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June 21, 2007

Faith and Reason

Posted by David Corfield

Back last October I mentioned a book – The Good Life in the Scientific Revolution: Descartes, Pascal, Leibniz, and the Cultivation of Virtue – which describes

how three key early modern scientists, René Descartes, Blaise Pascal, and Gottfried Leibniz, envisioned their new work as useful for cultivating virtue and for pursuing a good life. Their scientific and philosophical innovations stemmed in part from their understanding of mathematics and science as cognitive and spiritual exercises that could create a truer mental and spiritual nobility.

Someone closer to our times who considered the relationship between scientific and spiritual enquiry was the chemist turned philosopher Michael Polanyi. In Faith and Reason, The Journal of Religion, Vol. 41, No. 4 (Oct., 1961), pp. 237-247, Polanyi establishes the central concept of his epistemology:

Understanding, comprehension – this is the cognitive faculty cast aside by a positivistic theory of knowledge, which refuses to acknowledge the existence of comprehensive entities as distinct from their particulars; and this is the faculty which I recognize as the central act of knowing. For comprehension can never be absent from any process of knowing and is indeed the ultimate sanction of any such act. What is not understood cannot be said to be known. (p. 240)

Like many other philosophers I enjoy, Polanyi stresses the movement of a knowing mind rather than any static state of knowledge.

The dynamic impulse by which we acquire understanding is only reduced and never lost when we hold knowledge acquired and established by this impulse. The same impulse sustains our conviction for dwelling in this knowledge and for developing our thoughts within its framework. Live knowledge is a perpetual source of new surmises, an inexhaustible mine of still hidden implications. (p. 244)

Elsewhere he defines his conception of reality as “that which may yet inexhaustibly manifest itself”. This conception informs the next passage.

To hold knowledge is indeed always a commitment to indeterminate implications, for human knowledge is but an intimation of reality, and we can never quite tell in what new way reality may yet manifest itself. It is external to us; it is objective; and so its future manifestations can never be completely under our intellectual control.

So all true knowledge is inherently hazardous, just as all true faith is a leap into the unknown. Knowing includes its own uncertainty as an integral part of it…

The traditional division between faith and reason, or faith and science…, reflects the assumption that reason and science proceed by explicit rules of logical deduction or inductive generalization. But I have shown that these operations are impotent by themselves, and I could add that they cannot even be strictly defined by themselves. To know is to understand, and explicit logical processes are effective only as tools in search of a problem, commitment by which we expand our understanding and continue to hold the result. They have no meaning except within this informal dynamic context. Once this is recognized, the contrast between faith and reason dissolves, and the close similarity of this structure emerges in its place. (p. 244)

This prepares the way for an assertion about the similarity of faith and reason.

The very act of scientific discovery offers a paradigm of this transition. I have described it as a passionate pursuit of a hidden meaning, guided by intensely personal intimations of this unexposed reality. The intrinisic hazards of such efforts are of its essence; discovery is defined as an advancement of knowledge that cannot be achieved by any, however diligent, application of explicit modes of inference. Yet the discoverer must labor night and day. For though no labor can make a discovery, no discovery can be made without intense, absorbing, devoted labor. Here we have a paradigm of the Pauline scheme of faith, works and grace. The discoverer works in the belief that his labors will prepare his mind for receiving a truth from sources over which he has no control. I regard the Pauline scheme therefore as the only adequate conception of scientific discovery. (pp. 246-7)

One might question whether spiritual or moral enquiry approaches a reality which inexhaustibly manifests itself, but, as a description of research in mathematics, Polanyi appears to me to have captured something important. Having learned via John of the n-categories program, and seen his commitment to it over so many years, it would be hard to overlook this aspect of faith.

Posted at June 21, 2007 9:38 AM UTC

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Re: Faith and Reason

I hope I’m not off-subject here, but your thoughts intrigued me and sent me off on the following tangent…

I think it is important to distinguish between the act of seeking, which is unquestionably a spiritual endeavor, and that of condescending (co-descending, in a positive sense) to popular theory in order to grasp and conform to the discoveries of others.

Much of what occupies our time in science is not the pursuit and acquisition of pure knowledge (which requires no language whatsoever), but rather the deciphering of the knowledge of our peers, and the formatting of our own discoveries for the purpose of presentation. We are trying to hang our ornaments on the same tree, which is understandable and necessary if we are to build upon a corporate foundation. But this is where we must delineate between the carnal and spiritual quest (if I may use such language on this board).

Consuming ourselves disproportionately in popular conformity of pure discovery can be spiritually and creatively draining. We may find it distracting us from fundamental inquiry. There is no spiritual fulfillment in the formatting. It is hidden in simplicity… in the lonely isolation of personal discovery.

“If thou wilt receive profit, read with humility, simplicity and faith, and seek not at any time the fame of being learned.”

- Thomas a Kempis

Posted by: Rose on June 21, 2007 3:44 PM | Permalink | Reply to this

Re: Faith and Reason

It’s a tricky balance to strike for a scientist or mathematician. I suppose the ideal is to be recognised as brilliant so people will make much more effort to understand you. But even someone like Bill Thurston had to put in a huge effort to build infrastructure for people to understand him, see section 6 here.

Posted by: David Corfield on June 21, 2007 9:20 PM | Permalink | Reply to this

Re: Faith and Reason

Perhaps your distinction is too stark. If your good is wrapped up with the good of mathematics, you should want to broadcast your ideas to others.

In the words of the proto-blogger himself:

But, I like getting people interested in math: for me, that’s a crucial part of doing math. If I can’t get lots of people interested in what I’m doing, I have trouble staying interested in it.

We should do more to cultivate this virtue, as Thurston suggests.

Posted by: David Corfield on June 23, 2007 7:42 AM | Permalink | Reply to this

Re: Faith and Reason

Rose said:

Consuming ourselves disproportionately in popular conformity of pure discovery can be spiritually and creatively draining. We may find it distracting us from fundamental inquiry. There is no spiritual fulfillment in the formatting. It is hidden in simplicity … in the lonely isolation of personal discovery.

That isn’t true for me, and I think it isn’t true for a lot of scientists and mathematicians. Yeah, explaining things can be dispiriting and emotionally draining, but so can research, or anything that takes a lot of effort and has a significant probability of not succeeding. But I’m of the school that says that if you can’t explain something, you haven’t understood it. I also doubt I’m alone in finding that a major motivator for learning about something properly is the desire to explain it; and that getting something across successfully is rewarding in itself.

Posted by: Tim Silverman on June 23, 2007 1:01 PM | Permalink | Reply to this

Re: Faith and Reason

Imagine how many important (yet initially unpopular) mysteries would remain unsolved today had researchers found their motivation exclusively in the fashionable.

The Pauline concept of “work” is poorly translated. Allowing that which virtually swells up within to manifest itself in detailed thought and action should seem effortless… not at all like work. Maybe it’s still about the payoff, but I remain convinced that it is personal discovery that brings the deepest and most profound satisfaction. That is not to suggest there is no satisfaction in evangelizing the discovery, only that it is secondary and inferior to the revelation. It clarifies, it corrects, but public interest should never be the primary motivation for the quest.

Thank you, James Dolan. You have formed my theology!

Posted by: Rose on June 29, 2007 2:05 PM | Permalink | Reply to this

A.S.K.; Re: Faith and Reason

“distinguish between the act of seeking… and … co-descending…”

As Martin Gardner pointed out, there’s this nice acrostic:

Luke 11:9

Ask and it will be given to you;
Seek and you will find;
Knock and the door will be opened to you.

First letters of each line spell “ASK.”

Good job of translation, as this was a standard poetic structure on old Hebrew, and had numerological consequences inappropriate for this blog.

Theodore Sturgeon invented a symbol, sort of a question mark overprinted with a diagonal functorial arrow, and wore it around his neck (well, on a chain) that symbolized: “Ask the next question!”

Posted by: Jonathan Vos Post on June 24, 2007 6:17 PM | Permalink | Reply to this

Re: A.S.K.; Re: Faith and Reason

numerological consequences inappropriate for this blog

Oh, come now. We talk about numerology in string theory all the time, and my advisor pointed out to be the connection between string theory and Kabbalistic numerologies.

Posted by: John Armstrong on June 24, 2007 8:02 PM | Permalink | Reply to this

Re: A.S.K.; Re: Faith and Reason

Okay, I can’t find my files on this, so quick outline off the top of my head. Alright, I’m double checking against:
http://www.i18nguy.com/unicode/hebrew-numbers.html

(1) The great majority of modern Hebrew texts today use Arabic numerals (0, 1, 2, 3…9) to represent numbers. See my summary, quoting Fibonacci, at:
http://mathworld.wolfram.com/ArabicNumeral.html

(2) Some religious or biblical texts, and calendars in Hebrew, still use the traditional form which depends on an isomorphism between Hebrew letters and numeric values, per the table below.

(3) There is no representation for zero (0). This system was popularized by Hillel II in the fourth century A.D., when he prescribed the rules for the Hebrew calendar system.

(4) The isomorphism maps each letter in the Hebrew alphabet (or aleph-bet) to a numerical value. The first 10 letters (consonants actually) have the values 1-10. The next 9 letters are valued 20, 30, … 100. The remainder are valued 200, 300, and 400.

(5) Well after Hillel II the final forms of the letters kaf, mem, nun, pe, and tzadi were used for the missing values 500, 600, 700, 800, and 900.

(6) Table (letter, unicode, integer):

Alef 05D0 1
Bet 05D1 2
Gimel 05D2 3
Dalet 05D3 4
He 05D4 5
Vav 05D5 6
Zayan 05D6 7
Het 05D7 8
Tet 05D8 9

Yod 05D9 1 * 10^1
Kaf 05DB 2 * 10^1
Lamed 05DC 3 * 10^1
Mem 05DE 4 * 10^1
Nun 05E0 5 * 10^1
Samekh 05E1 6 * 10^1
Ayin 05E2 7 * 10^1
Pe 05E4 8 * 10^1
Tzadi 05E6 9 * 10^1

Qof 05E7 1 * 10^2
Resh 05E8 2 * 10^2
Shin 05E9 3 * 10^2
Tav 05EA 4 * 10^2

Later:
Qof 05E7 1 * 10^2
Resh 05E8 2 * 10^2
Shin 05E9 3 * 10^2
Tav 05EA 4 * 10^2
Final Kaf 05DA 5 * 10^2
Final Mem 05DD 6 * 10^2
Final Nun 05DF 7 * 10^2
Final Pe 05E3 8 * 10^2
Final Tzadi 05E5 9 * 10^2

(7) Hebrew numbers are syntactically differently from Western or European (Indian or Arabic) numbers. In the West, only 10 digits are used, and the position of the digit indicates its value in powers of 10 beginning at 1, so the digit value is multiplied by 1, 10, 100, 1000, etc. as the position increases from right to left. (Being position-based, a zero digit is an absolute requirement.)

(8) Hebrew numbers on the other hand, simply add the values of each letter together and the position doesn’t matter. Hence a permutation group operator is implicit. Still, numbers are generally written from largest to smallest, which in the right-to-left Hebrew script, means the largest is right-most. For numbers greater than 799, tav (ת 400) is repeated.

(9) Numbers are formed by choosing the hebrew letter with the largest value that doesn’t exceed the number and then selecting the next largest valued letter that reduces the remainder. For example, to represent 765, the largest valued letter is tav (400 ת) leaving a remainder of 365. Adding the letter shin (300 ש) leaves 65. Adding somekh (60 ס) and he (5 ה) eliminate the remainder. So 765 is represented by tav, shin, somekh, he: תשסה.

(10) There is one exception. Numbers ending in 15 or 16 would be written as yud-he (10+5) and yud-vav (10+6), but the letters “yud he vav he” spell out the name of God and for religious reasons are not used. Instead, by convention, tet-vav (9+6 טו) and tet-zayin (9+7 טז) are always used.

(11) Well gosh, there is a second exception. Some numbers spell out a word with strongly negative or positive connotations. I’ll cuicrcle back to this later. In these cases, the order of the letters might be changed. The number 18 for example, yud-het, uses the same letters as the word for life het-yud. So instead of יח, you may see חי.

(12That’s like the Chinese mapping of 4 into Death, and 3 into Life, and 8 into proseperity.

(13) Thousands are represented by the same letters as the unit values, sometimes a character similar to an apostrophe is appended. The character is a punctuation mark called geresh. When geresh is not available, the single quote (U+0027) is often substituted. A space (U+0020) often separates thousands, millions, etc. The pattern for numbers 1-999 is repeated for each thousand from 1,001- 999,999. Millions and Billions etc. are formed by extending and repeating the pattern.

(14) Using letters for numbers, there is the possibility of confusion as to whether a string of letters is a word or a numerical value. Therefore, when numbers are used with text, punctuation marks are added to distinguish their numerical meaning. Single character numbers (numbers less than 10) add the punctuation character geresh after the numeric character. Larger numbers insert the punctuation character gershayim before the last character in the number.

(15) So strings of numbers and strings of letters map to each other, with the twists described.

(16) So things described by strings of numbers and things described by strings of letters map to each other, with some extra transformations thrown in.

(17) hence the hermetic theorem: “As above, so below” which defines a natural isomorphism between the Book of the Bible (or pentateuch, anyway) and the Book of Nature (physical reality, including astronomy/astrology).

(18) Galileo expounded on this, in language designed not to annoy The Church too much. Hate to be executed like Bruno, who said that tyhere were an infinite number of worlds in the cosmos, and thus some which were just like ours except the Mass was said in the vernacular. He only knoew of one cardinal infinity, sad to say. Cantor extended his results.

(19) Greek letters equate to numbers in a similar way, including letters now vanished.

(20) Forgetful functors applied to some mappings leads to the whole 666 notion.

(21) Some people memorized the Torah as a 3-D array of letter/numbers. They could tewll you what sequence of letters one would encounter if (theire metaphor) one plunged a needle through the Torak, page by page, from a given initial position on the 1st page.

(22) Some people visualized the Torah as a 4-D array.

(23) My great^n ancestor Rabbi Aharon Eliakum Wertheimer (born ca. 1600) was into Kabbalah, but his main fame rested on his leading the Freethinkers, thus spreading The Enlightenment to Jewish eastern Europe. I cannot read the standard biography of him, as it is in German. He held that one should be skeptical of all dogma, secular or religious. Way ahead of his time!

(24) “As above, so below” is a mapping frm which deductions were made as to the nature of infinity (hotly debated in Arabic, Christian, and Hebrew theology) in space and in time. The space-time continuum was modeled differently from today.

Posted by: Jonathan Vos Post on June 25, 2007 1:39 AM | Permalink | Reply to this

Re: A.S.K.; Re: Faith and Reason

Sorry for all the typos. Typing fast, interrupted by barbecuing dinner, in the teeny box with tiny characters.

Of course, I meant concatenate the Hebrew letters; I’d said “add” because it’s Abelian. But really, more like a star-algebra.

One should be explaining the deeper structure of sequences involving concatenation of digit strings, and digital reversal, the latter generalizing to adjoint operators which reverse the order of multiplication of products
(abc…xyz)* = z* y* x* … c* b* a*.

And we know what Adjoints are really about.

So here’s where the Category Theory should start, in numerology…

See:

A127698 Sum of n-th triangular number and its reversal (leading zeros not truncated). for an example of how I like to play with reversal-like sequences.

Posted by: Jonathan Vos Post on June 25, 2007 2:12 AM | Permalink | Reply to this

Re: Faith and Reason

The idea goes back to Plato, doesn’t it? The philosopher king was schooled in mathematics.

From a Judeo-Christian POV, if humankind is created in the image of the diety, then our capacity for reason is among one of the few functions that is awe inspiring. (Yeah, yeah, yeah, what about our capacities for jealousy, capacity to wage war, capacity for hatefulness, and the capacity to shoot at one another just because your sister-in-law’s prophet raped the wife of my brother’s prophet?)

My favorite argument for the spirituality of the mathematical endeavor remains the following. You go back to your room, examine the axioms, definitions, and theorems, and think up consequences. Your mathematical inspiration comes from nothing but introspection. It is therefore a subjective reality. Then after a few hours, days, or weeks you tell me about your deep inner thinking — either in person, in your blogs or by posting the result on the arXiv. I take that material, digest it, examine it, and through my own process of introspection come to the same conclusions that you did, and I can draw my own conclusions with which you and others will agree. Thus the subjective realities of our own minds match the objective reality described by the axioms. But the axioms have no objective reality. They are just axioms. Finally, these mathematical truths can be used to give approximate models to the physical world.

Apparently objective truth is obtained from instrospection. Is that cool or what?

Finally, there are a few things that make the passage of time seem unimportant. Thinking about mathematics, playing ball in the swimming pool, and making rude noises that masquerade as music with an electric guitar are the main things. The time spent doing mathematics is spiritual time.

Posted by: Scott Carter on June 21, 2007 4:07 PM | Permalink | Reply to this

Re: Faith and Reason

[…] Is that cool or what?

Yes! Wonderful comment. The best about spirituality and reason that I have seen in a while.

A related aspect, maybe: I have come to think that a couple of human endeavours in history which have come to be counted as belonging to the world of religion and/or spirituality are really an expression of a desire for mathematical reasoning which hasn’t found a way to express itself “properly”.

I am thinking for instance of

- parts of the program of alchemy

- parts of Cabbala lore and practices

- various attempts at “ontological proofs”, etc.

Posted by: urs on June 21, 2007 4:22 PM | Permalink | Reply to this

Re: Faith and Reason

The philosopher king was schooled in mathematics.

Yes, but it’s just one stage in his training, running according to the Republic between ages 20 and 30.

Mathematics is always an intermediate kind of activity. It is to awaken “the eye of the soul” to what is permanent, but it itself is not the highest activity. The objects of the world both are and are not. Mathematics points us to what is. Ultimately there are the forms, and the highest form is that of the Good.

After mathematics one must study dialectic:

What do you mean? I said; the prelude or what? Do you not know that all this is but the prelude to the actual strain which we have to learn? For you surely would not regard the skilled mathematician as a dialectician?

Assuredly not, he said; I have hardly ever known a mathematician who was capable of reasoning.

But do you imagine that men who are unable to give and take a reason will have the knowledge which we require of them?

Neither can this be supposed.

And so, Glaucon, I said, we have at last arrived at the hymn of dialectic. This is that strain which is of the intellect only, but which the faculty of sight will nevertheless be found to imitate; for sight, as you may remember, was imagined by us after a while to behold the real animals and stars, and last of all the sun himself. And so with dialectic; when a person starts on the discovery of the absolute by the light of reason only, and without any assistance of sense, and perseveres until by pure intelligence he arrives at the perception of the absolute good, he at last finds himself at the end of the intellectual world, as in the case of sight at the end of the visible.

Of course, we here at the Café don’t believe that mathematicians can’t engage in dialectic.

Posted by: David Corfield on June 21, 2007 9:39 PM | Permalink | Reply to this

Re: Faith and Reason

If we were to have a dialogue about dyadic numbers and we both learned something, would that be a “didactic dyadic dialectic?”

Reading the n-cafe entries reminds me, at least, of reading the more classical dialectics. We probably need to work on our expository techniques ;-)

Posted by: Scott Carter on June 21, 2007 11:28 PM | Permalink | Reply to this

Re: Faith and Reason

My inner Wittgenstein-ian materialist cries out at reading this article. How can we know of what we cannot speak when our thoughts are always in a language (counting math as a language ;) )?

(Arguably one could say “But what about thinking in pictures?” Is this not expressible through a language when effectively using it? Einstein claimed to think in pictures, and his concepts were translated into math or some other language rather coherently and well.)

Admittedly, all three referred to in the first title (Descartes, Pascal, and Leibniz) were idealists (as is Polanyi), which is probably the source of my malcontent. There really isn’t any convincing argument for idealism.

(Actually, there was an interesting book out nearly a decade ago by M. Balaguer: Platonism and Anti-Platonism, in Mathematics; he points out that it isn’t really possible to argue for or against Platonism, or as I generalize it to idealism, in mathematics.)

But, to be honest, it doesn’t really matter with how you came up with an idea (voodoo, throwing darts at a board, praying, etc.) insomuch as whether the idea is correct or not.

You could use the “correct” epistemological method and still come up with just blatantly incorrect ideas (it’s not hard to use any method incorrectly).

Newton is a great example of this; no one seems to care that Newton was a mystic. As a matter of fact, as Charles Webster points out in From Paracelsus to Newton: Magic and the Making of Modern Science, Newton was a lot more questionable a fellow than we’d like to admit.

But we ignore that characteristic because it doesn’t change the final product that he (kind of) came up with (calculus and classical physics, of course).

David Corfield wrote:

“After mathematics one must study dialectic”

Oh good God, not dialectics!

Posted by: AngryPhysicist on June 22, 2007 9:55 AM | Permalink | Reply to this

Re: Faith and Reason

Oh good God, not dialectics!

Plato is not Hegel or Marx.

His criticism of mathematicians (although note it’s in the voice of Glaucon, so one might argue whether or not its his view) is that they merely reason from hypotheses and don’t put them into question.

This seems obviously inaccurate to us as we’ve witnessed so many presuppositions being put into question by mathematicians - the parallel postulate, etc.

Even so, it is not uncommon to hear the charge that there has been a lack of readiness to seek better founding principles. Here, for example, is Grothendieck in ‘Sketch of a Programme’:

This situation, like so often already in the history of our science, simply reveals the almost insurmountable inertia of the mind, burdened by a heavy weight of conditioning, which makes it difficult to take a real look at a foundational question, thus at the context in which we live, breathe, work – accepting it, rather, as immutable data. It is certainly this inertia which explains why it took millennia before such childish ideas as that of zero, of a group, of a topological shape found their place in mathematics. It is this again which explains why the rigid framework of general topology is patiently dragged along by generation after generation of topologists for whom “wildness” is a fatal necessity, rooted in the nature of things. (259)

Posted by: David Corfield on June 23, 2007 8:01 AM | Permalink | Reply to this

Re: Faith and Reason

angryphysicist makes an important point here but Newton’s crime is not being a mystic or an alchemist or having written more about religion than about science, or being a serial plagiarist. No. Newton’s crime is exactly in his science. Newton took Kepler’s rule and added occult and mystical labels to it and repackaged it as Newton’s laws. See this simple graphic showing the labels Newtonian physicists attached to Kepler’s rule over the years.

Posted by: Pioneer1 on June 24, 2007 2:49 PM | Permalink | Reply to this

Re: Faith and Reason

See this simple table before you take your windmill-tilting to yet another innocent weblog.

Posted by: John Armstrong on June 24, 2007 4:20 PM | Permalink | Reply to this

Re: Faith and Reason

That’s an iconic list, of course, and it’s hard to match it but its transpose the genius index has some value as well.

Posted by: Pioneer1 on June 26, 2007 12:29 AM | Permalink | Reply to this

Re: Faith and Reason

Excuse the intrusion of an amateur (both in math and philosophy) but is it not the case that a lot of confusion arise from conflating “reality” with descriptions of reality?
Maths as well as philosophy texts are just our attempts to model reality, why do want to surmise that the model is the thing modelled?
This is why I am anti-platonist in the sense that existence assumptions pertain to the models we are currently working on not to reality, except for the fact the model too belongs to reality since it does “exist” somewhere in our brains/minds/discourses.
Any ontology is a handy tool to build models but I am quite sure that reality as such is not made of any animals, triangles, molecules, particles, fields, colors, whatever, for short not of objects nor of concepts.

As for spirituality I see this as within the realm of psychology rather than philosophy, just some peculiar form of qualia.
And faith though sometimes conducing to interesting hypotheses is certainly not an appropriate way to build models.
Please note that my position is not “just another faith” but an absence of faith about the nature of reality.

Posted by: Jean-Luc Delatre on June 22, 2007 2:11 PM | Permalink | Reply to this

Re: Faith and Reason

This goes back to Korzybski’s aphorism, “the map is not the territory”. It’s a useful thing to have in mind for physics, but I’m not so sure for mathematics.

The difference is that physical reality seems to exist “out there”, and we are attempting to describe it. We can make observations, which may or may not agree with our predictions. Mathematics, on the other hand, doesn’t seem nearly so constrained. In mathematics the “model” is all there is.

One nitpick, though: as I understand it (maybe David will correct me) “anti-Platonism” in mathematics would be the position that there is no objective mathematical “reality” out there, and that it’s just in our collective head. Your position seems to be that mathematicians are trying to model a “real” mathematics which is independent of the mathematicians themselves, which seems to be a Platonist viewpoint.

Posted by: John Armstrong on June 22, 2007 4:23 PM | Permalink | Reply to this

Re: Faith and Reason

This goes back to Korzybski’s aphorism, “the map is not the territory”

Yes, I didn’t care to mention it.

Your position seems to be that mathematicians are trying to model a “real” mathematics which is independent of the mathematicians themselves, which seems to be a Platonist viewpoint.

No, I do mean that there are no “mathematical objects” nor any objects “in reality” if you read me carefully.
All objects pertain to some model and their “existence” is only a property of the model not of reality.
I deny that we ever “see” or “reach” reality in maths or otherwise.
We never deal with the territory only with maps, the most basic of which come from our built-in perceptions (qualias) and over which we have no control nor any information about their “real nature”.
This is why I see no exception in maths and I am truly anti-platonist, though I am not an idealist or nominalist in that I acknowledge that there is “something out there” which is usually called reality in that it constrains our observations, however, against Plato I don’t think it adequate to suppose that this “reality” could be fully modelled by entities and concepts and that we should or could look for this “ultimate” description.

Posted by: Jean-Luc Delatre on June 22, 2007 5:42 PM | Permalink | Reply to this

Re: Faith and Reason

I have never enjoyed debates in the philosophy of mathematics as to whether mathematical objects exist or not. What interests me is the nature of this tradition of enquiry (“one of man’s longest conversations.” Mazur). What is to understand mathematically? Where does the rationality lie?

Posted by: David Corfield on June 23, 2007 8:26 AM | Permalink | Reply to this

Re: Faith and Reason

What is to understand mathematically?

My best guest about understanding (mathematically or otherwise) is that it’s when one have the “feeling” of holding together all the pieces needed to make sense of a chunk of some “structure”, in mathematics like “seeing” all the lemmas and (major) proofs steps which validate a theorem.
The drawbacks :
- one could be wrong and lacking a critical building stone.
- in order to be able to encompass all the parts of the “big picture” the subdividing into primary pieces must be done cleverly. I guess this is why some theorems are “interesting”, they are crucially located along many usefull paths of reasoning.
- beyond a certain complexity, no matter how smartly you try to slice down the problem there will be too many “primary parts” for a given mind to hold together, depending on individual capabilities.

Where does the rationality lie?

Do you mean rationality in the process of finding proofs and “interesting structures”?
I am afraid there is no possible rationality for this, randomness, serendipity and intuition seem to always have played a role in any major discovery in mathematics or elsewhere.
An “always safe” method would go against Godel, the halting problem and equivalents.

Posted by: Jean-Luc Delatre on June 23, 2007 1:01 PM | Permalink | Reply to this

Re: Faith and Reason

But why do you suppose that rationality must take the form of an “always safe” method?

Surely it’s worth distinguishing between the activity of isolated unskilled guessing, and that of working very hard with excellent teachers, being open to ideas which challenge your own, and taking part in well-focused discussion (such as sometimes happens on this blog).

Polanyi’s point was that however well you do at maximising your chances of doing something important, there must be something ungrounded about your approach, a “doing the best you can”, which can be characterised as a kind of faith. You can receive indications of being on a good track, but ultimately it comes down to an ungrounded judgement.

Posted by: David Corfield on June 23, 2007 2:46 PM | Permalink | Reply to this

Re: Faith and Reason

As a Christian virtue, wouldn’t that count as hope rather than faith?

Posted by: Tim Silverman on June 23, 2007 5:19 PM | Permalink | Reply to this

Re: Faith and Reason

I’m no theologian. The history of the terms faith, hope and charity is long and complex. But I think ‘hope’ was rather used to describe one’s expectation of eternal happiness. ‘Faith’ has more the sense of an assenting.

Polanyi’s thought was that the scientist must make some form of assent with regard to their life as a researcher. Which isn’t to say that what was so assented to can’t latter be something either reasoned to or given up. But at any given time there’s a “Here I stand, I can do no other” to one’s stance.

Posted by: David Corfield on June 23, 2007 7:41 PM | Permalink | Reply to this

Re: Faith and Reason

But why do you suppose that rationality must take the form of an “always safe” method?

Sorry, my misunderstanding of your use of the word.

Surely it’s worth distinguishing between the activity of isolated unskilled guessing, and that of working very hard with excellent teachers, being open to ideas which challenge your own, and taking part in well-focused discussion (such as sometimes happens on this blog).

Guessing is not necessarily unskilled, of course this is the common way of the crackpots, quite often delusions of grandeur about simplistic ideas.
OTOH, do you really think “working very hard with excellent teachers, being open to ideas which challenge your own, and taking part in well-focused discussion” is likely to drive you out of the “insurmountable inertia of the mind” as you quoted yourself from Grothendieck?
I suggest randomness and serendipity are necessary ingredients by analogy to the case of the Travelling Salesman Problem where it seems that the best and most robust method is simulated annealing.
Introducing some random and even counterintuitive tries is probably a way out of local optima provided of course that rationality is applied to the evaluation of the results.
For an outside observer like me it is obvious that mathematicians are driven by some aesthetics they are absolutely at loss to explain, so they can be unknowingly trapped in local “aesthetic optima”.
Since you are both a philosopher and mathematician are you aware of any investigations in the role of the aesthetics of maths in discovery?

You can receive indications of being on a good track, but ultimately it comes down to an ungrounded judgement.

Ungrounded in any “obvious” rationale but probably not purely hapazard, this is called intuition and is a touchy subject even if people like Hadamard (The Psychology of Invention in the Mathematical Field) have exposed the non rational parts of mathematical creativity.
There is a tension between the requirements for rigor and creativity which makes mathematics especially difficult.
The late Jon Barwise and John Etchemendy have found that “real” reasoning does not follow at all the rules of logic even when the proof is ultimately formatted in a classic way.

Posted by: Jean-Luc Delatre on June 23, 2007 7:52 PM | Permalink | Reply to this

Re: Faith and Reason

Guessing is not necessarily unskilled

Of course. George Polya showed us this very clearly.

do you really think “working very hard with excellent teachers, being open to ideas which challenge your own, and taking part in well-focused discussion” is likely to drive you out of the “insurmountable inertia of the mind”

No, but it may help. It’s the ‘works’ part of Polanyi’s ‘faith, works and grace’.

This leaves the ‘grace’ to deal with. Returning to Grothendieck,

In our acquisition of knowledge of the Universe (whether mathematical or otherwise) that which renovates the quest is nothing more nor less than complete innocence. It is in this state of complete innocence that we receive everything from the moment of our birth. Although so often the object of our contempt and of our private fears, it is always in us. It alone can unite humility with boldness so as to allow us to penetrate to the heart of things, or allow things to enter us and taken possession of us.

This unique power is in no way a privilege given to “exceptional talents” - persons of incredible brain power (for example), who are better able to manipulate, with dexterity and ease, an enormous mass of data, ideas and specialized skills. Such gifts are undeniably valuable, and certainly worthy of envy from those who (like myself) were not so “endowed at birth, far beyond the ordinary”.

Yet it is not these gifts, nor the most determined ambition combined with irresistible will-power, that enables one to surmount the “invisible yet formidable boundaries” that encircle our universe. Only innocence can surmount them, which mere knowledge doesn’t even take into account, in those moments when we find ourselves able to listen to things, totally and intensely absorbed in child’s play.

Worth mulling over, seeing how profoundly he transformed ways of thinking, as his legacy continues to do so.

are you aware of any investigations in the role of the aesthetics of maths in discovery?

That’s something I mean to look at some day. Rota wrote a piece on beauty in mathematics, but there’s much more to be done.

probably not purely hapazard

Naturally.

Posted by: David Corfield on June 23, 2007 10:06 PM | Permalink | Reply to this

Re: Faith and Reason

No, but it may help.

Or it may impede thru an infusion of worn out ideas.
Which way will the balance go is anyone’s guess.
I don’t like either of ‘faith, works and grace’ except may be grace which I would rather call dumb luck.

Returning to Grothendieck,
…/…
Such gifts are undeniably valuable, and certainly worthy of envy from those who (like myself) were not so “endowed at birth, far beyond the ordinary”.

That’s very amusing that Grothendieck didn’t see himself as a “person of incredible brain power”.
In the late sixties I saw him at demonstration against a nuclear power plant (!) and he was looking real weird, may be a little nuttiness is also a requirement for genius.

Grothendieck : Only innocence can surmount them, which mere knowledge doesn’t even take into account, in those moments when we find ourselves able to listen to things, totally and intensely absorbed in child’s play.

It’s words like “innocence” which bring the spirituality overtone, if we are to make any progress toward the “understanding of understanding” it is probably good for us to resist our inclinations toward this emotion laden terminology because all thing sacred and spiritual are shrouded in “mystery” and it is mystery that we want to get rid of.
Though this doesn’t mean that the emotionality should be summarily dismissed, on the contrary it likely points to the “murky areas” to be clarified.

There has been lately some blog posts which seem relevant to the question of mathematical understanding:
Math texts as piano recitals (a rant) at GNXP.
Math as a Natural Language via Kenny Easwaran.

Posted by: Jean-Luc Delatre on June 26, 2007 8:38 AM | Permalink | Reply to this

Re: Faith and Reason

No, I do mean that there are no mathematical objects nor any objects in reality if you read me carefully.

I think that there are only mathematical objects. Objects exist in the sense that objects in Object Oriented languages exist, e.g. in Smalltalk. Atomic materialistic absolute objects do not exist. I am glad to read that someone else thinks in a similar vein. I have been writing about this subject in my blog and I collected a few of them here I would appreciate your comments, and of course, anyone else’s comments here as well. Thanks.

Posted by: Pioneer1 on June 27, 2007 1:35 AM | Permalink | Reply to this

Re: Faith and Reason

When we use a computer to solve a problem, we load some software, we input a bunch of data, it runs for a while, sometimes for a really long while, and then it outputs an answer. The answer is the result of the computation. You can check your power bill and use thermodynamics and find that the theory that the computation produced the answer adds up.

When we use mathematics to solve a problem, we learn some math techniques, we read about some problems, we work for a while, sometimes for a really long while, and then output an answer. Apparently exactly the same as the case of the computer? I know that the majority of people believe that humans have some special quality that computers can never have. Is contact with the mind of omniscient God the quality that they think we have? Computers can have contact only with the internet, which is like the mind of a god who is usually thinking about barely legal teens, copyrighted downloads, wholesale software and imported drugs.


I’m not quite sure what the Pauline scheme of faith, works, and grace is all about. It seems to say that God is the source of all knowledge. If we think that we learned something, all that actually happened is that God revealed the knowledge to us. That doesn’t sound any different from Platonic idealism. Does it mean that if you don’t believe in the existence of Platonic Ideals and you work hard on math, your answers will be wrong?

It seems like most people here are rejecting Platonic Idealism. John’s faith isn’t in universals, it’s just in himself and the program of categorifying everything, which is a much smaller thing to have faith in. David Hilbert and a lot of others had faith in Hilbert’s Program to give math a solid foundation, and they ultimately failed. I don’t mean to rain on your parade, though. Godspeed!

Posted by: David Lyon on June 23, 2007 3:55 AM | Permalink | Reply to this

Re: Faith and Reason

In this piece Polanyi merely wants us to reconsider the relationship between faith and reason. We have passed through various cycles of setting one against the other: reason only takes us so far, then faith is necessary; reason gives us certainty, faith is not to be relied upon; and so on. We may have wanted to free ourselves from the faith-driven religious wars to rely instead on reason, but all we did was drive ourselves into the horrors of the 20th century.

Grand themes indeed. But what caught my eye was that the main evidence for his claim that faith and reason should not be viewed as antagonistic is that scientific work bears the hallmarks of the ‘Pauline scheme’. A scientist would get nowhere without the sustained belief that there is a story which will make sense of what to others may look like unconnected phenomena. And this belief must be sustained through adversity.

Posted by: David Corfield on June 23, 2007 8:22 AM | Permalink | Reply to this

Re: Faith and Reason

When I read Polyani’s reference to the Pauline scheme of faith, works, and grace I immediately recognized something I knew (but hadn’t verbalized) about how I’ve learned Mathematics. I’m no Mathematician, I just meddle, but this is what I understood:

I never learn something without recreating it for myself. I can sit in a class and learn that the the lecturer made certain statements, or read a book and know that the author made certain claims. But I don’t feel like I know something until I’ve come up with it myself. That takes work.

I understand “faith” here to be what I usually call “interest”. When I’m interested in a subject, I don’t understand it, but somehow I have a strong belief that I will be able to put the pieces together and make sense of it. Or I believe that some ideas are likely to produce “interesting” results. I’ve never embarked on any ambitious research projects, so I can’t say I have deep experience here.

The grace is the moment of comprehension, often very surprising. Not always reliable.

As an aside, I thought “Oh, maybe that’s what they were going back and forth about at church. It’s just like Math.” If our minds develop this way as we learn Math and Science, why wouldn’t they develop this way spiritually too?

Posted by: Rolfe Schmidt on June 24, 2007 11:47 PM | Permalink | Reply to this

R.I.P. John “Jack” Todd; Re: Faith and Reason

I don’t know what thread this belongs in, but the man who taught me the beauty of algorithms as mathematical objects, in the late 1960s, has just died. He was a friend of Turing, and a walking history of 20th century Math/computers..

Todd, former Caltech math professor, dies
By Elise Kleeman Staff Writer
Article Launched: 06/25/2007 09:29:37 PM PDT
http://www.pasadenastarnews.com/news/ci_6227870

PASADENA - John “Jack” Todd, a Caltech professor emeritus and one of the pioneers of 20th-century mathematics, died June 21 at his home in Pasadena. He was 96.

Todd, who started his career in the days before computers or hand-held calculators, specialized in
understanding how to find numerical answers to complicated equations.

“The methods that he developed to solve all kinds of equations had to be really, really efficient because you couldn’t just punch a few keys on a computer,”
Gary Lorden, a Caltech mathematician, said Monday.

Now, although most people use computers to solve complex math, “what goes on behind the scenes is very much the application of the kind of mathematics that Jack developed,” Lorden said.

Todd was “a very fine gentleman of the old school,” Lorden said.

In a statement, Caltech officials said Todd was born in Ireland in 1911 and grew up near Belfast, Northern Ireland. He earned his bachelor’s degree from Queen’s University, Belfast, in 1931 and attended Cambridge University for graduate studies.

At King’s College in London he met and wed Olga Taussky, one of the most prominent female mathematicians of the century.

“They just loved mathematics - that was the center of their life, that was their great love,” Lorden said. In 1939, when Britain declared war on Germany, Todd took a post with the British Admiralty, studying ways to protect ships from
enemy fire.

In his Caltech oral history, Todd - referred to by some as the “Savior of Oberwolfach” - recalled his wartime rescue of the Mathematical Research Institute at Oberwolfach in Germany as “probably the best thing I ever did for mathematics.”

Near the war’s end, he and his colleagues investigated rumors that mathematicians were being held as prisoners of war in Germany’s Black Forest. There, they discovered that the University of Freiburg was protecting the mathematicians at the institute. Todd
claimed the building for the Admiralty and prevented Moroccan troops from destroying the school and its work.

He and Olga Taussky-Todd came to the United States in 1947 and took posts at Caltech in 1957, where he remained a professor until his retirement.

She was the first woman to receive a formal Caltech teaching appointment, and, in 1971, a full professorship. She remained active in research until her death in 1995.

The couple, who lived simply and saved their money, donated a seven-figure endowment to Caltech to support future generations of mathematicians.

Services have not been announced.

elise.kleeman@sgvn.com

(626) 578-6300, Ext. 4451

===========================

Posted by: Jonathan Vos Post on June 26, 2007 7:15 AM | Permalink | Reply to this

Re: Faith and Reason

Polanyi’s talk of faith’s function in science seems almost naive in today’s scientific world, increasingly dominated by micro-incremental scientific research, small and large scale professional management, and reliance on funding. Maybe a return to the ‘naive’ is called for.

Dennis

Posted by: dennis on June 29, 2007 10:40 AM | Permalink | Reply to this

Re: Faith and Reason

Gauss invokes Grace:

Finally, two days ago, I succeeded - not on account of my hard efforts, but by the grace of the Lord. Like a sudden flash of lightning, the riddle was solved. I am unable to say what was the conducting thread that connected what I previously knew with what made my success possible.

Quoted here.

Posted by: David Corfield on October 14, 2008 10:50 AM | Permalink | Reply to this

Re: Faith and Reason

cf. Poincare’ getting on the bus

Posted by: jim stasheff on October 14, 2008 1:22 PM | Permalink | Reply to this

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