{"id":45,"date":"2006-06-25T11:59:00","date_gmt":"2006-06-25T11:59:00","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2006\/06\/25\/validity-of-mathematical-modeling\/"},"modified":"2006-06-25T11:59:00","modified_gmt":"2006-06-25T11:59:00","slug":"validity-of-mathematical-modeling","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2006\/06\/25\/validity-of-mathematical-modeling\/","title":{"rendered":"Validity of Mathematical Modeling"},"content":{"rendered":"

In the comments to one of my earlier [Demsbki][demsbki-info-theory] posts, I was responding to a comment by a poster, and realized that what we were discussing was probably interesting enough to promote up to a front-page post.
\nA poster named [Garote][garote] said:
\n>I think PaulC said exactly what needs to be said, that grants us all
\n>hearty permission to ignore Dembski’s work. I think it bears repeating:
\n>
\n>”If I am betting on coin flips and notice any kind of algorithmic pattern then
\n>there are legitimate reasons to conclude that the person flipping coins is not
\n>playing fair. However, this method does not allow one to discriminate between
\n>the product of evolution and the product of “intelligent design” since neither
\n>of these processes in any way resemble random coin flips.”
\n>
\n>Dembski’s feverishly comparing apples to oranges, to prove that one is an
\n>artichoke.
\nNow, my contempt for Dembski is pretty clear. But this argument against him struck me as making a mistake – and it’s a mistake that strikes at something near-and-dear to my interests: mathematical modeling. Modeling can be sorely abused, [as I’ve discussed before][modeling]; but I don’t want to throw out the baby with the bathwater.
\nMy whole viewpoint on mathematics is that much of its value comes from **abstraction**: the ability to look at something complex; to identify a set of relevant properties for understanding some feature of it; and to use that to create a simpler mathematical model of the complex system based on its fundamental principles. That kind of abstraction isn’t just useful for diddling around in the world of pure math, but for much of how math is used in science. There are a lot of interesting things that we can’t grasp in their entirety; but that we can understand by breaking them down through abstraction into a set of simpler facets that we can study.
\nI don’t want to throw away the validity of the concept of abstraction and mathematical modeling; but one way of reading that comment is as an argument against the concept of working with a simplified abstract model to understand something complex. (Note that I’m not saying that that’s what Garote meant; but I thought that focusing on this could produce an interesting discussion.)
\nIn Dembski’s favor, I can see the “coin-flip” thing is an abstraction of a complex process to something simple that contains a relevant feature. The thing is, there are things in nature that produce highly random results; and there are other things that produce interesting patterns<\/em>. Recognizing that there’s something different about processes that produce patterns<\/em> could be an interesting thing; and recognizing the distinction between randomness and pattern is<\/em> an interesting subject which probability theorists have spent time studying; it’s right at the intersection between probability theory and information theory, and it’s produced some interesting insights about what information is and what it means.
\nThe problem WRT to Demsbki isn’t that he’s reducing a problem to a mathematical model through abstraction and then analyzing the model. The problem is that when you build a mathematical model, you *need to make sure that your abstraction captures all of the relevant features of what you’re modeling*. And when you draw conclusions based on the analysis of the mathematical model, *you need to make sure that your abstraction isn’t the source of the conclusion*. Both of those are different ways of stating the fundamental rule of mathematical modeling:<\/p>\n

**Mathematical models must be validated against the thing that they model.**<\/p>\n

Demsbki’s reduction of the process of analyzing certain features of the universe to the a metaphor involving recognition of patterns in coin flips is fine with me. The problem is that he creates a model that *omits important elements* of the real world, and then insists that the *conclusions drawn from his abstract model are applicable to the real world*. But in fact, his conclusions are the result of properties of his model, *not* of the real world. The validation of his model fails: the specific conclusions are the result of eliminating things from his model that might permit a different conclusion.
\nTo continue with the example of the coin flip metaphor: if you’re studying patterns, you could abstract the problem of observing patterns to one of observing flipping coins in an idealized universe, where the coin is perfectly even, and nothing about the surface that it lands on can affect the outcome of a flip. That might be a useful model for trying to understand the difference between random sequences of flips, and patterns in sequences.
\nIf you then try to use that model to determine the *cause* of a pattern, you’ll conclude that the only possible cause of a pattern *is the actions of the coin flipper*. If you’ve abstracted away everything that could influence the outcome of the coin-flip except the influence of the coin-flipper, then *in your model*, concluding that the flipper is the only possible source of a pattern could be a reasonable conclusion.
\nThat doesn’t mean that you can then say that in the real world, the *only possible cause* of a pattern in a sequence of coin-flips is some action taken by the coin-flipper. You can’t apply that simplified abstraction of your model to the real world without showing that it’s a valid model with respect to the the properties that affected your conclusion.
\nThe ultra-simple coin flip model isn’t valid for the real world, because it deliberately omits factors of the real world that could influence the conclusion. In the real world, there are unfair coins (both deliberately unfair coins, and coins that have become unfair either through minting flaws or through wear and tear). There are magnetic fields. There are irregular surfaces.
\nThe problem with many of the conclusions that bad mathematicians draw from mathematical models is that *the models don’t represent what they claim to represent*. They abstract away relevant features of reality, and then demand that reality doesn’t possess those features, because they aren’t in the model.
\n[demsbki-info-theory]: http:\/\/scienceblogs.com\/goodmath\/2006\/06\/dembskis_profound_lack_of_comp.php
\n[garote]: http:\/\/garote.bdmonkeys.net\/
\n[modeling]: http:\/\/goodmath.blogspot.com\/2006\/03\/on-how-not-to-apply-mathematical.html<\/p>\n","protected":false},"excerpt":{"rendered":"

In the comments to one of my earlier [Demsbki][demsbki-info-theory] posts, I was responding to a comment by a poster, and realized that what we were discussing was probably interesting enough to promote up to a front-page post. A poster named [Garote][garote] said: >I think PaulC said exactly what needs to be said, that grants us […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[24],"tags":[],"class_list":["post-45","post","type-post","status-publish","format-standard","hentry","category-goodmath"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-J","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/45","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=45"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/45\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=45"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=45"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=45"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}