I hate mathematics. I was terrible at it in both high school and college. Nevertheless, the more I read on the history of philosophy the more it seems to be the one sure rational approach to a “way of knowing”. I am not saying we can’t know something outside of mathematics but it does appear we cannot approach it on a rational basis alone.
Or am I wrong about this? Is there something as “sure” as math? Is there a relativity to math that I should consider?
the archives near Emmaus 
Often what one discovers as he or she enters the field of analytical philosophy is the predominance of mathematics. Several of the greatest philosophers were also mathematicians, in our terminology at least (e.g., Davidson, Russell, Wittgenstein, Quine, etc.). The role of mathematics is foundational in the study of Logic, and scholasticism certainly elevated our sensitivity to our epistemological relationship to math. Wittgenstein’s attempt to create a metalanguage is very interesting in that it parallels quantificational logic: in other words can ~◇A have the same meaning as A couldn’t be true. This attempt to reduce language to logical forms (i.e. Russell) does, without being too reductionistic, raise an interesting question: what are we to expect from language and knowledge. If we find in mathematics a level of epistemic certainty that moves us beyond skepticism, our tendency seems to be to claim that what is not mathematically grounded is therefore unknowable (JTB sense). However, as you also raise the question, why should the epistemological certainty of mathematical theories converge upon other systems of beliefs? And moreover what epistemological theory unquestionably asserts that such mathematical knowledge cannot be questioned itself.
Surely, an answer to such a question is not capable in such a sort space and in such a short time. However, I think your question raises an interesting corollary as it pertains to theology: in what system of logic shall we “do” theology? Should a priori reasoning (i.e. akin to mathematical logic) take precedence over a posteriori, or vice versa, or even should they both have equal roles in formulating theological knowledge? I think such a question is at the heart of the current discussions of post-foundationalist theology.
On these and related questions, I read with much profit R. W. Howell and W. J. Bradley (eds.), Mathematics in a Postmodern Age: A Christian Perspective (Grand Rapids: Eerdmans: 2001), and wholeheartedly recommend it.
When I took music theory, the reason I was good at it is because it kind of looked like math to me. A lot of counting and numbers was involved. There were also quite a few rules to form chords, scales, etc. But, as we continued to progress in learning music, we learned more and more that the rules can be broken with newer rules we were learning. And, in a sense, if it sounds good, you can do it, whether there is a specific rule or not (though there probably is some deep embedded rule).
I don’t know if this adds more confusion but, all’s to say, there seems some sense of relativity to music, which is a bit connected to math. And, as I remind our music team in the church, it is not all about getting down according to rules (though I challenge them to grow in their music knowledge). But music is very much ‘felt’ as well. You can sense what is good and right, and what isn’t.
Ok, enough from me.
Even math requires a certain set of expectations to work within. Ignoring these “rules”, one can prove silly things – like 1=2, or that rabbits are always female. So there is a sense where mathematics still depends on the methods used to practice it, just as in any branch of study.
Additionally, Godel suggests that even for consistent systems (let’s just call this a gross over-generalization), there are true things that are unprovable. And “complete” systems that cannot be proved consistent. The basic take-home of this is that it is not clear that math (and that would include logic itself) can prove all things, not even things within its own domain – on its own terms.
While Godel’s theorem does not apply to all systems, only those which match a certain set of assumptions, it begs the question of what might be proven “logically”. Should we discount entirely something that appears true, simply because we cannot prove it on first principles? Are we actually rational thinkers at all, or just approximation engines based on a set of internal rules which could in fact be inconsistent?
Mathematics is not a closed book. And it is not entirely clear it ever can be. It can prove or disprove possibilities, but it does not necessarily give us answers. That our “natural laws” appear to obey mathematical rules sometimes gives us a more-than-warranted certainty about our understanding of the universe. For hundreds of years Newton’s mathematically and empirically-validated theorems governed our understanding of the physical world – but when we take into account small details, we find he is wrong, and that the math is only an approximation of reality. The mathematics of physical reality that have replaced it – namely relativity and quantum mechanics are not even today consistent with one another. Nor the interpretations of what the math means about reality, the role of observation, time, etc.
So while math has a solidity about it, where you can prove one thing from another, we need to realize that math is not God. And math will not answer all our questions. It makes no claim to. More…what if mathematics’s main value is in driving us to new questions?
Logic is from the Greek word, and uses some basic math. Aristotle brought forth the discipline in philosophy. Also the medieval education had the trivium (Latin for three ways): grammar, logic and rhetoric.
Math cheats, because it’s thoroughly abstract. You can’t show me “two”. And two somethings can’t be summed together as completely as two “ones”. Likewise, no pair of somethings can ever be “equal”. We say “1 + 1 = 2” in purely abstract terms. But congruent triangles are not said to be “equal”, and even split-zygote twins are merely referred to as “identical”.
Scripture, however, tells us that God is “One”. And John’s Gospel says Jesus once claimed to be “equal” with God – “exactly the same as”. Mindblowing.
Counting systems are relative, btw – base 10, binary, etc – but equivalent values relate consistently in all systems, as far as I’m aware. That may be the most amazing thing about mathematics. Even though the ‘world’ of math is completely abstracted from our physical universe, its applications describe and predict real world phenomena with remarkable accuracy.
These characteristics of Math may explain why Philosophers think they can calculate aspects of God. Math’s on a higher plane. Mathematical order seems to govern the cosmos, even where its chaotic. But of course, or at least imho, God has never been summed up precisely by anyone’s Theological calculus.
Just another thought to bring some relevance to my pensées: I said, “in what system of logic shall we “do” theology? Should a priori reasoning (i.e. akin to mathematical logic) take precedence over a posteriori, or vice versa, or even should they both have equal roles in formulating theological knowledge?” How a seminary dean either answers this question or assumes his or answer will have ramifications on what is to be encompassed within an entire MA/M.Div/or Th.M program.
James: It would assume that a priori and a posteriori approaches would benefit us. I don’t think it should be an either/or though personally it is very, very difficult for me to do anything analytic. I am terrible at math and logic.
This is one reason why the more I hear of so-called “Continental” philosophy with narrative, symbols, and so forth the more it interest me because I can “understand” and “know” things this way and it seems to me that Christianity is birthed and presented as a narrative type of logic with a phenomenal way of “knowing”.
That being said, I am a novice to this whole discussion, so who is to know if I even know what I am saying!
Esteban: That book sounds interesting. I will have to take a look at it.
Scott: I never could figure music out either which makes sense since I don’t do well in math. There is part of my brain that hasn’t had much exercise over the years!
George: So are you saying that there is a sense in which mathematics is as unstable and unsure as other approaches to knowledge?
Bill: When you say “Math cheats because it is thoroughly abstract” is this kind of like some forms of theology (e.g. Trinitarian theology where we cannot show God, prove God, present God, yet we believe his nature based on various intangible principles)?
Brian, I have always loved math. I studied Mechanical Drafting and actually worked as a Draftsman for a number of years before switching careers. At one time in my life I was very good with Trigonometry, but today I can’t hardly remember a thing.
I have placed an emphasis on math with my son, as I believe that learning math especially the higher aspects of it such as Calculus is extremely beneficial for all other disciplines. He is on course so far, thank God. Math makes you think logically and teaches you to solve problems.
I don’t know about unstable, but “unsure” gives room for pause. It is not that we doubt the math, but we have no certainty in our own calculations. The application of math to reality (including logic and philosophy) has inherent difficulty as scale is dealt with. Thus, with either smaller or larger “measurements” we might find that our math no longer accurately represents reality, though it is sound in itself, and possibly even practical at certain scales.
Relating this and what I said before to philosophy or religion, there may be things that are true that are not provable absolutely, though there may be great evidence or consistency to support them. Things that require faith or “common sense”, appalling as that is to some of our sensibilities. Additionally, there may be truths that are verifiable or derivable at a certain scale, but that at either micro- or macro- scales we find to no longer work, to no longer represent reality.
Our assumptions about how reality works can also cause us to misuse math – just like we misuse other approaches to knowledge-gathering. Quite against our natural expectations, it turns out parallel lines can “touch” in the right mathematical framework, or maybe never touch, but not be equi-distant either. 1+1 = 0 in the right framework. Similar approaches to problem solving are also not always portable. Try to test the length of a line by subsection – with both triangles and half circles, and you will come up with different and contradictory results.
I love math. But though it has much descriptive power, it has its limitations.
George,
Would you say math is the most trustworthy of “rational” ways of knowing?
Robert,
Your son is off to a good start. I hate math from the beginning until now. I can sit and stare at an equation and it makes as much sense to me as Mandarin Chinese symbols.
Yes. Once you know something to the level of a rigorous mathematical proof, you know it. Often, however, that restriction will mean what you can claim to “know” is much more limited than what can be guessed at or reasoned from the collected evidence.
When you say “Math cheats because it is thoroughly abstract” is this kind of like some forms of theology…
By “Math cheats” I meant that Math’s certainty is easy to uphold in the mathematical realm. To predict actual phenomena with accuracy, the real world equations get very complicated very quickly. It’s the difference between “knowing” that a tangent is always 90 degrees to a radius, and whether an engineer for NASA “knows” that nothing’s going to go wrong at the next launch.
So, would I compare that to any Theologies? Absolutely, and not in a flattering way.
“The difference between theory and practice is smaller, in theory, than in practice.”
😉
After suffering thru the demise of western thought, the logic of absolutes, even the postmodern church is cast back upon a consensus of the ancient faith, “once delivered to the saints.”