Monday, May 7, 2012

Mathematics the Handmaiden of Theology: Augustine and Cantor


Golly, I had thought I could proceed in a nice orderly sequence about belief, knowledge, the limits of science, but articles keep appearing that I have to discuss.   Here's one by Adam Drozdek (Associate Professor of Computer Science at Duquesne University) that has great insights on mathematics and its relation to theology: Beyond Infinity: Augustine and Cantor.   Although I'll try to summarize the main points of his article, I urge the reader to go to the original article for a detailed exegis.
First, here is  Drozdek's summary of St. Augustine's (Hippo) ideas about mathematics, infinity and God.
"To summarize, there are three important aspects of Augustine's discussion of the problem of infinity. First, infinity is an inborn concept which enables any knowledge. Second, infinity can be found in the purest form in mathematics, and thus mathematics is the best tool of acquiring knowledge about God. Third, God is neither finite nor infinite and his greatness surpasses even the infinite. Augustine is original in combining these three aspects in his philosophy ; some of them can be found in other philosophers and theologians, but also in mathematicians."
Augustine anticipates later developments in mathematics, the mathematics of infinity put forth in set theory:

"God's infinity would still be of a higher magnitude, an infinity of different kind. His infinity is above all possible temporal (and spatial) infinity ; it is an infinity of infinities, whose magnitude can be dimly imagined by means of mathematical infinity. It is an infinity of infinities also in that "all infinity is in some ineffable way made finite to God," since no infinity is incomprehensible to God (De civ. Dei), he can count numbers without succession of thought. God is even able to count without numbers, which assumes that there is no number equal to the quantity of all numbers, that is, no number, to use modern parlance, expressing cardinality of integers (which is aleph zero). This is no hindrance to God who is able to see the entire sequence of numbers without looking at these numbers one by one.  Infinity of these numbers can be grasped in one act of comprehension."

Drozdek points out that Augustine's view on God's infinitude differs from that of later Catholic theologians and philsophers--Roger Bacon, Thomas Aquinas, Duns Scotus--who emphasized the infinite nature of God.

"Augustine is an exception to this rule. To him, God encompasses infinity, himself not being infinite."
However, Augustine does have a worth successor, Georg Cantor (1845-1918) who proposed new ideas about infinity as a mathematically rigorous subject, by use of set theory.   Cantor's motivation was theological and philosophical:

"mathematical statements are not divorced from reality, and, for instance, set theory makes certain pronouncements about things in themselves, about 'true being,' and 'the general set theory [...] belongs entirely to metaphysics' and is its servant."
Like Augustine, Cantor believes that knowledge of infinity is innate:
"...abstract knowledge is already in us, implanted and dormant, enlivened by our quest for it. In particular, infinity cannot be recognized unless it is inborn, since infinity "even inhabits our mind (Geiste)".21 Therefore, mathematics has not only a purely theoretical interest, but it is also of philosophical and theological bearing."
God, according to Cantor is the Absolute (what in modern mathematics is designated by the Greek upper-case omega):
"But whereas different transfinite levels can be known (erkanni), the Absolute can only be recognized (anerkannt), not known, not even approximately ; however, an 'absolutely infinite sequence of numbers,' i.e., sequence of all infinities, can be considered 'a suitable symbol of the Absolute'.25 Set theory shows that there is no set encompassing all sets, and yet God is able to comprehend all these infinities, hence he is above infinity, he is the Absolute. The transfinite, unlike the Absolute, 'clearly appears to us as limited, capable ofbeing augmented and thus related to the finite'.  With this statement Cantor returnsto Augustine's conviction that 'all infinity is in some ineffable way made finite to God.' "
Like Augustine, Cantor believed that the concept of infinity is put within us, as a Divine implant:
"The transfinite numbers are not pure creations of our mind, they are only discovered in the mind and in the world. They cannot be our creations since they precede our very existence and the existence of the world. As Augustine, whom he quotes, Cantor believes that God utilized numbers to create the world."
My wife, who is NOT a mathematician, in reading all the above, recognized that there is at  the basis a Platonic philosophy, that is to say, an assumption that there is a reality to mathematical ideas that is different from the reality of concrete things, the world of sensation.   Other mathematicians (not all of whom are theists) are also Platonists, for example Roger Penrose, who proposes three worlds, the platonic (ideas), the physical, and the mental. (see The Emperor's New Mind )
If Drozdek's article stirs you up, you can also go to Rudy Rucker's fine book on the same subject, "Infinity and the Mind", which gives more mathematical detail than Drozdek's article.

Thursday, May 3, 2012

Fr. Barron on "Faith and Reason"

Here's a link to a video by Fr. Barron on "Faith and Reason".   Powerful and elegant!   As he puts it, we're all "Doubting Thomases" whom the Church leads to faith.

http://www.thinveil.net/2012/05/faith-and-reason-why-both-are-needed.html

Enjoy!   Ponder!   Spread the Word!

Tuesday, May 1, 2012

Science: a Subset of Rationality


This is from a review by Michael Potemra (see NRO, "The Corner", April 28,2012) about a new book by a noted paleontologist, Michael Asher: Evolution and Belief:Confessions of a Religious Paleontologist.   Potemra has written an excellent review, to which I can't add anything, but I'd like to quote his quote from the book, because it expresses my attitude about science and religion to a T:
“The absence of a scientific proof for God is more indicative of the limits of science than the lack of a deity. . . . Evolutionary biology is not about the origin of life or the existence of God. It is about how living things are interconnected through a specific, natural mechanism, one which we can understand through the fossil record, individual development, and molecular biology.”
and
“it is rational to believe that an entity beyond our comprehension was the agency by which something was derived from nothing at the beginning of time. . . . Although I acknowledge my belief to be non-scientific, it is entirely rational. Science is a subset of rationality; the former has a narrower scope than the latter. To ignore rationality when it does fall beyond the scientific enterprise would be an injustice to both reason and humanity.”
Most people aren't aware of the distinction between science and other rational enterprises--science requires theory (usually mathematical and linked to other theories, more basic and fundamental), confirmed or falsified by observation/reproducible experiment.
More of this later.

Thursday, April 26, 2012

Belief, Knowledge, Faith--Rational and Irrational





“Alice laughed: "There's no use trying," she said; "one can't believe impossible things." 

"I daresay you haven't had much practice," said the Queen. "When I was younger, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast." 

Lewis Carroll, Alice in Wonderland and through the Looking Glass 



In order to justify religious faith as a rational endeavor,  I will examine what we usually mean by belief and knowledge, and to show, by example, the difference between rational and irrational faith.   My motivation for proceeding from this starting point is to show that the arguments of evangelical atheists (those published and those posting on web-sites) ignore their own faith assumption in choosing science (or perhaps more properly the scientific method) as the only source for justifying belief and gaining knowledge.  From my background of 50 years as a practicing physicist, I will try to draw examples to show that faith is integral to a scientific world-view, as well as a religious one.

  • BELIEF  
    "Lord, I believe.  Help Thou my unbelief."  Mark, 9:24


Let's talk about belief first.   Clearly there is a difference between the statements "I believe in one God..." (the Credo) and "I believe it's going to rain tomorrow".   An obvious difference is what one is willing to do or to pay in order to act on one's belief.   The Christian martyrs were willing to suffer and to die for their beliefs;  you might be willing to bet five dollars that it will rain tomorrow, but not your life, no matter what the weather forecast is.

Accordingly, there are degrees of belief, which in fact can be quantified using various techniques in subjective probability and decision theory (see "Probability and the Art of Judgment" and "Subjective Probability--The Real Thing" by Richard Jeffrey).  To explore these methods in detail would require a book, not a blog, but those interested can go to the Stanford Encyclopedia of Philosophy articles .     

 Possibly the most famous example of quantifying belief and using outcome probabilities as a guide to action is Pascal's Wager, in which the argument is made that belief in God is the prudent choice, given the existence of an afterlife (even though the probability of that existence might be infinitesimally small).   Objections, most of which are substantial to Pascal's arguments, have been raised and countered (see Pascal's Wager Revisited--The Pearl of Great Price and Pascal's Wager--Insurance for Agnostics .)   Nevertheless, it is clear that we don't always act rationally in terms of our beliefs.   We will bet on the lottery, even if the odds--expectation value for a likely gain--are not in our favor.   We will take on insurance, even if, again, it will in the long run be a losing proposition.   As Pascal himself argued  (Pensees, #233), it is not always easy to believe on the basis of prudence and rationality:
I am so made that I cannot believe. What, then, would you have me do?”
Pascal replies:
“Endeavor then to convince yourself, not by increase of proofs of God, but by the abatement of your passions.   You would like to attain faith and do not know the way;  you would like to cure yourself of unbelief, and ask the remedy for it...There are people... who are cured of an ill of which you would be cured.   Follow the way by which they began:  by acting as if they believed, taking the holy water, having masses said, etc.”
Now can one “fake it until you make it” as Pascal suggests?  Or will the sacraments be ineffective, because the motive of the recipient is mercenary?  Which of the Catechism dicta are appropriate,
(1131)”The sacraments are efficacious signs of grace....They bear fruit in those who receive them with the required dispositions.”
or
(1128) “The sacrament is not wrought by the righteousness of either the celebrant or the recipient, but by the power of God.”

The second suggests that if one prays for faith, then the “top-down” approach will work, starting from the head  and eventually through to the heart, or, as Pascal suggests:
“...at each step you take on this road you will see so great certainty of gain, so much nothingness in what you risk, that you will at last recognize that you have wagered for something certain and infinite, for which you have given nothing.”
So we can see that belief is not an absolute, a two-valued yes or no, but a quality that is measured on a continuous scale, and that can be implemented by means that may be other than rational.


The next segment will deal the bases for belief, knowledge and faith,  and, in particular, what constitutes scientific knowledge.

See ya....
Duhem

The header picture shows again, that although empirical evidence (e.g. what's happened in the past) should rationally be a basis for belief and thus for action, we don't always follow through--we're all Charlie Browns at times.


yet another blog from a rational catholic scientist



“It is also necessary—may God grant it!—that in providing others with books to read I myself should make progress, and that in trying to answer their questions I myself should find what I am seeking.
Therefore at the command of God our Lord and with his help, I have undertaken not so much to discourse with authority on matters known to me as to know them better by discoursing devoutly of them.”
St. Augustine of Hippo, The Trinity I,8. 


This is to be a blog about the consonance/compatibility of science and the teachings of the Catholic Church.     If you ask why yet another blog about science and religion, I'll answer that I hope to bring a different perspective, as a late convert to the Church (at the age of 64, 18 years ago) and as a physicist (now retired after 60 years in academia and medical physics).

Being a physicist (since 1951), I should, according to popular opinion, be an atheist, or at worst an agnostic with no clear idea of whether God exists or that He acts in the world.   That opinion, given loud voice in the media and on the internet, is of course not correct.   There are many physicists, among them Nobel Prize winners, who are believers  (to be listed later) just as there are many who are not.

So, scientific achievement is not in itself a basis for crediting or discrediting belief in God, nor should it be on rational grounds.    There are intelligent people who are atheists, and there are intelligent people who are theists.    And it is not true, despite claims of evangelical atheists to the contrary, that one either lacks intellectual acuity or has to suppress one’s critical faculties in order to believe in God.

What then are the roots of faith, and in this context, by faith I mean belief in God?  The purpose of this blog is to explore (but not necessarily answer) this question in both a general and personal way.    To begin,  I offer a general apology (not apologia):  I am not a professional philosopher although I have done much undirected reading in this last decade.   What philosophical discourse I’ll attempt will be distilled from such reading and, of course, can be subjected to critical analysis by those more academically versed in philosophical arguments.

First, I’ll discuss what might be rational (and sometimes irrational) grounds for belief, particularly belief in God.     Next, I will give a personal account of my own (rather later) road to belief, which was, unlike St. Paul’s, a top-down conversion.    Finally, I will    examine what the world around us tells us about the existence and intervention of God, in both a scientific and supra-scientific context.

I will also try to show (as a quondam practicing scientist) the "Limits of a Limitless Science" (the elegant phrase used by Fr. Stanly Jaki) and, in particular, that my faith as a Catholic is entirely consonant with what science tells us about the world.

I hope comments will be constructive and, based on unhappy experience with evangelical atheists on Facebook and other web sites, I will be ruthless in deleting comments and barring posters who are rude, uninformed and clearly not seeking a meaningful dialog.


One word about the header picture:  the left hand image is that of William Blake's beautiful painting, "The Ancient of Days";  the middle image is that of the hypothetical digrammatic history/timeline of the universe, including its inflationary phase (the bell shape at the beginning);  the right hand image is a WMAP image of microwave radiation from the original creation event, the embers left of the initial gigantic energy of "the Big Bang", the observation of which is crucial evidence for that initial event.  Together these illustrate what a synthesis of faith and reason can achieve.

More will be forthcoming, at an irregular rate....Thanks for looking.
Duhem



About Me

My photo

Retired, cranky, old physicist.   Convert to Catholicism in 1995.   Trying to show that there is no contradiction between what science tells us about the world and our Catholic faith.   Intermittent blogs and adult education classes to achieve this end (see http://rationalcatholic.blogspot.com/   and http://home.ptd.net/~rkurland)

Extraordinary Minister of Communion volunteer to federal prison and hospital; lector, EOMC.
Sometime player of bass clarinet, alto clarinet, clarinet, bass, tenor bowed psaltery for parish instrumental group and local folk group.

And, finally, my motivation:
“It is also necessary—may God grant it!—that in providing others with books to read I myself should make progress, and that in trying to answer their questions I myself should find what I am seeking.
Therefore at the command of God our Lord and with his help, I have undertaken not so much to discourse with authority on matters known to me as to know them better by discoursing devoutly of them.”
St. Augustine of Hippo, The Trinity I,8.