that were built into reality and which speak of their roots in such a design.

    The founding era modern scientists put it in a much simpler way: we are thinking God’s [creative and sustaining] thoughts after him. (Hence, the legitimate role of natural theology and exploration of its links to the rise of science above in the OP etc. Not on a “proof” basis, but on inference to best explanation.)

    Doubtless, others will have a different view, so let us compare and see what is best.

    I do note that Wigner clearly captures what seems to be a dilemma for today’s common a priori materialism: the “unreasonable effectiveness” of mathematics, seems to reflect problems grounding such on this view. A view that is ever so dominant or even rampant nowadays, in ever so many “scientific” circles – to the point that many think adherence to such is a criterion of being rational. (When the very opposite actually holds.)

    KF

  39. per kf’s request:

    It is interesting to note that ‘higher dimensional’ mathematics had to be developed before Einstein could elucidate General Relativity, or even before Quantum Mechanics could be elucidated;

    The Mathematics Of Higher Dimensionality – Gauss and Riemann – video
    http://www.metacafe.com/watch/6199520/

    The Unreasonable Effectiveness of Mathematics in the Natural Sciences – Eugene Wigner – 1960
    Excerpt: We now have, in physics, two theories of great power and interest: the theory of quantum phenomena and the theory of relativity.,,, The two theories operate with different mathematical concepts: the four dimensional Riemann space and the infinite dimensional Hilbert space,
    http://www.dartmouth.edu/~matc.....igner.html

    One peculiar thing about the higher dimensional 4-D space time of General Relativity is that it ‘expands equally in all places’:

    Where is the centre of the universe?:
    Excerpt: There is no centre of the universe! According to the standard theories of cosmology, the universe started with a “Big Bang” about 14 thousand million years ago and has been expanding ever since. Yet there is no centre to the expansion; it is the same everywhere. The Big Bang should not be visualized as an ordinary explosion. The universe is not expanding out from a centre into space; rather, the whole universe is expanding and it is doing so equally at all places, as far as we can tell.

    Thus from a 3-dimensional (3D) perspective, any particular 3D spot in the universe is to be considered just as ‘center of the universe’ as any other particular spot in the universe is to be considered ‘center of the universe’. This centrality found for any 3D place in the universe is because the universe is a 4D expanding hypersphere, analogous in 3D to the surface of an expanding balloon. All points on the surface are moving away from each other, and every point is central, if that’s where you live.

    Centrality of Earth Within The 4-Dimensional Space-Time of General Relativity – video
    http://www.metacafe.com/w/8421879

    And higher (infinite) dimensional quantum mechanics is also very mysterious in that consciousness is found to be the ‘ultimate universal reality’:

    “It will remain remarkable, in whatever way our future concepts may develop, that the very study of the external world led to the scientific conclusion that the content of the consciousness is the ultimate universal reality” -
    Eugene Wigner – (Remarks on the Mind-Body Question, Eugene Wigner, in Wheeler and Zurek, p.169) 1961 – received Nobel Prize in 1963 for ‘Quantum Symmetries’

    Four intersecting lines of experimental evidence from quantum mechanics that shows that consciousness precedes the quantum wave collapse of material reality (Wigner’s Quantum Symmetries, Wheeler’s Delayed Choice, Leggett’s Inequalities, Quantum Zeno effect):
    https://docs.google.com/document/d/1G_Fi50ljF5w_XyJHfmSIZsOcPFhgoAZ3PRc_ktY8cFo/edit

    Of related note; there is also a mysterious ‘higher dimensional’ component to life:

    The predominance of quarter-power (4-D) scaling in biology
    Excerpt: Many fundamental characteristics of organisms scale
    with body size as power laws of the form:

    Y = Yo M^b,

    where Y is some characteristic such as metabolic rate, stride length or life span, Yo is a normalization constant, M is body mass and b is the allometric scaling exponent.
    A longstanding puzzle in biology is why the exponent b is usually some simple multiple of 1/4 (4-Dimensional scaling) rather than a multiple of 1/3, as would be expected from Euclidean (3-Dimensional) scaling.

    “Although living things occupy a three-dimensional space, their internal physiology and anatomy operate as if they were four-dimensional. Quarter-power scaling laws are perhaps as universal and as uniquely biological as the biochemical pathways of metabolism, the structure and function of the genetic code and the process of natural selection.,,, The conclusion here is inescapable, that the driving force for these invariant scaling laws cannot have been natural selection.”
    Jerry Fodor and Massimo Piatelli-Palmarini, What Darwin Got Wrong (London: Profile Books, 2010), p. 78-79

    Of related interest is that mathematics was shown to be incomplete by Godel:

    Kurt Gödel – Incompleteness Theorem – video
    http://www.metacafe.com/w/8462821

    i.e. the ‘truth’ of a mathematical equation is not within the mathematical equation itself but the ‘truthfulness’ of the equation must be imparted to it from God:

    BRUCE GORDON: Hawking’s irrational arguments – October 2010
    Excerpt: Rather, the transcendent reality on which our universe depends must be something that can exhibit agency – a mind that can choose among the infinite variety of mathematical descriptions and bring into existence a reality that corresponds to a consistent subset of them. This is what “breathes fire into the equations and makes a universe for them to describe.” Anything else invokes random miracles as an explanatory principle and spells the end of scientific rationality.,,,
    Universes do not “spontaneously create” on the basis of abstract mathematical descriptions, nor does the fantasy of a limitless multiverse trump the explanatory power of transcendent intelligent design. What Mr. Hawking’s contrary assertions show is that mathematical savants can sometimes be metaphysical simpletons. Caveat emptor.
    http://www.washingtontimes.com.....arguments/

    Moreover, Godel, who was perhaps Einstein’s closest confidant at Princeton, also had this to say

    The God of the Mathematicians – Goldman
    Excerpt: As Gödel told Hao Wang, “Einstein’s religion [was] more abstract, like Spinoza and Indian philosophy. Spinoza’s god is less than a person; mine is more than a person; because God can play the role of a person.” – Kurt Gödel – (Gödel is considered one of the greatest logicians who ever existed)
    http://www.firstthings.com/art.....ematicians

    And when one allows God into math to make it ‘complete’ then one finds a very credible reconciliation between General Relativity and Quantum Mechanics into the infamous ‘theory of everything’:

    General Relativity, Quantum Mechanics, Entropy, and The Shroud Of Turin – updated video
    http://vimeo.com/34084462

    Romans 11:36
    For from Him and through Him and to Him are all things. To Him be the glory forever. Amen.

    Natalie Grant – Alive (Resurrection music video)
    http://www.godtube.com/watch/?v=KPYWPGNX

  40. BA77, thanks. I think this is worth pondering. I find it interesting who and who, having demanded addressing of natural theology issues are absent. As of now, I think Euler’s eqn is a part of a wider pattern that I am thinking of calling the Ex Nihilo principle. Imagine, starting from a set that collects nothing, thus creating a conceptual entity, and next getting a whole number sequence, then going continuum, dropping in the i* operator to get the plane and the ijk to get the 3-d space, and time added, then kinematics and dynamics, then FIAT LUX. All, delivering a case where we see the unreasonable effectiveness of math as being rooted in God’s creative and sustaining thoughts that — per the founders of modern science — we think after him. KF

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