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Archive for June, 2011
Is mathematics religious?
29 Jun 2011
What a strange question, I hear you say!
But if everyone's beliefs about everything are shaped by their own religious beliefs, as has been claimed, then that must include mathematics.
Chapter 7 of Roy Clouser's The Myth of Religious Neutrality (1, 2, 3, 4, 5, 6) faces this challenge head-on: how do religious beliefs affect mathematics?
You (both of you) may recall the basic shape of the argument. Religious beliefs are beliefs about what is independently and unconditionally real (Chapter 2). And our beliefs about what is fundamental about reality will affect the kinds of hypotheses we will entertain when we are thinking about any particular area of reality, including mathematics (Chapter 4).
Here are some examples.
1. The Number-World Theory (Pythagoras, Plato, Leibniz). In this theory,
the numerals and other markings of mathematics stand for real entities in another world or dimension of reality (p.133).
In order for this to be true,
it would have to be the case that the quantitativeness of things relates to the other kinds of properties and laws true of them by being utterly independent of them all. Thus, the quantitative aspect is (at least part of) what things and their other kinds of properties depend on for existence (p.134).
The quantitative aspect as one of the aspects of our experience mentioned in Chapter 4, and is one of fifteen aspects identified by Herman Dooyeweerd. (The full list given in Clouser's book is: fiduciary, ethical, justitial, aesthetic, economic, social, linguistic, historical, logical, sensory, biotic, physical, kinetic, spatial and quantitative.)
Examples of the practical difference these religious beliefs have had on mathematics have been the resistance of the Pythagoreans to the idea of irrational numbers, and that of Leibniz to the idea of negative numbers.
There are more practical differences when we consider the intuitionists (Brouwer, Weyl, Poincaré), who make even the logical aspect to be dependent on the quantitative aspect. This forces them to deny "the existence of actual infinite sets" and therefore to "reject an entire branch of mathematics, the theory of transfinite numbers developed by Georg Cantor" (p.141).
2. John Stuart Mill, whose "theory was that numerals symbolise sensory perceptions" (p.134).
Mill defended this view of math[s] by arguing that not only the quantitative aspect, but all other aspects of our pre-theoretical experience are actually identical with its sensory aspect. That is, Mill's theory was that the nature of all reality is sensory (p.134).
This approach is similar to the number-world theory in selecting one (or two) of the aspects, and asserting that it is the non-dependent reality on which all of the other aspects depend.
3. Bertrand Russell, who took the logical aspect to be non-dependently and unconditionally real:
The logical laws, he says, are not only those to which all reality — actual or possible — must conform, but they are "the heart and immutable essence" of all things (p.144).
Thus mathematics "is nothing other than a short-cut way of doing logic" (p.135).
4. Instrumentalism, e.g., John Dewey, for whom the physical-biological aspect(s) have the non-dependent and unconditional reality:
through all his theorizing, he regards all other aspects as dependent on the physical-biological and never regards them, in turn, as dependent on anything else (p.144).
Under his theory, "humans are to be understood as essentially biological beings struggling to survive in a certain environment", and thus "all human cultural products are instruments" (p.136), helping us to survive.
Just as it is inappropriate to ask whether a hammer is true or false, it is equally inappropriate to ask that of mathematical tools. 1 + 1 = 2 is thus neither true or false, says Dewey, though it performs certain tasks well (137).
5. Belief in God, "which should lead us to the view that no aspect of creation is self-existent, nor does any generate any other since all are dependent on God alone" (p.145). Under this view of mathematics,
the abstractions we arrive at, numbers, sets, etc., will never be seen as independently existing realities. The are never more — or less — than the properties, relations, functions, etc., of the quantitative aspect true of the things and events of ordinary experience (p.146).
In summary, it would have been good to explore the practical implications of these different views more fully. While all of the views differ greatly on their understanding of the nature of mathematics, it is so far clear only in the cases of the number-world theory and of intuitionism that any specific theories of mathematics are affected by one's religious beliefs. But, for example, if I switch between the theories of Mill and of Russell, would I then be forced to change my beliefs about any specific theories of mathematics?
But I think enough has been said to demonstrate the point: that religious beliefs do indeed exert a controlling influence on mathematical theories.
How do religious beliefs control other beliefs?
25 Jun 2011
If I believe something because it is in the Bible, then how might that affect my beliefs about some other area of reality, such as geology or history?
First, I might take a scholastic approach (see yesterday's episode), by which I formulate my theories of geology just as any other geologist would, but at the end of the process I whip out my Bible and check for any inconsistencies. If necessary, I do some more digging and some more theologising until those inconsistencies disappear (which hopefully they will).
Or, second, I might take a fundamentalist approach. This is the subject of Chapter 6 of Roy Clouser's The Myth of Religious Neutrality (1, 2, 3, 4, 5). In this approach, I scour the Bible for anything it might say about geology, and explicitly include that information in my theories.
Despite some rather strong language, Clouser doesn't seem to have a fundamental problem with either of those approaches, per se. (His main problem with the fundamentalist approach is when the Bible is misinterpreted, and forced to speak on matters about which it doesn't actually speak.) These two approaches
[focus] on either the logical compatibility of specific religious beliefs and specific theories [the scholastic approach], or on the inclusion of biblical teachings in the content or confirmation of theories [the fundamentalist approach]. But while not denying that revealed truth can, at times, act as "control beliefs" for theories in those ways, this position [see below] denies that those are the only or most important ways divinity beliefs impact theories (p.127).
This third position is the radically biblical position, which we encountered last time. In this, the religious beliefs control other beliefs by acting as presuppositions:
By acting as a presupposition to all theory making, rather than by being part of, or by confirming part of, the content of any particular theory, belief in God can guide every theory and do it in a more pervasive and important way (p.121).
The next chapters form a "casebook", giving examples of how this works in mathematics, physics and psychology. So watch this space...
A brief history of faith and reason
24 Jun 2011
1. Religious belief controlling theoretical reason
So far (1, 2, 3, 4) in The Myth of Religious Neutrality, Roy Clouser has been arguing for a view of the relation between religious belief and theoretical reason as follows (p.96):
Religious Belief
- guides and directs the use of reason in all of life
Theoretical Reason is:
- not neutral because controlled by religious belief
- not final court of appeal
- not able to decide all matters
In Chapter 5 he now labels this as the radically biblical position, saying
The position is this: there is no knowledge or truth that is neutral with respect to belief in God. The [Bible] writers who assert this do not also specify exactly how belief in God impacts "knowledge of all kinds" or "all truth," but they are clear that they regard beliefs in other (putative) divinities as partially falsifying all that is taken to be truth or knowledge, and that knowing God enables us, in principle, to avoid that partial falsehood (p.94).
He then quotes some biblical texts and concludes
that the cumulative effect of these texts is to teach that no sort of knowledge is religiously neutral (p.95).
However, most people throughout the history of thought have taken different views on the relation between religious belief and theoretical reason.
2. Theoretical reason controlling religious belief
First, there is the view that reason is autonomous and "trumps" religious belief. This may be religious rationalism, which "was the dominant influence in ancient Greco-Roman culture" (p.94). In this view, reason can be used either to justify or to refute religious beliefs. In some ways this may be seen as the reverse of the radically biblical position (p.93):
Theoretical Reason is:
- neutral respecting all matters
- final court of appeal in all matters
- able to decide all matters (?)
Religious Belief is:
- a theory or conclusion of reason
- optional
(Closely related to this is religious irrationalism, which states that religious belief is completely isolated from theoretical reason.)
3. Religious belief and theoretical reason in harmony
When the radically biblical position clashed with religious rationalism, religious scholasticism emerged, which
devised a compromise between the all-encompassing claim pagan rationalism made for reason, and the equally all-encompassing biblical claim that right faith is a necessary prerequisite for knowledge of every sort. This was done by limiting the scope of each claim (p.99).
This position "had permeated the whole of European thought by the sixth century" (p.104), and "Among thinkers who believe in God, [it] is still by far the most popular position in the world today" (p.105). Diagrammatically, it looks something like this (p.101):
Realm of Supernature or Grace
Faith accepts revelation as supreme authority concerning God and the soul and related matters.
Natural Realm
- Reason is neutral and final authority concerning nature;
- Reason harmonizes religion with the theories of science and philosophy;
- Reason proves the existence of the supernatural and systematizes revealed doctrines.
4. Faith and reason post-1500
The change in Western thought came
in the sixteenth century when scholasticism was simultaneously challenged by two movements. One of these, the Renaissance, advocated a return to pagan rationalism by insisting on the autonomy and neutrality of reason in all matters, so that it dispensed with faith imposing any limit to reason. The other was the Reformation, which rejected limiting faith to only supernatural matters and argued that reason is intrinsically guided by faith in all matters (p.105).
Calvin was strongest on the latter point, taking "the view that human reason is not neutral because it is affected by sin, where sin is understood as false divinity belief which produces deleterious effects on reason's attempts to interpret reality" (p.106).
However, the Protestant church soon sank back into scholasticism, having a "general view of the relation of faith and reason [that] was largely the same" as that of the Catholic church.
Their main difference over faith and reason came to be that while Catholic thinkers tended to harmonize their faith with theories about nature derived from Aristotle (due to the influence of Thomas Aquinas), Protestant thinkers felt free to harmonize their faith with whatever theories about nature were currently fashionable (p.107).
So where is the radically biblical position today? It does have its adherents, but generally doesn't have a good press, "owing to the specific interpretation of it which has been advocated by the largest single group of its adherents, the fundamentalists" (pp.108f), to whom (or against whom) the next chapter will be devoted.
What are universities for?
22 Jun 2011
I've had this lecture opened in my browser for months, and have just got round to listening to it. Here's a sentence to whet your appetite:
And yet, if it is true that university education — especially in the arts and humanities — is not about the growth of certain intellectual and social virtues, then it does become very hard to see why the study of landscape painting or medieval North African history or Byron’s reception among women is anything but a private and rather frivolous indulgence.
Lots of stimulating stuff there. And a full transcript. What are Universities For? by Professor Nigel Biggar, a talk for to the Oxford Graduate Christian Union on 17 January, 2011.
How do religious beliefs affect all beliefs?
10 Jun 2011
Roy Clouser's contention in The Myth of Religious Neutrality is that anyone's understanding of anything is strongly affected by their religious beliefs.
Chapter 4 of the book unpacks this by looking at the connection between our theories about any specific aspect of reality and our theories about the whole of reality (which are closely linked with our religious beliefs).
It gets a bit technical, but the essence is this:
- When we formulate a theory to explain something specific, we do so by focusing on a narrow set of related properties possessed by the things we are trying to understand. This involves making hypotheses: guesses about some entity that might exist, or some way in which different properties relate.
- The sorts of hypotheses we are willing to consider will be strongly dependent on the bigger theories we have: theories not about just one or two sets of properties that things might have (such as physical properties or aesthetic properties), but theories about how all of the different sets of related properties relate to each other. (A set of related properties is also known as an aspect of experience, or simply an aspect.) These bigger (philosophical) theories are expressions of what we believe about what is "unconditionally and non-dependently real". That is, they are expressions of our religious beliefs.
The rest of the book will develop this further (and I'm looking forward to seeing how it works in practice). But if the above points are correct, it can be seen that religious beliefs (which we all have) will always have a strong influence on what we believe about any area of reality.
I live in York and I 
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