Archive for the ‘Philosophy’ Category

Some Thoughts Religious Neutrality in Math

Andrew has spent quite a bit of time defending the idea that one cannot have a religiously neutral math education in the public school system. Benjamin has insisted that the functional math taught at this level simply is religiously neutral because it is simply formulaic. So who is right? I think that Andrew is correct because no subject can be made religiously neutral – including public math schooling. I will now demonstrate this claim.

I begin by pointing out that there is no widespread agreement among philosophers on exactly what math is. Some agree with Benjamin and view it as a functional system (formal system). In some cases, they believe that math can be reduced to logic. Others disagree. They believe that is not a formal system at all. They believe that it describes the underlying order of the universe. Since one cannot simply assume that one side is right, one cannot simply take one point of view when constructing a public school math curriculum. Now is this the only problem.

There is also no agreement among mathematicians on exactly what the most basic truths of math are. You might think that the most basic truths have to do with integer addition and subtraction. Some mathematicians agree. Others believe that set theory is actually more basic. Still others believe that logical operations are more basic. Naturally this affects education. You may have heard about the “new math” that was tried in Ontario a while ago. Some educators believed that students should begin with set theory rather than addition. This program is no longer in effect.

To make things even more complicated, these problems are related. What you think about the nature of math determines what sort of mathematical truths are most basic. If you believe that math is reducible to logic, then logical truths are most basic. If you believe that math is a formal system based on rules, then you will focus on the use of rules. If you believe that math is an expression of underlying order, then you will focus on order without necessarily involving formal rules. So what you believe about math determines how you teach it.

If you are sympathetic to American ideals (pragmatism) then you will just want to scream and suggest that we simply teach traditional math because it works. Even then, you are not being neutral. You are promoting a philosophy known as pragmatism in your teaching. Pragmatism will either lead you to promote a particular philosophy of math without carefully considering it or it will lead you to try some kind of “equal time” approach. The first option allows you to avoid being neutral and not know that you are doing it. The second option implies that all views of math are either equally true or equally suited to practice. So pragmatism in math is not religiously neutral.

Every kind of schooling about math in the public schools will require setting out what topics students are to learn, when they are to learn them and in what order. Every setup reflects a philosophy of math.  Therefore every possible kind of public school math education reflects a philosophy of math. I assume that all parties agree that philosophies of math are not religiously neutral. Therefore, public school math education is not religiously neutral. QED.

Re: At What Point Do Poythress' Concerns Matter

Since we can’t interface the two examples perfectly, so I don’t think applying equal time to both situations is appropriate.

I still think the time difference, and the relation of time to personal formation, makes it a relevant difference. Do you think that watching an hour of TV a day (for 5 days a week, 10 months a year, for twelve years, etc.), even though it takes far less effort than learning math, or even having a conversation with a friend about television, would have a small effect on someone’s belief system? (Edit: I had forgotten where this line of thought came from when I wrote this response; to add a bit, I’d ask: if one were to watch that much TV, and God was never mentioned, would God be thought of as more or less relevant to life than if one had instead watched Christian movies or something similar?)

Computers perform operations, and they have no spiritual capacity, no conscience, no sapience, etc.

Computers do not use concepts or words like people do (as in, “addition”); they are just circuit boards with electricity running through them, projecting light onto a screen so we can see what the effects of that electricity is. There is no understanding of function or of meaning, if you distinguish those things; there is no understanding it all. It’s just clockwork.

But I think one would be hard pressed to prove that a certain spiritual meaning is required for function.

What does required mean, though? And what does “spiritual” mean?

But the electrical impulses being sent and the processing that takes place is entirely unknown to me, and is irrelevant — my awareness of these frameworks do not modify my ability to press the @ key and have an @ show up on screen.

But your understanding of the meaning of the ink-mark @ on a piece of plastic set in a larger, rectangular piece of plastic, linked by a wire to a metal box, does affect your ability to do so.

In the same way, I can perform functions on numbers — very much in the way a computer does — without awareness of what principles might make operations possible, even though they are at work in me or enabling me.

As I mentioned above, I think its highly misleading to speak of computers as if they are aware of the use of anything. And you do need to know what the meaning of “+” or “2” is, to be able to do any functions with those concepts.

For the sake of interest, practically, what does a Faith based math education look like? What is the Biblical significance of 2+2=4?

I think Poythress’ article would give you a better answer than I ever could there.

While I was thinking about this issue over the weekend, I had another thought which I think will make my original point more clear.

Consider this: what would happen if, based on your position that the relevance of God to mathematics is so low that it is virtually insignificant in elementary through secondary school, you suggested to the local public school board that you incorporate Poythress’ insight that mathematics makes no sense except in a theistic universe. Do you think their answer would be: “Well, since God’s relevance is so slight to math, it won’t harm our children to teach them this. We could mention it only very rarely and it would therefore have no impact, and we could still make sure our children turned out as good secularist democrats.”? I doubt that very much. Not for one second would they allow God’s relevance to math to enter the classroom, and I suggest that’s because they would recognize it’s not insignificant to the education and formation of children and young adults. Even one lesson showing how math relates to God would be too much, I suspect.

Re: At What Point Do Poythress' Concerns Matter

I’m not sure if the math reviewed in school borders on anything that could be considered “entire.”

I guess we’re getting into murky territory here, but I figure when you talk about the same subject for 12 years, that’s close enough to the “entire” category to be an appropriate analogy.

I think to be fair to my analogy, we need to restrict “discussion of the spouse” to a single subject (going to the movies), just as we are exploring God as related to a single subject (math).

I’m not so sure this is the case. If you add up the amount of hours that we spend talk about one subject in the course of public education, it’s a lot higher than the total amount of time it would take to discuss “going to the movies with friends”. There’s a lot more time and deliberate thought and formation (through repetition) in math than in any casual discussion about the movies. And surely, if you talked for 1 hour x 5 days x 10 months x 12 years about movies, and never mentioned once that, for example, the whole reason you love movies and view them so much is that your spouse has taught you to love them, and explained the significance of movies to you, I still think it would seem very wrong and disproportionate.

 Yes. And again, I agree with you as far as meaning or significance is concerned. But I’m simply focusing on function

I don’t see what the distinction between function/use and significance/application is. If you don’t know how to use something, you don’t know its meaning, and vice versa. Thus, the more you know about its meaning, the more you know how to use it.

I feel that since this discussion has become somewhat separated from my original point, that I should bring it up again: even if we think the effect of teaching math in a religiously-neutral” way is negligible, certainly this could not be the case about, say, history.

Re: At What Point Do Poythress' Concerns Matter

Ben said:

I think this might be overstating the case: that excluding information is indicative of belief.

Obviously you’re correct, excluding information is not always tantamount to disbelief. But note what I said: “when we discuss a subject entirely… .” The “entirely” was deliberate. I can see that teaching for a few days, months, etc. might be okay without mentioning God directly, but if you teach someone math for 12-13 years of education and never mention God, it would be akin to you telling your friend about your life, in general, for years, and never mentioning you were married. In that context, it would be indicative that you were trying to hide something, or else had somehow forgotten your wife (which itself would be very problematic).

But is He at every moment, monitoring the Laws of Physics, seeing that they continue on unwaveringly? I don’t think so. I think that physical principles are his unconscious servants, accomplishing His will, albeit in a different way than humans or angels.

Well, that would reflect one view of providence, one which I don’t happen to share, but I don’t feel like getting into that right now…

You are not saying that humans will be able to carry out mathematical functions better per se. You are saying that they will have a better understanding of the significance or place of math.

I think I’m making a stronger claim.

Consider an analogy in logic. People all the time use the basic laws of logic that Aristotle discovered, such as the law of non-contradiction. But most people don’t know it in the technical form, nor do they know all the various reasons why it is true that things can’t be x and non-x at the same time in the same way. A philosopher who understands this has a better grasp of why reality cannot be contradictory, and for that reason better understands why the law non-contradiction is a true law. And to understand the cause or explanation of a thing is to better understand the thing.

So, while I agree that understanding math’s relation to God gives a greater understanding of God (a spiritual knowledge), I also think that it illumines the meaning of mathematical truths for us, as well.

Popular skepticism about science

Keith raised an interesting issue in his post about Global warming skepticism:

 An unfortunate consequence of the evolution-creation debates is that evangelicals have become skeptical of science in general. This is ironic given how dependent we moderns (and wannabe pre-moderns) are on science.

There are a few issues here. Firstly, I don’t think skepticism about global warming is limited to evangelicals, as most right-wing groups seem to take a skeptical stance toward this issue. Also, as Dan pointed out, it’s not just anti-market groups who support a non-skeptical stance.

What this seems to signal to me is that skepticism over this issue is not primarily because of religious or laissez-faire ideological motivations, but perhaps I’m wrong.

As well, I’m not sure that evangelicals are “skeptical” towards science in general; it seems to me they are more skeptical on matters that have no (immediate?) practical payoff. Evangelicals are not Amish, they seem to understand that advances have been made in medical, transportation, communication and other forms of technology, and they take those as basically good and helpful things. If there is any skepticism about science, it seems to relate to either the unrepeatable past (evolution), or predictions of a generally highly uncertain future (the weather). There are notably different kinds of scientific activity; in fact, both evolution and global warming predications are really more fields of history than of science, at least by the Modern scientific method. So I’m not sure it’s entirely inconsistent on their part, to be skeptical of the latter kind of science and not the former.

Re: At What Point Do Poythress' Concerns Matter


What I wonder about is, on a practical level, do these philosophical differences really impact on elementary and secondary level math education?

I’ve had a bit of time to chew on this, and I think I have a bit more to say.

One of the obvious problems with secularism in general is its feigned attempt to be religiously neutral. If there is anything that postmodernism has taught us, it is that there is no such thing in the world of humanity.

In a similar manner, when we discuss a subject entirely, and never mention God, I think there’s a tacit assumption that is being conveyed in instruction, that God is functionally non-existent when it comes to math. God is irrelevant to this part of life.

If anything, Poythress’ points have undermined that unreflective assumption.

So, as to the pragmatic question you asked, my answer would be: I imagine it would have some kind of religio-psychogolocial effect on the students who are taught math in such a way that God is seen to be absolutely disposable when it comes to doing it. This may be hard to quantify, but I think the likelihood of it having an effect seems relatively high.

For what it’s worth, too, math has a strange way of making people reflective, or at least making some people who are already just a little reflective a bit more reflective about philosophy and related things. The only time in high school I remember someone discussing philosophy directly (and not in a historical fashion) was a reference to a priori versus a posteriori knowledge (an issue in epistemology) in a grade 11 math class. Math is deeply related to philosophy in many ways, and I don’t doubt that that will have some kind of practical psychological effect on students.


Could we enhance our Spirituality by including God in the study of arithmetic or algebra? Yes. But can we learn all we need to about arithmetic and algebra just by studying those topics? Of course. 2+2=4 is not modified by any other information.

A few things:

1) I think part of Poythress’ point is that we don’t just enhance our spirituality, but our knowledge of mathematics itself by understanding its relation to God.

2) “All we need to know” is highly vague, and hard to discuss.

3) If one piece of information sheds light on the meaning of another piece of information, I would say that it modifies our understanding of that latter piece. And if, as Poythress argues, believing in the Christian God regulates the meaning of 2+2=4, then I would say that it modifies our understanding of that equation.

Mathematics and God

Ben and I discussed below whether math is “spiritual”.

Vern Poythress has an essay on A Biblical View of Mathematics in which he argues that mathematics is not religiously neutral:

The neutrality postulate holds special attractiveness as applied to mathematics, because of the apparent widespread agreement about mathematical truths. “Everybody knows that 2 + 2 = 4.” If religious beliefs really have an influence, why is there such widespread agreement, cutting across religious lines? We intend to answer this question on several levels: (1) by showing that the agreement in mathematics is not so widespread, nor so uncorrelated with religious beliefs, as the textbooks would have you believe (§§2-7); (2) by showing that non-Christian philosophy of mathematics is involved in deep-set cleavages and antinomies, in its understanding of even so simple a truth as 2 + 2 = 4 (§§11-18); (3) by showing that only on a thoroughgoing Biblical basis can one genuinely understand and affirm the real agreement about mathematical truths (§25).

So, first of all, what differences have arisen in mathematics in connection with religious belief? Differences have arisen over arithmetical truth, over standards for proof, over number-theoretic truth, over geometric truth, over truths of analysis, over mathematical existence-not to mention the long-standing epistemological disputes over the source of mathematical truth. Let’s consider these areas one at a time.

A couple additional comments:

1) Even if one does not buy Poythress’ strong claim that mathematics ultimately requires a trinitarian theistic grounding, his point that math is not religiously neutral still stands.

2) It seems to me that if math is not religiously neutral, that is essentially an a fortiori argument that every other field of knowledge is not religiously neutral. If something so (apparently) universally accepted is still religiously biased, how could something like history or biology fare any better?

(And for what it’s worth, Poythress has a PhD in mathematics, a ThM in apologetics, a ThD in New Testament, and also has studied and taught in linguistics, so he’s not out of his field here.)

Re: Faith-Based Schooling

Still going:

 The same struggles and influences are to be found in Christian schools as are in public schools (yes, possibly to a lesser degree, and with diminished regularity). But there is no guarantee that this is the case.

Obviously there is no guarantee of anything, but there is at least an increased probability. Since all these kinds of decisions are based on probabilities, why would one not go with the best odds?

Yes, but it’s silent in matters of mathematics, grammar, and biology. The fact is that these subjects (though grounded in the Living God) are not particularly or exclusively Spiritual, and the Scriptures are silent regarding them.

On the contrary; I think every subject is related to God in some way. What makes mathematical truths, true? How does eternity relate to time? What are the purposes of living things? Where did they come from?(!) What is the best way to use one’s words? Further, given what God says about the purpose and nature of children, what pedagogical methodologies for teaching these subjects would be best?  Answer any of these questions, and eventually God is related. And further, nothing in the Deuteronomy text limits the education to “Spiritual” matters. Even if it did, the question would be: what parts of the world is the Holy Spirit not related to?

Of course, it’s not as if every word coming out of the instructor’s mouth must be “God” or a quotation of scripture. But I think the previous point is true, notwithstanding.

Re: Pruning

Just to complicate matters with some further problems:

Dan asks:

What do we do?

I think the answer has to be divided based on who the “we” is. Is the “we” the church qua church? Then the discussion will go in one direction: because from the beginning Christianity has been a missionary religion commissioned by its founder to convert the whole world. The plan in that sense is to eliminate the competition by means of persuasion. I realize that there are some modern and postmodern theologians who would argue with this, but I don’t find them very convincing, so I don’t think that’s a real option in play (for those who wish to be consistent).

If the “we” is Western nation-states, then the discussion will go in a very different direction. Now we are asking about using force on people, and in which circumstances we should do so, essentially. I’m not sure what to say about that. One of the problems of thinking from this perspective, I believe, is that “we” are still a living tradition, with a particular point of view and non-neutral values. There is no view from the outside of matters related to religion, as it were.

A further problem is that those who are part of both “we”s (or have a different religious tradition roughly equivalent in nature to the Christian one) sometimes feel the conflict between the “we”s internally; there is then a pressure to compromise or compartmentalize, or alternatively become ghettoized or zealous (for either “we” identity, violently against the other one). There’s a fine line to walk to maintain one’s integrity before God and others, and only a few find it, it seems. And of course, these attitudes cannot fail to have some communal/corporate correlation.

So, now that I’ve added to the problems, what do we do? :-)

Pruning but carefully

I don’t know why we’re persisting in these agrarian metaphors, but here goes another one. I think that on a fundamental level I agree with what Andrew is saying. The great difficulty is in the fact that most religious and cultural systems don’t really treat pluralism as a practical concern. It’s all about either being an oppressed minority or a conquering majority (sometimes both in one religion). There isn’t much about tolerating or accepting the sustained presence of the other as an equal.

What do we do? My understanding is that, despite you might have heard, moral relativism is pretty much passe in philosophy of ethics circles, and I’ll count that as a good thing. The tension in a plural society is between respecting what you are convinced is universally applicable and respecting that those around you have a different definition of the same. I think we have to identify what practices ought to be regarded as wrong regardless of tradition (female circumcision, sati, witch burning, I can go on, give me a culture/religion and I’ll find something).

But in criticizing various traditions and identities we must be careful that we don’t focus so much on the shortcomings of others that we do not consider the beams in our own eyes.