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April 30, 2009

Taming the Boundless

Posted by David Corfield

In his article – The Invisible Link Between Mathematics and Theology, in Perspectives on Science and Christian Faith, Vol. 56, pp. 111-116 – Ladislav Kvasz argues for the thesis that

…monotheistic theology with its idea of the omniscient and omnipotent God, who created the world, influenced in an indirect way the process of this mathematicization. In separating ontology from epistemology, monotheistic theology opened the possibility to explain all the ambiguity connected to these phenomena as a result of human finitude and so to understand the phenomena themselves as unambiguous, and therefore accessible to mathematical description.

This thesis is explored through five notions: infinity, chance, the unknown, space and motion. Taking the first of these, we read

What we refer to today as infinite was in Antiquity subsumed under the notion of apeiron (απϵiρoν\alpha \pi \epsilon i \rho o \nu). Nevertheless, compared with our modern notion of infinity, the notion of apeiron had a much broader meaning. It applied not only to that which was infinite, but also to everything that had no boundary (i.e. no peras), that was indefinite, vague or blurred. According to ancient scholars apeiron was something lacking boundaries, lacking determination, and therefore uncertain. Mathematical study of apeiron was impossible, mathematics being the science of the determined, definite and certain knowledge. That which had no peras, could not be studied using the clear and sharp notions of mathematics.

Modern mathematics, in contrast to Antiquity, makes a distinction between infinite and indefinite. We consider the infinite, despite the fact that it has no end (finis), to be determined and unequivocal, and thus accessible to mathematical investigation. Be it an infinitely extended geometrical figure, an infinitely small quantity or an infinite set, we consider them as belonging to mathematics. The ancient notion of apeiron was thus divided into two notions: the notion of the infinite in a narrow sense, which became a part of mathematics, and the notion of the indefinite, which, as previously, has no place in mathematics.

So,

While for the Ancients apeiron was a negative notion, associated with going astray and losing the way, for the medieval scholar the road to infinity became the road to God. God is an infinite being, but despite His infiniteness, He is absolutely perfect. As soon as the notion of infinity was applied to God, it lost its obscurity and ambiguity. Theology made the notion of infinity positive, luminous and unequivocal. All ambiguity and obscurity encountered in the notion of infinity was interpreted as the consequence of human finitude and imperfection alone. Infinity itself was interpreted as an absolutely clear and sharp notion, and therefore an ideal subject of mathematical investigation.

Evidence for the change from the Ancients is provided by Kvasz in his book Patterns of Change where he quotes Nicholas of Cusa on page 77:

It is already evident that there can be only one maximum and infinite thing. Moreover, since any two sides of any triangle cannot, if conjoined, be shorter than the third: it is evident that in the case of a triangle whose one side is infinite, the other two sides are not shorter. And because each part of what is infinite is infinite: for any triangle whose one side is infinite, the other sides must also be infinite. And since there cannot be more than one infinite thing, you understand transcendently that an infinite triangle cannot be composed of a plurality of lines, even though it is the greatest and truest triangle, incomposite and most simple… (De Docta Ignorantia, 1440, trans. J. Hopkins).

It may seem odd to us that Nicholas could not imagine the limit as an isosceles triangle of fixed base is extended, but the point is that such a discussion of an infinitely large object would have been unthinkable for the Greeks.

The fundamental differences between early modern mathematics and the mathematics of the Hellenistic period can be perhaps characterized as breaking of the boundaries of the unequivocally given and opening of the world of mathematics to the ambiguously given phenomena such as infinity, randomness, or motion. This is a fundamental change, perhaps the most important one since the discovery of proof and of the idea of an axiomatic system. And this fundamental change, this radical break towards modernity, is most likely linked with monotheistic theology.

We may wonder how infinity, chance, the unknown, space and motion have been treated mathematically in other cultures. The Wikipedia article on infinity has the Jains as the first to distinguish between different infinite sizes.

Posted at April 30, 2009 8:56 AM UTC

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Re: Taming the Boundless

Thanks for this extremely illuminating post.

I recall reading Stanley L. Jaki’s book The Road of Science and the Ways to God on the role of Christianity in the formation of modern empirical science. If I recall correctly, Jaki held that “law” in the sense of universal invariant was tied to monotheistic notions of the transcendental universal law given by God.

Joseph Needham judged this Judaeo-Christian-Islamic understanding of divine law to be one reason that the Chinese, in spite of their more highly developed culture in other respects, did not develop empirical science - they could not view nature as imbued with universal invariants because they hated the memory of their first emperor, a tyrant who propounded such laws with the sword.

Regards,
Mike Gogins

Posted by: Michael Gogins on April 30, 2009 1:35 PM | Permalink | Reply to this

Re: Taming the Boundless

I’ve also heard it claimed, in various places, that (a) China at its cultural height was more thoroughly bureaucratized than Europe, reducing the impetus for technological change; (b) China was far more unified than the quarrelsome states of the European continent, preventing the occurrence of localities more favourable than average to science (i.e., no Chinese Holland); (c) other than the voyages of Zheng He, China was isolationist while Europe was colonial and expansionist, again knocking away an impetus for discovery; (d) printing with movable type took longer to develop in China, partly because of the language’s large symbol set, making the dissemination of knowledge more difficult. I doubt that these possibilities are mutually exclusive.

Would that history provided us with controlled experiments!

Posted by: Blake Stacey on April 30, 2009 2:12 PM | Permalink | Reply to this

Re: Taming the Boundless

A further reason I have heard given is that glass making techniques in Northern Europe, stimulated by the demand for windows in the cold, dark winters there, were sufficiently sophisticated to allow for high quality lenses, vital for astronomy and biology. One shouldn’t underestimate the impact of Micrographia on Western science.

Ah yes, Glass: A World History. Of course we Brits can cope with trickier titles – The Glass Bathyscaphe: How Glass Changed the World.

Posted by: David Corfield on April 30, 2009 3:17 PM | Permalink | Reply to this

Re: Taming the Boundless

These kinds of connections remind me of that highly entertaining TV series, appropriately called Connections. Unfortunately I think it had the opposite of the intended effect on me - it convinced me that most such claims of causality in history are little more than propaganda and the main thing to do with such claims is to look for whose agenda is being furthered by them.

Posted by: Dan Piponi on April 30, 2009 6:08 PM | Permalink | Reply to this

Re: Taming the Boundless

If you haven’t seen it, The day the world took off (downloads at link for those interested) is an interesting 6-part TV series that attempted to come up with some sort of analysis for why developments happened in some areas of the world and didn’t in others. Of course there’s always the risk of becoming just-so story-like. It was years ago I saw it, so I only remember the contention being put forward that because pretty much until pretty much the 1900s Chinese landowners paid labourers for acheiving tasks (eg, plant this field) rather than the late European model of paying for time spent working (eg, I’m paying you for 8 hours work today, and if you haven’t done the expected amount of work in that time you’re fired) they never developed clockwork for precise timekeeping, which stopped other engineering advances dependent on skills from that intricate engineering. But I can’t remember if there was any reason given why the payment model differed.

Posted by: bane on May 1, 2009 12:53 PM | Permalink | Reply to this

Re: Taming the Boundless

Thanks. The Macfarlane responsible for this series is the same as the one who wrote on glass.

Most surprising from episode 1: Due to its antiseptic qualities, because the British drank lots of tea, they could congregate in large towns without infections wiping them out. So could the Japanese, but there the wheel was barely even used in 1830.

And that dampness of the Lancashire and Yorkshire valleys was not only useful for water power, but also to prevent thread from breaking.

Posted by: David Corfield on May 1, 2009 4:32 PM | Permalink | Reply to this

Re: Taming the Boundless

Su-Sung’s clock was the best in its day (the late eleventh century). His technology just didn’t get used on a wide scale — what matters isn’t just whether somebody can think up an idea or build a single machine, but how the idea can spread and how the machine can be employed. To use a horribly hypermodern term, the idea has to “go viral”, which greatly multiplies the number of possible factors involved in its eventual success or failure.

Posted by: Blake Stacey on May 1, 2009 10:44 PM | Permalink | Reply to this

Re: Taming the Boundless

I saw the program on broadcast in 2000 so I’m probably misremembering big chunks.

I think part of the argument was that it’s not necessarily availability of the technological object itself but having a widespread base of the skills and knowledge involved in creating it, so that there’s a wide pool of knolwedgable workers around, some of whom will make the advance. (IIRC there was another argument that having a large pool of canon makers was helpful Britain for engine development because a canon is essentialy a “single-shot” piston :-) and the working-knowledge of what to reinforce, where and how was important in addition to the high level concept of “a steam driven piston”.) So it’s not just the having a widespread deployment of the end-product but a widespread manufacturing base (according to the show’s argument). [I sometimes wonder what the fact that there are only five-ish computer chip manufacturing companies in the world means for the variety of new ideas being tried.]

Of course, I should probably rewatch the program to see precisely what was argued.

Posted by: bane on May 2, 2009 9:22 PM | Permalink | Reply to this

Re: Taming the Boundless

Although it’s not precisely in the same vein, no discussion of this sort is complete without bringing up Jared Diamond’s excellent book Guns, Germs, and Steel, which traces Eurasian dominance of the world back to issues of geography and biology. The Europe/China difference is discussed much more briefly, in an appendix, with a number of geographical factors proposed.

Posted by: Mark Meckes on May 1, 2009 2:32 PM | Permalink | Reply to this

Re: Taming the Boundless

Connections, in the episode “Distant Voices”, spends a little time on the “why didn’t the Chinese conquer the world with science?” question. Burke proposes both the “transcendental universal law” idea and the “bureaucracy” one. I bet he would have added the glassmaking one if he’d heard about it.

Oh, and the first episode of Connections predicted 9/11. Really.

Posted by: Blake Stacey on April 30, 2009 7:01 PM | Permalink | Reply to this

Re: Taming the Boundless

Shouldn’t we? I mean, ignoring Hooke is a tradition unto itself! (-;

Posted by: Blake Stacey on April 30, 2009 3:40 PM | Permalink | Reply to this

Theomathematics; Re: Taming the Boundless

In earlier threads I have several times deemed such arguments: “Theomathematics.” Googling Theomathematics and Theophysics will gind many of my ideas on the subjects.

See also:

An interesting temporalization of Gödel’s ontological proof, by Gavriel Segre.

Posted by: Jonathan Vos Post on April 30, 2009 2:20 PM | Permalink | Reply to this

Re: Taming the Boundless

This is the kind of mindless Eurocentric speculation that gives philosophy and theology a bad name. Let us just take one of the main assertions:

“The fundamental differences between early modern mathematics and the mathematics of the Hellenistic period can be perhaps characterized as breaking of the boundaries of the unequivocally given and opening of the world of mathematics to the ambiguously given phenomena such as infinity, randomness, or motion. This is a fundamental change, perhaps the
most important one since the discovery of proof and of the idea of an axiomatic system. And this fundamental change, this radical break towards modernity, is most likely linked with monotheistic theology.”

What about zero? Absence, nothingness and nonbeing are as ambiguous as infinity. But Indian mathematicians - from a culture as non-monotheistic as possible - figured zero out before anybody else.

I mean, c’mon, the guy cannot have it both ways. You cannot celebrate the modern West as the only culture that broke away from the shackles of religious dogma and then simultaneously claim that very dogma as the reason for successful modernization.

Here is another, equally speculative reason why the Greeks did not concern themselves with ambiguous entities. It is because the derived most of their intuitions about mathematics from geometry, while the kind of abstractions that lead to mathematical conceptions of zero and infinity are more easily motivated by studies of language since human languages quite often make nonbeing and absence explicit rather than ambiguous. Hence classical Indian mathematicians, who worked in a culture where the study of language was highly advanced were able to think of zero.

My “language vs geometry as the origins of mathematical ideas” thesis is as speculative as Kvasz’s claim. And equally irrelevant to genuinely interesting questions in the philosophy of mathematics. These attempts by Kvasz and others to valorize the West/Christianity/Secularism/Modernity etc etc are both false and repugnant.

Posted by: Rajesh Kasturirangan on April 30, 2009 9:30 PM | Permalink | Reply to this

Re: Taming the Boundless

Or, alternatively:

In the book later known as the Almagest, Claudius Ptolemy used a zero symbol as a placeholder in sexagesimal notation. This innovation did not, however, gain acceptance among Hellenistic mathematicians. Perhaps Indian mathematicians, having readier access to paper, were more driven to record steps in abacus calculations on paper, or to work directly on paper itself, and were therefore more eager to employ a symbol to stand for an untouched abacus row.

‘Twill come to pass
That nowhere can a world’s-end be, and that
The chance for further flight prolongs forever
The flight itself.

— Lucretius, De Rerum Natura, Book I

Posted by: Blake Stacey on April 30, 2009 11:08 PM | Permalink | Reply to this

Re: Taming the Boundless

See also:

Recently, however, evidence has come to light which suggests that not all ancient Greek mathematicians felt constrained to deal only with the potentially infinite. The authors of [32] have noticed a remarkable way that Archimedes investigates infinite numbers of objects in The Method in the Archimedes palimpsest:

… Archimedes takes three pairs of magnitudes infinite in number and asserts that they are, respectively, “equal in number”. … We suspect there may be no other known places in Greek mathematics - or, indeed, in ancient Greek writing - where objects infinite in number are said to be “equal in magnitude”. …

The very suggestion that certain objects, infinite in number, are “equal in magnitude” to others implies that not all such objects, infinite in number, are so equal. … We have here infinitely many objects - having definite, and different magnitudes (i.e. they nearly have number); such magnitudes are manipulated in a concrete way, apparently by something rather like a one-one correspondence. … … in this case Archimedes discusses actual infinities almost as if they possessed numbers in the usual sense …

At the moment, I don’t have full access to Netz, Saito and Tchernetska (2001), but it’s probably worth retrieving later.

Posted by: Blake Stacey on April 30, 2009 11:15 PM | Permalink | Reply to this

Re: Taming the Boundless

Russo (2004) argues that the Greek word apeiron had “a long and complicated history” and “was eventually used in mathematics in its current meaning of ‘infinite’ (by Apollonius of Perga, for example).” Quoting a footnote:

See, for instance, Conica, II, proposition 44, where Apollonius, after showing how to construct a diameter of a conic, concludes: “In this way we will find infinitely many diameters” (απϵιρνς\alpha\pi\epsilon\iota\rho\omicron\nu\varsigma διαμϵτρνς\delta\iota\alpha\mu\epsilon\tau\rho\omicron\nu\varsigma). The use of απϵιρν\alpha\pi\epsilon\iota\rho\omicron\nu in the sense of actual infinity in a mathematical context appears already in Plato’s Theaetetus, 147d. Theaetetus reports a conversation between the mathematician Theodorus and his students (of which he was one), dealing with squares that are multiples of the unit square but whose sides are not multiples of the unit length (and therefore are incommensurable with it). They remark that such sides are infinite in number (απϵιρι\alpha\pi\epsilon\iota\rho\omicron\iota τ\tau\omicron πληθς\pi\lambda\eta\theta\omicron\varsigma).

Having but little Latin and less Greek, I’m not particularly fit to judge whatever subtleties may be at work here.

Posted by: Blake Stacey on April 30, 2009 11:54 PM | Permalink | Reply to this

Re: Taming the Boundless

“But Indian mathematicians - from a culture as non-monotheistic as possible…”

That gives an occasion to distribute the link to an interesting article on specifics of “indian thinking”.

Posted by: Thomas on May 1, 2009 7:56 AM | Permalink | Reply to this

Re: Taming the Boundless

Such vehemence! To cut Kvasz a little slack, he didn’t say the only way a culture could begin to mathematicise concepts such as the boundless was via monotheism. I read him as giving an account of how a form of monotheism allowed those who took Greek thought as their starting point to break free from certain restrictions.

An issue to raise to this reduced claim would be to point out that the Greeks already had a theological metaphysics with a single supreme being. E.g., Aristotle in part 7 of Book XII of the Metaphysics:

If, then, God is always in that good state in which we sometimes are, this compels our wonder; and if in a better this compels it yet more. And God is in a better state. And life also belongs to God; for the actuality of thought is life, and God is that actuality; and God’s self-dependent actuality is life most good and eternal. We say therefore that God is a living being, eternal, most good, so that life and duration continuous and eternal belong to God; for this is God.

So next question, is there something in the marriage of Christian theology to Greek metaphysics which allows ways of treating the infinite unavailable to the Greeks? A lifetime might be spent answering it.

That the courses of theology and the study of the infinite have interacted importantly over the centuries is indisputable. For some late nineteenth century interaction take a look at Joseph Dauben’s work.

Posted by: David Corfield on May 1, 2009 10:51 AM | Permalink | Reply to this

Re: Taming the Boundless

There is also the ‘unmoved mover’ of Physics book 8, whose powers are infinite. There is a good case to be made that infinity as ‘inexhaustibility’ is a natural outgrowth of ordinary human cognitive capacities, a la Lakoff and Nunez. Still doesnt explain Cantors imaginative leap though.

Posted by: Rajesh Kasturirangan on May 1, 2009 12:06 PM | Permalink | Reply to this

Re: Taming the Boundless

I must agree with Rajesh here. Thinking that Abrahamic theologies provided some kind of advantage in math or science for being monotheist is somehow shauvinistic.

What I believe it is by the end of the Ice Age, humans were free to roam and test new technologies. As usual, the advances were many times scattered and random, sometimes there were coincidences, sometimes there weren’t, but overall, the process was cumulative. So, the place, or culture where it would show up first was random among cultures near a certain technological threshold. It could happen in China, India,Middle East or anywhere in Europe, during the middle ages.

Iraq won the lotto, but the method flourished just 4 centuries later in Europe, because the muslim and christian worlds were too busy in wars against each other.Europe emerged in a better shape, with a surplus due to the then recent ocean trading routes, so, more people could get educated. And, of course, science provides a smashing advantage to any culture, so they were able to impose themselves to the rest of the world.

Posted by: Daniel de Franša MTd2 on May 1, 2009 6:58 PM | Permalink | Reply to this

Re: Taming the Boundless

These comments appear to me a bit of an overreaction. I don’t know Kvasz’s work at all other than through the casual perusal of the linked article. Does he really

‘celebrate the modern West as the only culture that broke away from the shackles of religious dogma (*)’?

Nor could I gather from my superficial reading any obvious attempt to

‘valorize the West/Christianity/Secularism /Modernity.’

Isn’t the article in fact trying to contribute a counterpoint to the naive model (*), rather than have it both ways? My impression is that there is non-trivial literature of that general nature floating around nowadays, vaguely interesting, correct or not.

I like to joke sometimes that the notion of a universal object is very monotheistic.

Posted by: Minhyong Kim on May 1, 2009 11:46 AM | Permalink | Reply to this

Re: Taming the Boundless

To clarify my point a bit, one corrective to (*) is to argue against the ‘only.’ But it’s not uninteresting to subject the other portions as well to some critical scrutiny.

Posted by: Minhyong Kim on May 1, 2009 12:05 PM | Permalink | Reply to this

Cantor, Chaitin, Wolfram; Re: Taming the Boundless

Sorry that I can’t give pinpoint citations to this, but I believe that Cantor, late in his life when he was accused of madness, did indeed suggest religious implications of transfinite arithmetic.

For instance, he is said to have suggested that even if there were aleph-null atoms in the world, there were at least aleph-one particles of mind or spirit in a human being, and that God was of an even greater infinity.

But we should not assume that we fully understood Cantor’s intent or philosophy. I had quite an interesting conversation with Gregory Chaitin and Stephen Wolfram at the conference in Boston that I attended last year (chairing the Physics sessions). Chaitin was quite convincing in pointing out things that Leibniz wrote that only now make sense to a modern Mathematician, and still in possible conflict with some deep musings of Feynman.

The key issue being whether we live in a LAWFUL universe. Determined by the complexity of the system of natural laws. Which presupposes that there are a finite number of natural laws, does it not? Feynman was not sure of this. Could there be an infinite number of natural laws, but finitely generated from metalaws?

Posted by: Jonathan Vos Post on May 3, 2009 4:34 PM | Permalink | Reply to this

Re: Cantor, Chaitin, Wolfram; Re: Taming the Boundless

Or if `natural laws’ formed an algebra,
there could easily be infinitely many generated by finitely many or is that what you meant by `meta’ as opposed to `of some higher order’.

Speaking of theology, there is a (?substantially different?) notion of natural law.

What is the evolution of the notion of `law’ from Hammurabi on? To what extent is
the legal notion of `law’ hiding in `natural law’?


Posted by: jim stasheff on May 3, 2009 7:20 PM | Permalink | Reply to this

Law and Truth in different magesteria; Re: Cantor, Chaitin, Wolfram; Re: Taming the Boundless

My inventor/engineer friend Forrest Bishop comments, first: “‘Metalaws’ is vague to the point of useless, imo. From another side, their are no natural ‘laws’ at all that can be reduced to a mathematical statements, for much the same reason a cave painting cannot capture or replace its subject. If there were an infinite number of natural laws, there would also have to be an infinite number of ‘weightings’ of those laws in order to prevent one law from capriciously superseding another. The latter sort of Universe would be an acausal kaleidescope.”

I am quite interested in your suggested formalism that Natural Laws Make an Algebra.

Yes, there is a weird mixture of the idea of Law (Theology), Law (National/international), Law (Mathematics), Law (Scientific, engineering, empirical).

To me, the philosophical problem is drowned out by language problems. The words “law” and “truth” and “proof” are used very differently, semantically and pragmatically, in each such domain.

Law (Theology) presume immanent or transcendental godhead as substrate, and “truth” as Revealed. And hidden assumptions of finite versus infinite for people and God(s) and space and time.

Law (National/international) is whatever Armies have enforced, and courtrooms, and is NOT either axiomatic nor empirical, but based on hierarchy and precedent in a way that I have explained as chaotic attractors in the space of all possible (precedent-based) laws.

Law (Mathematics) is split between Axiomatic (old school) and n-Category Theory (new school), greatly complicated by the explosive growth of computing power, and clouded by incoherent pedagogy.

I omit for lack of time the discussion of Law (Artistic/Aesthetic), and we are not yet able to penetrate Law (Scientific, engineering, empirical) without worrying about the Math and Theology.

Posted by: Jonathan Vos Post on May 3, 2009 7:33 PM | Permalink | Reply to this

Re: Taming the Boundless

The question concerning the number of natural laws can be answered by using the Goedel incompletness theorem: If there were a simple Theory of Everything (TOE)(simple means a finite number of laws,i.e. axioms) then such a theory should contain aritmethics, and therefore one can apply the Goedel theorem. This means that statements would exist which are undecidable but true. In other words, one could find phenomena that cannot be explaind by the TOE laws, and these phenomena can be added to the original list of laws (axioms). This argument against a simple TOE could be found in the works of father Jaki, as well as in a talk by Stephen Hawking. As far as the “weighting” of these laws is concerned, this is an interesting question, since it amounts to a question about relations between statements generated by various finite subsets of the laws.
Metaphysically, this all nicelly fits into a platonic scheme, where all abstract ideas exist, and then naturally the idea of a single God could be contemplated, as a being with maximal properties. In this sense I think that there is some truth in the hypothesis that monotheisam led to an increased interest in infinite, but I beleive that old Greeks gave us a mental framework which was transformed in Europe in modern science and mathematics.

Posted by: Aleksandar Mikovic on May 4, 2009 5:35 PM | Permalink | Reply to this

Re: Taming the Boundless

in re: the hypothesis that monotheism led to an increased interest in infinite

one or more of the classical arguments for the existence of God is the refusal to believe in various infinities, e.g. an infinite regression causes/movers

Posted by: jim stasheff on May 5, 2009 2:47 PM | Permalink | Reply to this

Re: Taming the Boundless

one or more of the classical arguments for the existence of God is the refusal to believe in various infinities, e.g. an infinite regression causes/movers

We even believe this in mathematics; it's the intuition behind the axiom of foundation.

One can prove that a first cause exists if one assumes that direct causality is a well-founded relation whose reflexive-transitive closure (not-necessarily-direct causality) is a co-direction.

(We haven't got some automatic markup for links to the Lab, have we?)

Posted by: Toby Bartels on May 5, 2009 5:35 PM | Permalink | Reply to this

Re: Taming the Boundless

Perhaps the anti-foundation axiom of non-well-founded set theory offers theologists new tools and imagery.

Posted by: David Corfield on May 6, 2009 10:23 AM | Permalink | Reply to this

Feferman vs. Quine and Putnam; Re: Taming the Boundless

This new arXiv paper supports the subthread here that the Axiom of Infinity does not belong to nor is needed by Foundations of Physics (the philosophical arena, not the journal of the same title).

“Feferman has observed that this severely undercuts a famous argument of Quine and Putnam according to which set theoretic platonism is validated by the fact that mathematics is ‘indispensable’ for some successful scientific theories (since in fact ZFC is not needed for the mathematics that is currently used in science)”

“I also make the point that even if ZFC is consistent, there are good reasons to suspect that some number-theoretic assertions provable in ZFC may be false. This suggests that set theory should not be considered central to mathematics.”

Posted by: Jonathan Vos Post on May 12, 2009 11:05 AM | Permalink | Reply to this

Re: Feferman vs. Quine and Putnam; Re: Taming the Boundless

Direct link:

http://arxiv.org/abs/0905.1680

Posted by: Tom on May 12, 2009 12:40 PM | Permalink | Reply to this

Re: Feferman vs. Quine and Putnam; Re: Taming the Boundless

Even directer link.

Posted by: David Corfield on May 12, 2009 1:00 PM | Permalink | Reply to this

Re: Feferman vs. Quine and Putnam

Interesting.

I think that various groups of ZFC-challengers don't talk to one another enough.

Posted by: Toby Bartels on May 13, 2009 1:16 AM | Permalink | Reply to this

Re: Feferman vs. Quine and Putnam

From the Nik Weaver paper (more directly linked by David Corfield – thanks):

I also make the point that even if ZFC is consistent, there are good reasons to suspect that some number-theoretic assertions provable in ZFC may be false. This suggests that set theory should not be considered central to mathematics.

Very odd statement. Section 7, where he argues this point, seems very thin.

Posted by: Todd Trimble on May 13, 2009 4:36 PM | Permalink | Reply to this

Re: Taming the Boundless

Interesting. Here’s Ernst Cassirer in An Essay on Man:

Giordano Bruno was the first thinker to enter upon this path, which in a sense became the path of all modern metaphysics. What is characteristic of the philosophy of Giordano Bruno is that the term “infinity” changes its meaning. In Greek classical thought infinity is a negative concept. The infinite is the boundless or indeterminate. It has no limit and no form, and it is, therefore, inaccessible to human reason, which lives in the realm of form and can understand nothing but forms. In this sense the finite and infinite, περας and απειρον, are declared by Plato in the Philebus to be the two fundamental principles which are necessarily opposed to one another. In Bruno’s doctrine infinity no longer means a mere negation or limitation. On the contrary, it means the immeasurable and inexhaustible abundance of reality and the unrestricted power of the human intellect. It is in this sense that Bruno understands and interprets Copernican doctrine. This doctrine, according to Bruno, was the first and decisive step toward man’s self-liberation. Man no longer lives in the world as a prisoner enclosed within the narrow walls of a finite physical universe. He can traverse the air and break through all the imaginary boundaries of the celestial spheres which have been erected by a false metaphysics and cosmology. The infinite universe sets no limits to human reason; on the contrary, it is the great incentive of human reason. The human intellect becomes aware of its own infinity through measuring its powers by the infinite universe. (p. 15)

So Cassirer appears to attribute the modern willingness to approach the infinite as brought about by natural science. Whether this brings him into disagreement with Kvasz is not immediately obvious though. We would probably have to look up Individuum und Kosmos in der Philosophie der Renaissance.

Posted by: David Corfield on May 12, 2009 1:23 PM | Permalink | Reply to this

Bruno and Infinite Worlds; Re: Taming the Boundless

“Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds.”

“An academician belonging to no academy”, an unfrocked monk, an excommunicated Calvinist, an expelled Lutheran, an avant-garde and marginal philosopher, a forgotten genius of the Renaissance - Giordano Bruno was also an inspired magus.”

The Trials of Giordano Bruno: 1592 and 1600
Selected Links and Bibliography
by Lawrence MacLachlan

I’ve enjoyed many dialogues about Bruno with Astronomy students, fellow scientists, high school science students, and a thousand or so Elderhostel adult students.

My discussions are based on secondary sources, as I do not read Latin or german, for instance. But someone here might see:

The infinite worlds of Giordano Bruno.
by Antoinette Mann Paterson, Springfield, Ill.: Thomas, 1970
[Change Cover Series: American lecture series no. 760. A monograph in the Bannerstone division of American lectures in philosophy
Language: English
Pagination: xi, 227 p.
LCCN: 71088393
Dewey: 195
LC: B783.Z7 P35
Subject: Bruno, Giordano, 1548-1600.
Cosmology — History — 16th century.]


Posted by: Jonathan Vos Post on May 12, 2009 11:17 PM | Permalink | Reply to this

Re: Bruno and Infinite Worlds; Re: Taming the Boundless

As long as we’re on cultural instances of infinity, I’ll comment on how it turns up in my culture. That passage you cite sounds very similar to one in “The Book of Moses”, an apocalyptic, rather-more-detailed account of the conversation Moses had with God that included the creation story in Genesis, revealed to Joseph Smith:

And worlds without number have I created; and I also created them for mine own purpose; and by the Son I created them, which is mine Only Begotten. And the first man of all men have I called Adam, which is many. But only an account of this earth, and the inhabitants thereof, give I unto you. For behold, there are many worlds that have passed away by the word of my power. And there are many that now stand, and innumerable are they unto man; but all things are numbered unto me, for they are mine and I know them.

And it came to pass that Moses spake unto the Lord, saying: Be merciful unto thy servant, O God, and tell me concerning this earth, and the inhabitants thereof, and also the heavens, and then thy servant will be content.

And the Lord God spake unto Moses, saying: The heavens, they are many, and they cannot be numbered unto man; but they are numbered unto me, for they are mine. And as one earth shall pass away, and the heavens thereof even so shall another come; and there is no end to my works, neither to my words. For behold, this is my work and my glory—to bring to pass the immortality and eternal life of man.

The creation story itself is explicitly figurative in the Book of Moses. The book dates from winter, 1830-31, around Smith’s 25th birthday.

The notion of the infinite in Mormon theology departs from classical Christianity in some important ways. First, that God has a physical body and is constrained by physical law; he did not create the universe, but rather organized the matter that was there into a world suitable for life and seeded the planet. He passed through a period of mortality like we did (though not on this world), and church members hope to become like Him in time. There’s a notion of “spirit matter” that is currently undetected but not in principle undetectable; human agency and perception of qualia stem somehow from the interaction of spirit and normal matter. The physics of spirit matter and matter we perceive now are presumably similar, in that this world is “patterned after” the spirit world, and spirits of men look like men.

Another revelation from March 1830 addressed the notion of infinite time, particularly in regards to eternal damnation:

Nevertheless, it is not written that there shall be no end to this torment, but it is written endless torment. Again, it is written eternal damnation; wherefore it is more express than other scriptures, that it might work upon the hearts of the children of men, altogether for my name’s glory… Eternal punishment is God’s punishment. Endless punishment is God’s punishment… For behold, I, God, have suffered these things for all, that they might not suffer if they would repent; but if they would not repent they must suffer even as I.

The implications are that this pain is a natural consequence of mortal life, that it dwarfs any mortal pain, but that there’s a way to avoid the largest part of it; that there’s a calculus of pain and moral laws are, in some sense, physical laws. Also, that when scriptures talk about infinity, it often just means “really really big”. God’s knowledge and power are not necessarily infinite, just comparatively so–rather like second order differences are negligible.

Another consequence of the passage above is that “eternal life” means “God’s life”, i.e. the Mormon vision of heaven is perpetual education: existing for an unimaginably long time, in a peaceful environment, spending our time learning how to become like God. Therefore, there’s a heavy emphasis on becoming educated in Mormon culture. We believe God is continually learning and progressing; at the funeral of Joseph Smith’s friend King Follett, Smith said,

First God Himself who sits enthroned in yonder heavens is a Man like unto one of yourselves–that is the great secret! … The first principle of truth and of the Gospel is to know of a certainty the character of God, and that we may converse with Him… that He once was a man like one of us… You have got to learn how to make yourselves Gods… and be kings and priests to God, the same as all Gods have done by going from a small capacity to a great capacity, from a small degree to another, from grace to grace… from exaltation to exaltation. [Jesus said], “I saw the Father work out His kingdom with fear and trembling and I am doing the same, too. When I get my kingdom, I will give it to the Father and it will add to and exalt His glory. He will take a higher exaltation and I will take His place and also be exalted, so that He obtains kingdom rolling upon Kingdom.” … All the minds and spirits that God ever sent into the world are susceptible of enlargement and improvement. The relationship we have with God places us in a situation to advance in knowledge. God Himself found Himself in the midst of spirits and glory. Because He was greater He saw proper to institute laws whereby the rest, who were less in intelligence, could have a privilege to advance like Himself and be exalted with Him, so that they might have one glory upon another in all that knowledge, power, and glory.

The Mormon theologian Eugene England examines theodicy and the Mormon notion of a God that’s continually learning and progressing in his essay, “The Weeping God of Mormonism”.

Posted by: Mike Stay on May 13, 2009 7:00 PM | Permalink | Reply to this

Re: Bruno and Infinite Worlds; Re: Taming the Boundless

P.S.— David H. Bailey (one of the discoverers of the spigot algorithm for pi and a proponent of experimental mathematics) is a Mormon mathematician at LBNL who has written rather extensively on science and Mormonism; you can find his articles here:

http://www.dhbailey.com/papers/index.html

Posted by: Mike Stay on May 13, 2009 7:07 PM | Permalink | Reply to this

Re: Bruno and Infinite Worlds; Re: Taming the Boundless

It’s probably well known to everyone, but there’s also a reference to non-reachability of 0\aleph_0 by finite addition in John Newton’s hymn Amazing Grace:

When weve been there then thousand years, Bright shining as the sun, Weve no less days to sing Gods praise, Than when weve first begun.

I won’t opine on if the difference between accepting infinite future “time” and disputing infinite regress is consistent.

Posted by: bane on May 14, 2009 10:05 AM | Permalink | Reply to this

Re: Bruno and Infinite Worlds; Re: Taming the Boundless

Great quote! Not to be confused with Sir Isaac Newton, who wrote that the rate of flow of time was one second per second, promised to get back and re-examine that assumption, but never did. Not that time actually flows.

Posted by: Jonathan Vos Post on May 14, 2009 3:08 PM | Permalink | Reply to this

Re: Bruno and Infinite Worlds; Re: Taming the Boundless

[Newton] …who wrote that the rate of flow of time was one second per second

Where did he write that? There is of course Newton’s Scholium on Space and Time, where he argues for a background (absolute) space and time. But the formulation “rate of flow of time is one second per second” sounds, on the face of it, like a parody.

Posted by: Todd Trimble on May 14, 2009 6:11 PM | Permalink | Reply to this

No joke: dt/dt = 1; Re: Bruno and Infinite Worlds; Re: Taming the Boundless

Not a parody. Nor would I naively ascribe to Newton something in Leibniz notation:
dt/dt = 1.

But see:
Time, quantum mechanics, and decoherence
Journal Synthese
Publisher Springer Netherlands
ISSN 0039-7857 (Print) 1573-0964 (Online)
Issue Volume 102, Number 2 / February, 1995
DOI 10.1007/BF01089802
Pages 235-266
Subject Collection Humanities, Social Sciences and Law
SpringerLink Date Monday, February 07, 2005

-or-

Time, quantum mechanics, and decoherence

Simon Saunders1
(1) Department of Philosophy, Harvard University, 02138 Cambridge, MA, USA

Abstract State-reduction and the notion of ‘actuality’ are compared to ‘passage’ through time and the notion of ‘the present’ already in classical relativity the latter give rise to difficulties. The solution proposed here is to treat both tense and value-definiteness as relational properties or ‘facts as relations’ likewise the notions of change and probability. In both cases ‘essential’ characteristics are absent: temporal relations are tenselessly true; probabilistic relations are deterministically true.
The basic ideas go back to Everett, although the technical development makes use of the decoherent histories theory of Griffiths, Omnès, and Gell-Mann and Hartle. Alternative interpretations of the decoherent histories framework are also considered.

Stirring up trouble

PC Davies - Physical Origins of Time Asymmetry, 1994 - books.google.com
… flows equably, without relation to anything external”-a notion from which Newton derived the idea … To assert that time flows at one second per second…

The physics of time travel

JZ Simon - Physics World, 1994 - isr.umd.edu
… travel forward in time at the rate of one second per second. … the obvious exception of the time machine link … an infinite number, which all obey Newton…

Groundhogs, Gravity and Loops through Time

J Williams - fizika.unios.hr
… into the future at the rate of one second per second, one year … majority of physicists
and mathematicians accept time as a … This was true for Sir Isaac Newton. …

[BOOK] In Search of Time: The Science of a Curious Dimension

D Falk - 2008 - books.google.com
… were around to mark its passing - but, as we will see, Newton does not … at the rate
of, say, a thousand gallons per second; time flows by … one second per second? …

I don’t have the ideal citation at my fingertips here at Lincoln High School. My rough recollection is that Newton wrote of “flow of time” as “flows equably” by reference to what? He considered 2 different times, T and t, and stated (in essence) that the default was dt/dT = 1.

But that this was not automatically true, and was subtle, and that he intended to say more about this later.

Posted by: Jonathan Vos Post on May 14, 2009 7:22 PM | Permalink | Reply to this

Re: No joke

It’s very hard to evaluate these textual extracts, one big reason being that in every case, there are ellipses which separate key phrases like “one second per second” from what Newton purportedly said. I’ve no idea if the gaps are narrow or wide.

The one by P.C. Davies strikes me as the most promising of the batch. A direct link would be most helpful.

My rough recollection is that Newton wrote of “flow of time” as “flows equably” by reference to what? He considered 2 different times, T and t, and stated (in essence) that the default was dt/dT = 1.

But that this was not automatically true, and was subtle, and that he intended to say more about this later.

If one could point directly to something Newton said that substantiates what appears after “flows equably” here, that would be great. I’m certainly no expert here.

Posted by: Todd Trimble on May 15, 2009 5:49 AM | Permalink | Reply to this

Isaac^3; Re: No joke

About Isaac Newton’s claim that “time flows equably” – i.e. if time “t” flows with respect to some more fundamental time “T” that dt/dT = 1. (not true by the way, once you extrapolate Newton’s Gravity to Einstein Gravity, and local time is the square root of the g_tt term of the tensor with respect to Proper Time).

It turns out that Isaac Newton got this notion from his teacher, Isaac Barrow. So I have a Brief History of Time based on the Triumverate of Isaacs: Barrow, Newton, Asimov.

Others objected to Newton’s statements refining Barrow in the Scholium. Leibniz, Kant, Berkeley, and the Edinburgh philosopher John Taggert McTaggert who refuted the very existence of time as a linguistic construct.

Newton had his reasons for Absolute Space and Absolute Time. Minkowski replaced that with Absolute Space-Time. Einsten used that for SR, and then went to GR which rejected it and again had a little t and a big T (or tau) as mentioned.

Endless commentary exists in Philosophical, History of Science, and other literature about what Newton meant by “time flows equably.” On this ancient PC running Windows 98 and with too little memory to have two browser windows open at once, I cannot cut and paste the texts or citations. But they are easily found on GoogleScholar and even vanilla Google.

Posted by: Jonathan Vos Post on May 15, 2009 10:32 PM | Permalink | Reply to this

Re: No joke

I’d prefer to keep the focus here on what you say Newton actually wrote. I’m not aware that he spoke anywhere about a little t versus a big T. It’s certainly okay by me if you want to interpret what he did say in such terms, as long as you make clear that’s what you’re doing. Which is different from ascribing to him words he never actually used.

Specifically, you said

[Sir Isaac Newton] wrote that the rate of flow of time was one second per second, promised to get back and re-examine that assumption, but never did.

I’ll guess for now I’ll let slide the loose ascription which appears before the first comma, but: I am curious where he made such a promise. Forgive me if I seem skeptical about that.

As I say, I’m no expert, but please respect the fact that I am not asking you to cite other people’s commentary on “time flows equably, without relation to anything external”. I can google that as well as the next guy. Much less am I asking you for your comments on special and/or general relativity.

Perhaps you may think I am nitpicking, but loose talk about promises made by Newton is the type of thing I don’t hold to or sit comfortably with. I do apologize if you turn up some factual support of this somewhere in Newton’s writings.

Posted by: Todd Trimble on May 16, 2009 8:54 AM | Permalink | Reply to this

Isaac^3; Re: No joke

About Isaac Newton’s claim that “time flows equably” – i.e. if time “t” flows with respect to some more fundamental time “T” that dt/dT = 1. (not true by the way, once you extrapolate Newton’s Gravity to Einstein Gravity, and local time is the square root of the g_tt term of the tensor with respect to Proper Time).

It turns out that Isaac Newton got this notion from his teacher, Isaac Barrow. So I have a Brief History of Time based on the Triumverate of Isaacs: Barrow, Newton, Asimov.

Others objected to Newton’s statements refining Barrow in the Scholium. Leibniz, Kant, Berkeley, and the Edinburgh philosopher John Taggert McTaggert who refuted the very existence of time as a linguistic construct.

Newton had his reasons for Absolute Space and Absolute Time. Minkowski replaced that with Absolute Space-Time. Einsten used that for SR, and then went to GR which rejected it and again had a little t and a big T (or tau) as mentioned.

Endless commentary exists in Philosophical, History of Science, and other literature about what Newton meant by “time flows equably.” On this ancient PC running Windows 98 and with too little memory to have two browser windows open at once, I cannot cut and paste the texts or citations. But they are easily found on GoogleScholar and even vanilla Google.

Posted by: Jonathan Vos Post on May 15, 2009 10:34 PM | Permalink | Reply to this

Re: Bruno and Infinite Worlds; Re: Taming the Boundless

Apparently, that verse was added later; no one knows by whom.

Posted by: Mike Stay on May 14, 2009 9:15 PM | Permalink | Reply to this

Naming the Boundless

A recent publication just announced: Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity

If anyone has seen it….
Zeilberger likes it!

Posted by: jim stasheff on May 13, 2009 4:33 PM | Permalink | Reply to this

Infinitely Euler; Re: Taming the Boundless

Leonhard Euler, Jordan Bell, On the infinity of infinities of orders of the infinitely large and infinitely small, May 14, 2009.

Posted by: Jonathan Vos Post on May 15, 2009 4:33 AM | Permalink | Reply to this

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