Misuse Of Math

Math is not always science. Sometimes a lot of equations and axioms can be derived from an idea, but that alone does not validate the idea itself. It is a nice bonus if useful ideas have a "math" behind them, but not necessarily a prerequisite. If in comparing two ideas, one has a nice math behind it and another doesn't, perhaps one should go with the mathified one if they otherwise seem equal.

In other words, the quantity of things that can be surmised or derived based on root axioms alone does not necessarily reflect the quality of the base axioms themselves.

Misuse of math tends to happen when the root ideas are unsound or still up in the air. Somebody takes these shaky or unproven ideas as givens and creates extensive formulations. Such work is a good thing to explore, but doing "interesting" things with simple ideas alone does not validate the root ideas in itself. A classic example from AddingEpicycles is the assumption that planetary orbits are all based on perfect circles. Elaborate systems of nested circles were often used to match the idea that "EverythingIsa circle" to the observed motions of the planets. Eventually it was replaced with Newtonian physics, which had better predictive value than nested circles and was based on the observable properties of gravity. Relativity later superseded Newtonian physics. Newtonian physics turned out not to be perfect (perhaps nothing known is), but at least serves as a UsefulLie because of its low complexity-to-reliability ratio.

I believe that is similar to what we need in the software design field: UsefulLies with low complexity-to-utility ratios. Every abstraction is probably going to turn out to have deviations and flaws, but we still need abstractions to help us manage overwhelming complexity. Thus, we need simple but powerful ideas. We can use math to help us find and test ideas, but it may not serve as the ultimate validation tool for such ideas.

-- top

Excellent observations, Top! Does this mean that you have now rejected the OoLacksMathArgument? After all, that silly dispute strikes me as just as much of a MisuseOfMath as anything else.

I have moved away from LaynesLaw issues around defining "math" and toward OoLacksConsistencyDiscussion. Relational's consistency advantage over OO may or may not be due to its math-based beginnings. I think it helped because math requires consistency, or at least specificity, to work, but the real "road test" is observed improved consistency. -- top

Procedural programming lacks consistency too, Top. Sometimes people use pointers, other times they don't. Sometimes they pass in a single struct as a parameter, and other times they split up the paramaters and don't use a single struct. This is inconsistent. Sometimes we return error codes as an OUT parameter, whereas other times we return the error as a function result. Again inconsistent. There are many inconsistencies in many programming practises. Tables in databases often contain inconsistent data: some is phone numbers, some is blobs, some is only positive integers and not negative, some is strings (which is why you need a TypeSystem, to handle different kinds of Types of things).

I meant that the rules were clear and available for inspection. Using different rules to achieve the same goal is another matter.

Also, did you know that object oriented programming is just structured programming with extensions? Since you dislike OO, do you also dislike structs? A struct in C with methods, is an object. Structs are objects. The OO people invented new snake oil terminology instead of sticking with existing terminology like "struct" or "record". They could have just called it "extended struct" instead of calling it "object". They introduced new terminology to sell something old: structured programming. So if you dislike OO, then you dislike extended procedural programming, because that's what OO is. Sometimes the extensions are not needed and programs can be done without them, other times these extensions come in handy. That's what you need to figure out, when they are handy and when they are not.

I agree that structured programming by itself is limiting. That's why I like to combine it with TableOrientedProgramming. TOP is more powerful than OOP. For example, it handles many-to-many relationships much nicer. But this is not the place for arguments over such. --top
PageAnchor: "Difference Between Math And Science"

Math is about producing an internally consistent model or notation. Science is about the applicability of that model to the external world. Math can be completely divorced from the physical world, while science cannot. Math can "lie" as long as the lies are consistent; science cannot because in the end it must be tested against the physical world.

I hate to ask what programming where programming exists then. . . Graffiti seen on a temporary barrier during construction on Evans Hall (the home of the math department at CalBerkeley):

"Math is God"
"No, Math is the set of all Gods"
"God is Physics"
"No, Physics is Math on earth"

Some Math misuses: Misuses? The first statement without context is probably useless. If it reflects an actual study, it could be interesting. The second statement is basically correct, except the inference not really useful in the issue of understanding why infinity is not a number in the usual system. The third statement is true, i.e. there *is* a good reason. How do any of these statements reflect the claims of the page?

Response: To clarify the above, let's restrict the discussion to some real function denoted by f(x) with an antiderivative denoted by F(x), both having as domain the closed interval (a,b), where a and b are distinct reals, with a < b. To avoid quibbles about the meaning of "under", let's also suppose that the function f(x) is positive (and finite) or zero. Let's also suppose (unless indicated otherwise) that "area under the curve" makes sense for the function f(x) (but we allow f(x) to have discontinuities). Let's further suppose that F(x) has no discontinuities, and that it has derivative f(x) everywhere in the open interval (a,b) except for a finite (possibly zero) number of values of x. If that "except" clause is too restrictive, let me know. Now let's consider (under the listed assumptions) various points raised above. No. Your restriction is arbitrary, and has no bearing on the original comments (sorry, I missed this bit last response). I was being generous - the usual (i.e., standard) definition of an antiderivative of f(x) would require that it's derivative is f(x) throughout their domain. Either way, the Dirichlet function doesn't have an antiderivative.

1. You made the point that "area under a curve" can make sense in places where the Riemann integral does not work. Can you give an example function f(x) for which the area under the curve for f(x) makes sense but is NOT obtainable as the Riemann integral of f(x)? (Note: the assumptions were designed to exclude the possibility of an area which "makes sense", but is infinite.) Dirichlet function. What continuous function F(x) has the Dirichlet function as its derivative? Irrelevant. The Dirichlet function is integrable, and that integral is interpretable as an area. It is not Riemann integrable. However, as noted above, it doesn't have an antiderivative. The point under discussion is the relationship between the concepts of antiderivative and area under a curve, so one has to restrict examples to cases where both exist, not just one of them.

2. You made the point that "the idea of an antiderivative, and the idea of an integral apply in places where 'area under a curve' does not make sense". Please give example functions f(x) and F(x), where your point applies, and f(x) is integrable (Riemann integral or Lebesgue integral - your choice). Please. Areas are strictly non-negative values. Or simply consider a path integral. Or complex analysis. We're discussing non-negative real functions of a real variable, so none of your responses are significant. I asked for example functions. See above. Look, this stuff is all readily available to you - it isn't my job to get you past intro calculus here. That doesn't apply. It's trivial that an antiderivative isn't related to anything if it doesn't even exist! Hence, the existence of a function without an antiderivative doesn't help.

3. For the purposes of relating the area under the given function f(x) (with antiderivative F(x)) to another function of one variable, I choose to consider a part of the curve having "start point" (A, f(A)) and "end point" (x, f(x)), where a <= A <= x <= b, A is a constant and x is a variable. Suppose that f(x) has a definite integral between the limits A and x. I assert that the fundamental theorem of calculus applies (but there may be some cases where it doesn't - I don't have a proof of the theorem under the stated conditions), that the value of the definite integral is equal to the area in question, and also equal to F(x) - F(A). Now define G(x) = F(x) - F(A) (where x lies in the closed interval (a,b)). This makes G(x) another antiderivative of f(x), since it differs from F(x) by a constant. This antiderivative function gives the area in question directly - it is not necessary to subtract G(A) because G(A) is zero. Note that I referred to "an antiderivative" originally, so I'm within my rights to choose this one. This was my reason for making my original correction. Will comment later.

4. I invite you to justify your objections to my original comments by providing example functions f(x), F(x) where the above reasoning breaks down - for example, where the fundamental theorem of calculus does not apply, or the area under the curve either does not make sense or cannot be found via the antiderivative. The basic objection is very simple... you attempted to 'correct' something that was not incorrect, and have continued to demonstrate a failure to grasp what fundamental means, among other things. Again, you're not providing examples. You're free to clarify "fundamental" as well if doing so helps to clarify how your comment applies to the examples. I provided a good example above. The fundamental point here is how these things are defined, and what those definitions mean. It seems to me you are bashing around a half-baked grasp of introductory calculus and some very vague idea that there is more too it than that. This is all standard intro analysis though, so if you are genuinely interested, you can learn about it. Trying to force things to conform to a somewhat confused surface understanding of analysis will not get you very far..... I'm aware that the concepts being considered are not defined for some functions. However, that doesn't imply that the concepts "fundamentally have nothing to do with each other".

5. You stated "The *integrals* agree, but an integral is not a measure." I invite you to give an example of any function f(x) (no presuppositions (except that f and x are real)) for which the Riemann integral exists but the Lebesgue integral either doesn't exist or is different in value. You are obviously confused about what a measure is. This has nothing to do with Lesbesgue vs Riemann. Integrals are defined in terms of measures, but measures *are not integrals*. Got it? Almost. What you said was "an integral is not a measure". If that's correct, at least give an example integral and prove that it can't be interpreted as a measure. Just look at the definitions. An integral (including the Riemann integral, technically, is defined in terms of a measure, which is a function with specific properties. There is no 'interpretation' here. Are you intentionally being obtuse? No. If you think it's trivial to give an example of a Riemann integral which doesn't have the properties of a measure, just do so.

Top, you should stop defending the relational model - because that isn't what you are defending at all. You are defending something else - your own table oriented model. The inventors of the relational model and the current maintainers do not speak of the relational model as you do (in other words, what you are describing is not the relational model, nor does it have anything to do with hierarchies as you have stated on other pages). When defending your model, defend it - but don't discuss your table model as the relational model in your defenses, please. It is very confusing and misleading.

I smell a brewing LaynesLaw debate over "relational" similar to the Nggard versus AlanKay OOP definition battles [insert link when found]. Who sanctioned the "current maintainers" anyhow? That is kind of a euro-centric view of "truth" where appointed people instead of peer consensus determine definitions. Plus, the issue of "using relational" and "using relational properly" may be two separate things. But I'll save debating such issues for another day.... --top

(Please do not put specific issues left over from other debates near the top of a topic if possible. The top should be reserved for introductions and generalities, not detail nits.)

See also: AddingEpicycles, TopOnTypes, ProgrammingIsMathDiscussion, SoftwareGivesUsGodLikePowers, SovietShoeFactoryPrinciple

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