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What If They Denied the Existence of Math?

June 13, AD 2013 63 Comments

When I wrote “What If I Denied the Existence of Science?” I was pointing out a contradiction. Science itself relies on abstraction, it is a body of knowledge not a material thing. It cannot walk right up to you and say “Howdy!”

Well it turns out people do question the existence of another body of knowledge — math. Thank you Mrs. Mary C. Tillotson, who writes at Ignitum Today, for sending me the video. It was posted at The Atlantic and played on PBS’s YouTube channel.

Here’s the short summary: Biology studies living things. Chemistry studies chemicals. Physics studies physical things. Math? Math just studies math. Math is the abstract study of something abstract, something disembodied from the material world. Therefore, some people say that math is just something we created, and is only meaningful in the story of human life — that is, it doesn’t really exist.

The video commentator mentions Alain Badiou, a professor of Philosophy at the European Graduate School in Saas-Fee, Switzerland. I cannot follow his philosophy, but this is what he says about truth, from his biography page.

As Alain Badiou explains in detail in his major work to date L’Etre et l’événement(1988), truths are militant processes which, beginning from a specific time and place within a situation, pursue the step-by-step transformation of that situation in line with new forms of broadly egalitarian principles. Only a pure commitment, one detached from any psychological, social, or ‘objective’ mediation, can qualify as the adequate vehicle for a truth, but reciprocally, only a properly universal truth qualifies as worthy of such a commitment. Only a truth can ‘induce’ the subject of a genuine commitment.

If I put my one-liner goggles on, I think that says, “There is no truth, we just think there is.” He claims to be an atheist and a communist. No truth but man’s truth.

I hear scientists say that too. They even say that science is not a search for truth.

On the contrary, I’m reminded of one of my favorite (and kind of funny) St. Thomas Aquinas explanations, Chapter 4 of Summa Contra Gentiles where he talks about what “awkward things” would happen if the search for truth were solely left to human reason.

What does St. Thomas say? He says that reason alone can only go so far in a lifetime, that ultimate answers about God surpass the ability of human reason. We needed Divine Revelation so that we could know and love God better. He names “three awkward consequences” that follow if someone searches for truth without searching for God.

Quick aside: St. Thomas uses the words “search for God,” but before this part he also explains (one-liner goggles again) that if you are searching for truth, you are searching for God because God is truth, which probably explains the dislike for the word “truth” among some atheists.

First, he says, few men would have any knowledge of God. A lot of people do not have the “physical disposition for such work” and are “naturally not fitted to pursue knowledge.” No matter how much they try, they wouldn’t be able to get very far. Others are cut off from it by the “necessities imposed upon them by their daily lives.” People have to work and cannot spend an entire lifetime in intellectual pursuit. There are also, he says, some who are “cut off by indolence,” which is his way of saying that some people are just too lazy to “labor for the mere love of knowledge.” They do not have that “appetite” for knowledge.

Second, he says, that the few who would come to discover knowledge would “barely reach it after a great deal of time.” Why? Because the truth is profound and the human intellect is weak. It needs training. In youth the soul is “swayed by the various movements of the passions” and isn’t ready for such high knowledge. He quotes Aristotle, “One becomes wise and knowing in repose.” (Was it Aristotle who said that we don’t reach prime intelligence until after the age of fifty?*) If the way open to knowledge of God is limited to the few who can work that hard and who spend a lifetime to reach it, then “the human race would remain in the blackest shadows of ignorance.”

You can’t argue with him, though I know some young people who would.

Finally, he says, the third effect is that with human reason alone we risk going astray. There is “falsity present” within our reasoning due in part to weakness in intellectual judgment and in part to the “admixture of images.” What is that? It’s imagination. We let our imaginations run wild too easily. The result is that many, “remaining ignorant of the power of demonstration,” would doubt what is true and believe what is false. People are capable of making sophisticated-sounding arguments and all too willing to neglect an intellectual examination of the demonstrations. This is why we needed “unshakeable certitude and pure truth” to be presented to us, to help us.

In other words, if faith guides our search for truth, we’ll get much further.

And that, I think, is exactly what happened in the video. Math is abstract, but if you approach it having accepted the truths of faith, then math makes sense. Math is beautiful, math is something you expect, discover, and put to use, “…but thou hast ordered all things in measure, and number, and weight.” (Wisdom 11:21) If you approach math without faith and dig into the deeper questions proposed by the imaginative man in the video, then bam, you might just run into a logical brick wall, and find yourself asking whether math really exists.

Tagged and filed accordingly.

*Yes, it was Aristotle. If you want to have some fun, go read what he says about it. Physics, Book Seven, XVI.

Hello, and thank you for reading. My name is Stacy Trasancos. I am a wife, mother of seven, and joyful convert to Catholicism. I write from my tiny office in a 100-year-old restored Adirondack mountain lodge that overlooks a small spring-fed lake. Read more about me here. Find me on Facebook, follow me on Twitter, or contact me by email. God bless you!
  • Luke Arredondo

    Nice!

  • Mary C. Tillotson

    Thanks, Stacy! I knew you’d have something interesting to say about it :)

  • Jeff_McLeod

    Absolutely. Back in the 20th century it was easy to zing the nonbelievers with math and science as meta-physical systems.

    Nowadays, the blokes look at you like you’re crazy. They wave you off with a sort of “well we just KNOW that math is true.”

    Oh dear, how they would have been treated during the middle ages (the actual renaissance).

    Scholars would have looked at them and said “what???” You don’t just “know” math is true. You have critical reason.

    Nowadays, our impoverished education system has reduced math to something that is true “because my college professor said it’s true.”

    And they whine that WE base our lives on FAITH?

    The good news is, we have a greater opportunity to convert people with limited exposure to the truth.

  • Howard

    Good analysis Stacy.

  • Jeff_McLeod

    Reason 3 given by Aquinas is heavily supported in cognitive psych research. It is in fact one of the hottest research topics in the field, understanding how formal reason is corrupted by imagery and associative learning.

    There is evidence that scientists and scholars are in fact more susceptible to the errors of Aquinas’ reason number 3 than uneducated people because scholars are overconfident in their intelligence and are thus likely to trust their first impulse.

    A reasoning problem like the following is often used:

    Boy bought a baseball bat and ball for $1.10
    The bat cost $1.00 more than the ball
    How much did the ball cost?

    This paradigm triggers the fast-but-false response: the ball costs 10 cents.

    Even PhD’s get it wrong because we rely on the visual imagery of the problem. We invest too much in the 2 pieces of numerical data.

    The ball does NOT cost 10 cents. There is an answer, and it is found using 8th grade algebra.

    This is exactly what Aquinas is talking about by the admixture of imagery.

    • http://stacytrasancos.com/ Stacy Trasancos

      OOOHHH! You’re right. I had to pull out my kids’ white board and a dry erase marker to work it out. It seemed obvious that it was 10 cents.

      Your last sentence sent me on a search for the Latin. It is phantasmatum permixtionem. I have to stop myself and do some other work, but I seem to remember that word also being used in describing angels. They don’t have imaginations since they don’t have bodies (senses) and thus their knowledge is immediate and pure.

  • Ib

    Your reading of St. Thomas on revelation is good, but you mistake him about Maths. Indeed he would most likely laugh at your reduction of math to a truth of the faith. What he actually held is elaborated in his commentary on Boethius’ De Trinitate. Maths and other logical truths are what is called second intentions. They are mind-dependent (ens ratione), but objectively true since they derive from First or generally accepted Principles by way of reason. So 5+7=12 not because of some Platonic Number mysticism, nor because of Kant’s synthetic apriori, but because our reason can make true judgments about things in general, including Maths. This stems from Aristotle’s approach to how we come to have knowledge in the first place. For an Aristotlean, Maths are in one sense discovered, but in another invented. The great Aristotle scholar, Hippocrates Apostle, wrote a book on this just before he died, “Mathematics As A Science of Quantities”. I highly recommend it.

    As for Badiou, he is one of the last gasps of French Postmodernism which is even disdained in France at this point. No need to spend much time on him.

    • Jeff_McLeod

      Stacy didn’t say Aquinas was a Platonist. She interprets St. Thomas correctly.

      You are right the truths of mathematics are a both created and discovered, but only from the aspect of the knowing subject. Math is objectively true but we must come to know it through the senses. Thus, we grasp the concept of number very early on in life by seeing “one cow” and “two chickens” and “two goats” etc. We need raw sense data. But from the raw data we form the concept of number and we come to know the objective reality of number because God created us to know the truth.

      I’m struggling to find out where Stacy said or even implied Aquinas was a Platonist or worse yet a Kantian?

      • Ib

        You misread me. I never said she was a Platonist or a Kantian. But she does attribute the truth of Maths to the holding of the faith … St. Thomas holds something completely different. I encourage you to read his commentary on Boethius, not just for this point, but for the very deep insights he has into the nature of rational investigation (what we would call the sciences in our times).

        • Jeff_McLeod

          You held up 3 positions,. Aquinas, Plato, and Kant. You said Stacy was not true to Aquinas. That leaves two alternatives. I don’t see a shred of Plato or Kant in what she said. That’s all.

          Nevertheless I think I see what you are saying but I think you have left out a segment of Aquinas’ argument.

          As you correctly said, the reason we can know math as necessary truth is because our reason can make true judgments in general.

          Absolutely yes.

          Why can our reason make true judgments in general? Because of an accident (i.e., we “evolved” to make true judgments?) or because God so loved us that he created us to know the truth?

          This is the heart of the matter.

          We know math because God so loved us as to desire we know the truth.

          To the pagan who denies this, mathematics is indeed a game. It can be nothing other than a game.

          Right?

          To the pagan, “I believe X” is a tenuous statement that bobs in the sea of doubt.

          To the Christian, “I believe X” rests on the foundation that “God created me to know the truth, therefore I can trust my cognition in believing X”.

          Are these not 2 distinct kinds of belief?

          The only way the pagan gets to trust mathematics is if he acknowledges God. He can’t do that. So he places his faith in the proxy of mathematics.

          I.e., the pagan believes in math by faith in (self, evolution, magic, etc.) Without God, what other ground is there?

          • Ib

            Jeff, I completely understand your generous desire to defend the author, but please read carefully to discern what someone has actually written, rather than what you may feel they wrote. I did mention three philosophically important approaches to the foundations of mathematics explicitly, and referred obliquely to a fourth, but only in the last case did I attribute the position to Ms. Trasancos. From her comment, it turns out I read her rhetorical “if you accept the truths of faith …” as a statement of her position, not as a hypothetical. But see my comment below on that misreading.

            Now about pagans and Christians: by pagan you seem to mean contemporary non-Christians and not the more technical use of the term to refer to polytheists of antiquity. That’s a good choice, because the ancient polytheists had no doubts about their faith. Read through some of the texts of the Ancient Near East or the pre-Socratic Eleusinian texts. You will never find any wavering doubt about the truth of their beliefs. Indeed, it is only among a scattered few ancient philosophers that any doubt concerning polytheism comes to our attention (see Lucian’s Herotimus for the paradimatic example). The vast majority of people who lived in the climate of ancient paganism seem unafflicted with modernist doubt.

            But even with contemporary non-Christians, there are many approaches to the foundations of mathematics. Conventionalism (as championed by Wittgenstein, Quine and the notable Alonzo Church), Formalism (put forward by Russell, Carnap and the early Tarski), Platonism (Kurt Gödel and many actual working mathematicians), Intuitionism (Poincare, Kronecker, Brouwer Heyting) and Constructivism (Nelson Goodman, Andrey Markov). None of the mathematicians and philosophers proposing these approaches sees any doubt in their particular approach. Indeed they see their arguments as grounded in reason (or logic, as most of them would put it). And each one has some good things to be said for it, and some major weaknesses. But for example, the weaknesses of Platonism, didn’t stop Kurt Gödel from being a life-long proponent of it. He had no doubts at all.

            Of course, none of them looked to Aristotle or St. Thomas since they were all children of the Enlightenment, which meant they were either materialists, platonists, or sometimes both at the same time! For Aristotle and St. Thomas, truths of mathematics were mind-dependent things (ens rationis). In our univocal modern way (derived from the Enlightenment) we want to shoehorn everything into materialist or platonist categories. In the medieval synthesis, there were many more ways that a thing could be than simply either made of matter or divine. In particular, ens rationis was an important way that something could exist. A great book explaining this is Jacques Maritain’s The Degrees of Knowledge. I highly recommend it. Another more advanced book which goes into this in great detail is John Deely’s Intentionality and Semiotics.

            Does a contemporary non-Christian believe in math by faith? Some do. Others would argue that they found mathematics on reason. Most mathematicians and scientists are of a basic pragmatic nature and just have never thought about it. This question doesn’t interest them.

        • http://stacytrasancos.com/ Stacy Trasancos

          “…she does attribute the truth of Maths to the holding of the faith.”

          No I do not. I said if you approach it having accepted the truths of faith, then math makes sense. You expect there to be order in the created world, and when you find it you apply math with no contradiction in epistemology.

          All the same, it’s nice to *meet* someone who appreciates St. Aquinas. Thank you for the reference to his commentary on Boethius’ De Trinitate. Is this it? http://dhspriory.org/thomas/BoethiusDeTr.htm#53

          • Ib

            Sorry, Ms. Transacos. From this comment, it turns out that I read your rhetorical “if you accept the truths of faith …” as a statement of your position, not as a hypothetical. For that misreading, I apologize.

            It’s wise of you not to hold this, since it would be hard to defend historically. Many Christians who accepted what they held to be the “truths of faith” have attacked reason itself (not just maths) as being incompatible with true faith. Even Martin Luther couldn’t help ridiculing it. Bishop Berkeley poked fun at Newton’s notion of the infinitesmals. I can’t think of any recent Roman Catholics who have done so, but I suspect some troll will trot out the tired canard of Hypatia and Peter the Reader. Sigh.

            The truths of mathematics are mind-dependent truths, and so depend not on revelation, but on the working of the human rational soul, apprehending, abstracting, and separating (or judging). As such, the “truths of the faith” don’t come into the actual doing of maths by mathematicians. [Aside: I write this as a Roman Catholic "paleo-Thomist" (as Jacques Maritain called himself), but Platonist Christians (such as many of the mathematicians in the 16th century circle of Isaac Barrow) would still see maths as coming from reason, not revelation].

            Yes, the link you have discovered is St. Thomas Commentary on Boethius’ De Trinitate. I was unaware that it was online. It makes sense for the DSPT to put it up, though. It’s strange that they have two rather different translators. I myself recommend the Maurer translation (which is what the DSPT text is for questions 5 & 6). One thing the online text lacks are the extensive footnotes providing the historical and philosophical background to the text. Get the book if you can: Aquinas, “The Division and Methods of the Sciences,” trans. Armand Maurer, Pontifical Institute of Medieval Studies.

            Jeff (above) argues that the human soul is created by God, so any truths of reason must be divine truths as well. St. Thomas doesn’t go so far, since we must recall that it is the work of the rational soul which derives truths from First or generally accepted principles. But the generally accepted principles, may be just-about-anything. That’s why we can have both Euclidean and non-Euclidean geometries, set theories with or without the Continuum Hypothesis, etc. All of these have theorems deduced through the use of reason, but starting with different generally accepted principles.

          • Jeff_McLeod

            Sorry I seemed to jump earlier to the defense of Dr. Trasancos (she is a PhD Chemist). She is definitely a dear friend, but she could run circles around me in theological discussions! She does quite well on her own. I know that!

            I am responding here because I am utterly surprised that you claim St. Thomas did not believe we can attain the truth in this life.

            I say this in the best spirit possible, but I wonder if you are inclined to assimilate certain views that are not Platonist toward Platonism because you find Platonism objectionable? I fully understand St. Thomas was not a Platonist. But he did teach that we can and do know the Truth because we were created to know it. Did he not? We have a difficult time coming to the truth, but our intellectual faculties are cut out for the job. What you are saying goes against much of what I was taught about St. Thomas.

            Any clarification you could give would be much appreciated.

          • Ib

            Jeff, I never wrote anything even remotely close to “St. Thomas did not believe we can attain the truth in this life”. In fact, in my first comment I wrote “our reason can make true judgments about things in general, including Maths.” Knowledge comes from true judgments (or separations, St. Thomas used these interchangeably). That’s how we know the form of an entity, whether an ens rationis or ens reale. We come to know the truth of a logical argument and the truth about some mind-independent thing by the same process of reasoning to true judgements about them.

            Thomists have always believed, as did Aristotle, that the mind easily makes true judgments about both mind-dependent and mind-independent entities all the time.

            One of the problems is that science as we practice it today bears only an attenuated relation to what the medieval natural philosophers thought science was about. Our way of practicing science was mostly forged in the Enlightenment, precisely by people who for the most part consciously rejected the medieval synthesis (with its reliance on Aristotle). Abandoning Aristotle and St. Thomas, and under the deleterious influence of nominalism, they went back to Idealism (DesCarte, Isaac Barrow, et al) or a radical empiricism (Francis Bacon, Locke). Because we “stand downstream” as it were from these thinkers, we take in much of that Enlightenment way of thinking. Retrieving a Thomistic approach to mathematics and the sciences has been the goal of many Thomists throughout the past century.

            These topics get complex very quickly and cannot be answered in a comment on a blog post. Again, the best book on this topic is Jacques Maritain’s “The Degrees of Knowledge.” Which, again, I highly recommend. I know it’s a lot to ask, but all I can say is “tolle, lege …”

            With all respect,

            Ib

          • Ib

            I will mention one other book, even though it is not exactly on-topic. It is an attempt to bring present-day science into dialogue with present-day Roman Catholic theology. I think readers of this blog will find it rewarding:

            “How Science Enriches Theology” by Benedict M. Ashley O.P. and John Deely

          • catholicscholar

            I think I’m with Paul Rimmer. I’m actually not too badly versed in the literature in this area and I have no idea what you are trying to teach us.

            I’m not complaining. I think you are clearly very specialized in some aspect of this problem but you seem very impatient with those of us whose PhD’s are not in this area.

            I respectfully bow out with Paul Rimmer :)

            Bless you Paul. I should have taken your lead.

          • Ib

            Sorry if I have come across as impatient. It’s probably due to the fact that I always feel frustration trying to provide enough detail to make a comment worth making, while having to type it into a smallish comment box … I do apologize.

            Briefly, the point I was originally making was that the Thomist position on mathematics was not reliant on revelation, but was a part of its account of reason and the rational human soul.

            The rest of my comments were all made to answer specific questions or challenges to what I had written and to apologize for my mis-apprehension of Ms. Transacos’ use of a rhetorical question as somehow implying where she stood.

            I am a mathematician, Deacon in the RCC, and a Dominican friar, so I guess I can get carried away on these topics which are in my areas … Thanks for your patience with me!

          • Jeff_McLeod

            Hello lb — I had my comment posted under “Guest”.

            Thank you for slowing it down a bit for me. What you just wrote helps a lot!

            I’ll stop interpreting and just listen more closely to you for a while! Please do stay around here with us. It’s a lovely community.

          • http://stacytrasancos.com/ Stacy Trasancos

            lb, I think I speak for the rest of us — we would love for you to continue to discuss these things. I certainly don’t mind being corrected, I welcome it, and I’m interested in your explanations. I sympathize with the challenge of trying to put complete thoughts into a comment box, but if you’ll keep doing it, we’ll keep reading. Go ahead — get carried away! A mathematician, Deacon in the RCC, and Dominican friar? Yes, we’re listening. :-)

            And, it’s Stacy. I appreciate the show of respect though. Thank you.

          • Ib

            Oops! I never thought I was correcting you (like some old crotchety schoolmaster), but just clarifying what St. Thomas actually thought. Thank you for the kind welcome to your blog. I had not known of it until I encountered a link to this post on the New Advent website. SInce it concerned a question that I have pondered for over 30 years, I thought I’d come over and read the post. Then I got sucked in … I usually refrain from commenting on blogs …

            I very much like what you’re doing here and hope I can add something to the conversation. Thanks for doing this! It is a very important ministry!

          • http://stacytrasancos.com/ Stacy Trasancos

            Your paragraph that begins, “One of the problems is that science…” is excellent. Having first studied science, that is exactly what blew me away when I started studying theology, which required studying philosophy and the history of science.

            “Retrieving a Thomistic approach to mathematics and the sciences has been the goal of many Thomists throughout the past century.”

            Aeterni Patris! Thank you Pope Leo XIII.

          • Howard

            lb, I think the line that Jeff responded to also startled me. Startled is the correct word that described my reaction.

            “…so any truths of reason must be divine truths as well. St. Thomas doesn’t go so far,..”.

            I was thinking of “Treatise on The One God” (Q16,A5)” at the time plus scripture. In the larger context God is truth, revealed directly by Him or revealed by human thought.

            Now I welcome being called chivalrous and admit having been so with Ms. Trasancos, but, Dr. Trasancos is very knowledgeable about principles of the faith – Jeff is a guy!

          • Ib

            Howard, you are using the word “revealed” in a way that blurs the Thomistic distinction between ordinary knowledge (obtained through the ordinary use of human reason; see Maritain’s book for a detailed account) and revelation. St. Thomas would not say that something was “revealed by human thought.” Revelation comes directly from God and cannot be obtained by human reason alone.

            Not all truths of reason are divine truths, since as I wrote before, we usually have some generally accepted principles (such as an axiom system) which can be relatively arbitrary. In mathematics, we mostly are careful that these generally accepted principles don’t contradict the fundamental first principles of reason, but outside of that condition they can be fairly arbitrary. Again, that’s how we can have a variety of geometries based on different axiom systems with absolutely no problem of contradiction. This is one of the great strengths of the Thomistic-Aristotlian approaches to maths (unlike the Kantian approach which pretty much refuted itself by staking itself to Euclidean geometry). Within a particular framework of First and generally accepted principles, we can reason to valid conclusions (i.e., true judgments). But these are ens rationis, mind-dependent being, not divine realities.

          • Howard

            lb, I am aware of the importance of the word “reveal” in Christianity,
            if you prefer then substitute “made known” in my second usage.

            “we can reason to valid conclusions (i.e., true judgments).”

            I don’t see how a “valid conclusion” and a “true judgment” CANNOT be due to God himself. You may not need His temporal aid in deriving to those conclusions, but the mechanism was given to you by Him.

            I refer to

            “Whether God is truth?

            I answer that, As said above (A[1]), truth is found in the
            intellect according as it apprehends a thing as it is; and in things according as they have being conformable to an intellect. This is to the greatest degree found in God. For His being is not only conformed to His intellect, but it is the very act of His intellect; and His act of understanding is the measure and cause of every other being and of every other intellect, and He Himself is His own existence and act of understanding. Whence it follows not only that truth is in Him, but that He is truth itself, and the sovereign and first truth.”

            My whole objection during this line of reasoning is the same
            one that is in the Descartes discussion. If we eliminate God (Theology) from our Philosophy we often end in a muddle. Our intellect is powerful but not without God. I do not deny the intellect has a function, I deny that God can be removed from that process without accepting sin as good.

          • Ib

            Actually, it’s not about removing God at all, in any way, but in considering what belongs where in the order of Being. It’s important to remember that — in stark contrast to Enlightenment ways of thinking — Thomistic treatments of truth, oneness, goodness, being (Aristotle’s four transcendentals) are analogical. Analogical treatments are neither univocal nor equivocal (which is the trap Enlightenment thinking falls into), but connect the substances or terms through an analogical relation. God is Truth, but not the truth that “I have coin in my pocket”. But these two do have an analogical relation (at least through remote causality). Mutatis Mutandis for many truths of reason, including maths. A great book introducing the authentic thought of St. Thomas is Fr. Benedict Ashley’s “The Way Toward Wisdom”. The late Fr. Ashley was one of the leading Thomists of our time, winner of the Pro Ecclesia et Pontifice Medal conferred by John Paul II. I am very glad to have been able to have him as a mentor. And I highly recommend his book.

            Of course, it is not necessary to be a Thomist and still be a faithful Roman Catholic. From what you’ve written, it seems to me you have a greater affinity for the generally more Platonic philosophy & theology of St. Bonaventure. Most Franciscans adopt his approach, even today! But of course after Aterni Patris, as Tracey notes, the majority of faithful Roman Catholic scholars have followed Aquinas.

          • http://stacytrasancos.com/ Stacy Trasancos

            My first theology course in 2010 was Philosophy for Theologians and it was an online course through Holy Apostles College and Seminary. Fr. Ashley gave the lectures on the DVDs. I can’t really say I studied under him I guess, but I feel like I did. I typed every word he said in the 6 hours of lecture and I promise you — I learned more about science from him than I ever did getting a PhD in chemistry. Maybe I learned more details and how-to’s in the lab, but Fr. Ashley taught me why we even do science and where we should be headed with it, or at least that we should be thinking in that larger view. He taught me why science and philosophy lost its way and why it needs to get back to the scholastic approach.

            Fr. Ashley studied under Mortimer J. Adler, and his friend Max Weismann comments here occasionally.

            Sorry that’s off topic, but it is exciting to me and such an honor to meet people who knew someone I so admire. I would have loved to meet Fr. Ashley, but I probably wouldn’t have gotten a word out. I do plan to read more of his books as soon as I can. You’re right, he is very readable.

          • Ib

            That’s super! I studied under him for my grad degrees in both philosophy and theology. He was on both my grad committees. I took a tutorial class from him on Moral Theology as well as a couple other classes. He was a very kind person, admired and loved by most of those he interacted with. Pray for him!

          • Howard

            I do appreciate your reply. I will add Fr. Ashley’s book to my reading list.
            Yes, my early reading in philosophy I was drawn to Plato but no real Catholic theology/philosophy until recent years – hence not much Aquinas. And I do admit to knowing a brilliant local Franciscan OFM and listen carefully to almost his every homiletic word.

          • Ib

            No worries. No one HAS to be a Thomist. My comments started out (as I wrote to Tracey above) as an attempt to clarify just where St. Thomas stood in his approach to maths. If the discussion had been about where St. Bonaventure stood, I hope I would have been just as anxious to clarify his position (as far as I know it; I only know his thought from general theological and medieval studies, I am not a specialist in his thought).

        • Jeff_McLeod

          Hey lb, I wouldn’t be surprised if I misread you.

          I will look at Thomas’ commentary on Boethius. I can tell by your passion that it must be very good.

          I also hope I don’t come across as argumentative. It sounds like I missed something in the video that stimulated your comment.

          Please hang out here with us. We love those who love St. Thomas.

          • Ib

            No problem. I hate writing comments and often don’t do it simply because saying anything nuanced takes a lot of work. It’s easy to squeeze off a few words which don’t really convey what I’m trying to say, but I just don’t have time to write a long comment … Thanks for hanging in with me …

      • http://stacytrasancos.com/ Stacy Trasancos

        Thank you Jeff. Maybe the confusion came from the short summary of the video at the beginning.

        I realized after reading what you wrote that math is actually a perfect example of Aristotelian epistemology, gathering data with the senses and reasoning about it with the intellect.

    • http://stacytrasancos.com/ Stacy Trasancos

      “As for Badiou, he is one of the last gasps of French Postmodernism which is even disdained in France at this point.” That’s good to know!

      • Ib

        Badiou is retired at this point, being 76 years old. The average life expectancy for men in France is 78.2 years. The reason he has become more noticed in his 70s is that all the major figures of French postmodernism have died (Lacan, Derrida, Lyotard, Foucault, Deleuze, LaPlanche), leaving the second string to puff out the last gasp of postmodernism. No one beyond a small clique of “professional postmodernists” pays these thinkers much heed.

  • Howard

    “I hear scientists say that too.” Really? I mean, are you in fact within earshot of scientists? I actually *am* a scientist — I teach physics at a mid-sized state university — and this is not the least bit typical of what I hear from my colleagues. I’d wager a good steak dinner that it would be easier to find professors of theology — yes, even in many of the larger “Catholic” universities — who think that “there is no truth” than professors of physics.

    Of course, if you search diligently you may “hear scientists say that too.” You may hear priests say it, I am sad to say; you certainly may hear a grocer, or a plumber, or a mechanic, or an architect say it, since fools may be found in all of these professions.

  • Paul Rimmer

    Math seems to be the science of quantities. Quantities seem to be abstractions from physical things. Physical things exist.

    Maybe I’m not deep enough for this sort of philosophy.

    • http://stacytrasancos.com/ Stacy Trasancos

      You have your one-liner goggles on I see. :-)

      • Paul Rimmer

        I would be very happy to go deeper, but it will be uncharted waters for me. Fun, but maybe not very helpful for you.

        I think that all that math requires is the physical world and minds to think about it. The physical world has a form, and we can think about the form without thinking about the stuff that it is made of.

        Math is man-made. But it’s made out of something that man didn’t make. It also seems to be governed by something or Someone in a way that cannot be accounted for in terms of human imagination.

        • http://stacytrasancos.com/ Stacy Trasancos

          That is helpful though Paul. Man-made but to describe something man didn’t make. Your the physicist, you would know. Thanks for the article recommendation.

          • Paul Rimmer

            You are welcome. It is more fun to talk about this with mathematicians than with physicists, probably. Mathematicians span the gamut from Platonism to radical nominalism. One math professor I know thinks that math just exists in people’s heads, and he spends his time looking for fundamental contradictions. He thinks that, given the origin of math, there probably are some.

            There are also some people out there who think that the success of math in physics is one sign that we are part of a computer simulation.

            In this case, the common-sense approach of Aristotle (and Aquinas) seems to be much more believable. Less interesting, though.

          • Howard

            I am not a math person but it seems to me that
            1 = .3… (recurring decimal)
            is a contradiction!

    • Jeff_McLeod

      For those who may not know Paul, this is a deeply ironic statement.

    • Micha Elyi

      “Math seems to be the science of quantities.”
      Paul Rimmer

      There are mathematics that have nothing to do with quantities. Some mathematicians describe their science as a study of relationships. And I struggle to understand mathematics as simply abstractions of beer mugs, tables, and chairs.

  • Loreen Lee

    Hi. I won’t presume I understood the whole video/talk but it is contributing to a ‘personal’ ongoing quest. I understand the ‘personalism’ of Acquinas has been related to the ‘functionalism’ which ascribes a ‘reality’ and ‘being’ to mathematical ‘figures?’. In exploring this idea I too came against the contrast that one can believe in the existence of God/mathematics’, or just be a ‘fideist’ and stick to Pascal’s wager on the efficacy of the belief only. This distinction seemed to parallel your argument. Then I tried to equate angels to the subconscious and ran into the contradiction that angels as they are form and existence only without material ‘substance?’ that we couldn’t regard the neurons in our heads in any way, shape of form as angels or ‘devils!”. But then, I would also find it very difficult to substantiate my idea of ‘person’, or the ‘personal’, just as in the ‘concept’ of a personal god, for the different ‘mysterious’ characterizations of the triune Godhead present very different descriptions etc. to our understanding, as is what happens if we think we know what produces all of those behaviors, motivations not only within others but of ourselves. So persons, angels and math seem to have something in common as far as I can figures it!!!!!

    • Howard

      All of this muddle can be made very clear by simply understanding and accepting the divinity of Jesus Christ.

  • Pedro Erik

    Fantastic post. Reminded me Edward Feser (author of the best book on the “new atheism”). Congrats.

    • http://stacytrasancos.com/ Stacy Trasancos

      Now that’s a compliment. Thank you. I am a huge fan of Dr. Feser’s.

  • TITO

    At University I chose all my “Electives” in Math. My Math Professor was a famous Nuclear Physicist. He started our first course in Advanced Math by saying “All mathematics is based on a convention, this convention is that there exists a number we will call “1″; that this number has a successor which is made by adding a “1″ to the first number to produce a number we will call “2″, and so forth and so forth. If we do not accept that convention then there is no mathematics.

    • Ib

      Yes, Conventionalism is one way that 20th century Anglo-American analytic philosophy coped with the foundations of math question. But here have been many others as well. See my above comments to Jeff and Ms. Trasancos for some details.

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  • John Darrouzet

    Stacy, sorry I am jumping in so late to this very thoughtful discussion. Please let enter it and let me turn your “what if” question into an issue for decision, like Aquinas might do.

    Issue: WHETHER WE WANT TO DENY THE EXISTENCE OF “MATH”?

    I answer that I do not. Why? I bring to the discussion Bernard Lonergan’s book “Insight: A Study of Human Understanding.”

    Lonergan takes on insight as an activity with elements and heuristic structures of classical and modern empirical methods that include a broad-brushed approach to math and science. He urges us to study insight into insight and insight into oversight.

    To reach the notion of what is meant by a “thing,” including what we call “math,” whether it does in fact exist or not, Lonergan takes his reader through the canons of empirical methods, the complementarity of classical and statistical investigations, descriptions of space and time, common sense and its subject and common sense as object.

    To deny the existence of “math” as part and parcel of the understanding of human insight, even in common sense undertakings, is clearly starting from a foreign starting point.

    To judge insight as an activity in the area of math Lonergan cautions the need of reflective understanding, where the material element of math “is what we have named the empirical residue” and the formal element “may be designated by abstraction as enriching.” Some may want only to deal with the former in their investigations, others in the latter. Those of us who are commonsensical students of math want to deal with the actual element that “lies in the conjunction of the material and formal elements”. As I can personally vouch for, the process of learning mathematics is laborious, but so much of life as we know it depends on math, even at its most abstract. Consider what your life would be without a deep understanding of zero as a number. Is it merely a matter of convention or is there such a thing as zero?

    Writes Lonergan: “…it does seem to be true that the empirical residue does supply mathematics with samples of the type of stuff on which mathematical ideas confer intelligibility and order. FOR UNLESS THE MATHEMATICIAN IS INVESTIGATING THE PURE INTELLIGIBILITIES THAT AQUINAS IDENTIFIED WITH ANGELS, THERE MUST BE SOME MATHEMATICAL MATTER; AND SINCE THERE ARE OTHER SCIENCES THAT DEAL WITH DATA AS OF DETERMINATE KINDS, THERE REMAINS FOR THE MATHEMATICIAN THE EMPIRICAL RESIDUE OF ALL DATA.” (My emphasis.)

    Lonergan concludes: “…it seems possible to identify the basic propositions of mathematics with serially analytic principles. For there is a material element in mathematical thought, and it bears some similarity to the empirical residue in the data of empirical sciences….It follows that the mathematician is concerned to establish generally, completely, and ideally, the range of possible systems that include verifiable scientific systems as particular, fragmentary, or approximate cases….The principal difference in our approach is that it goes behind concepts and affirmations to the grounding act of direct and reflective understanding. From this feature there follows its dynamic character, for it contains an invitation to mathematicians to explore the possibility of setting up the series of deductive expansions that would do as much for other empirical sciences as has been done for physics.”

    While clothed in many words in the heart of a tough read, Lonergan’s insights into math serve as a cautionary note for all of us. For, according to the second half of Lonergan’s book, we cannot rest in our fight against the flight from understanding.

    While it may be necessary to first entertain the proposal that math does not exist, it is not sufficient to do so. To know but not to act is unacceptable.

    We must ourselves take action with a view to the future, not just the present or the past as choice-making and judgment-making entail, respectively

    We must make a decision within a future orientation that involves the exercise of our free will and not our surrender to necessity. Otherwise we will end up arguing in endless circles of hypothetical reasoning. (See http://pinterest.com/pin/137500594845599992/)

    When “math” does not exist, i.e. is not connected to the empirical residue of data in a real way, but only to fictional data imagined to exist via concepts, there are dire consequences to human beings.

    The financial crisis we recently went through and seem bent on going through again in the not-too-distance-future, may be traced in part, if not in whole, to the schemes of credit-default swaps and the mathematical algorithms that detached mathematicians have invented to maximize profits in the world of fiat money. The embedded oversight, rather than insight, becomes a “thing” no longer deniable. Moreover, as we have come to understand, there are not good angels in the details of these instruments of mathematical wonder, but devils.

    I want to affirm the existence of math as a matter of a decision. By dealing with your “what if” question not as a hypothetical or rhetorical one, but as an issue whose answer is not readily apparent, I invite those who want to deny the existence of math now to argue in support of their side of the issue and help us all try to reach fitting decisions.

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  • TristanVick

    As a non-believer I am comfortable knowing that concepts such as math and God serve specific functions, but that they don’t necessarily require referential objects to prove practical in their application.

    • Martin Snigg

      Mathematical discoveries often well precede any practical application. Relationship with God, or any love relationship is often experienced as profoundly lacking in ‘practical’ value, assuming some reductive meaning on your part. The Crucifixion of Jesus Christ is the classic example, caring for a dying spouse or child or any kind of loyalty that demands a human and self sacrificing response.

      I don’t know how you came to your philistinism, but I hope you have a positive healing experience in these places. Best wishes.

      • TristanVick

        The Resurrection entails an entirely advanced theology. That was not my consideration.

        I was merely talking about the God concept.

        Philistinism? I do not understand your meaning. Are you under the misguided impression that it is something that needs curing? Your healing quote makes it sound as much.

        If so, I also wish you the best of luck on addressing your problem of being condescending to those who do not automatically prescribe to the same belief sets as you do.

        If not, I do not follow your meaning.

        • Martin Snigg

          Mathematicians are generally some kind of Platonist, they certainly don’t believe that mathematical entities can be reduced to some kind of social function or human utility as if they were a product of human ingenuity. They know they are discovered.

          The point about the crucifixion I was making is simply that Jesus’ self understanding and personal sacrifice was motivated by his love of God the Father. His ‘concept of God’ wasn’t merely ‘practical’ and to talk of love at all in terms of its practical usefulness completely misses the most important and humanising dimension of the experience. If you deny the reality of these human experiences as they are experienced and the transformation of the human person and cultures in which they are experienced then you are what is known as a ‘philistine’, which is a real lack, and certainly requires healing.

          • TristanVick

            Well, since the Resurrection never really happened, I am safe from being a philistine.

            I thank you for your concern though.

          • Martin Snigg

            I think we’ll leave it at that.

  • Micha Elyi

    All maths begin with “Fiat” – in English we write “Let”. Without such a declaration of things to be assumed, whether explicitly or implicitly made, there is no mathematics.

    If one wants to call me a conventionalist for that, I don’t mind. Placing me in the company of Henri Poincaré is more honor than I deserve.

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