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A Beka's hatred for set theory
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HawkbitAlpha Offline
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Star A Beka's hatred for set theory

For those of you that haven't read my thread about my friend writing a whole document on my school, this thread is a direct response to the math part of it. I'll repost the bit in question here. But let's focus on the bit from the A Beka website first:

Quote:Mathematics

Mathematics is the language God used in His creation of the universe, and thus it is logical, orderly, beautiful, and very practical in science and in daily life.

No subject matter better reflects the glory of God than mathematics. To study mathematics is to study God's thoughts after Him, for He is the great Engineer and Architect of the universe.

Unlike the "modern math" theorists, who believe that mathematics is a creation of man and thus arbitrary and relative, we believe that the laws of mathematics are a creation of God and thus absolute. All of the laws of mathematics are God's laws. Our knowledge of God's absolute mathematical laws may be incomplete or at times in error, but that merely shows human frailty, not relativity in mathematics. Man's task is to search out and make use of the laws of the universe, both scientific and mathematical.

A Beka Book provides attractive, legible, workable traditional mathematics texts that are not burdened with modern theories such as set theory. These books have been field-tested, revised, and used successfully for many years in Christian schools. They are classics with up-to-date appeal. Besides training students in the basic skills that they will need all their lives, the A Beka Book traditional mathematics books teach students to believe in the absolutes of the universe, to work diligently to get right answers, and to see the facts of mathematics as part of the truth and order that God has built into the real universe.

That thing about set theory? Matt tore into it and related topics in his analysis, specifically why A Beka dislikes set theory and how it affects the students.

Let's see his take on this. Starting off with math being relative, and a bit of a history lesson:

Quote:Some of our first records re maths are from the Sumerians, the Greek (via the Egyptians) then took off with things re geometry. The Pythagoreans actually made a bit of a religion out of geometry, including both the pentacle and--ironically--the "ichthus" (known as the vesica piscis (in fact, it's actually now thought the ichthus symbol was originally borrowed by the Christians from Pythagoreanism, as the number of fish Jesus catches is equal to one of the two numbers in the ratio formed (using whole numbers) in the vesica piscis and is thought to be a hidden reference; this is a pattern that would be repeated in alchemy years later). Much of the Pythagorean work (again, likely via the Egyptians, based on both historical records and traditions from alchemists) ended up as foundation for the principles of sacred geometry in alchemy as well as sacred geometry in Islam.

After the fall of the Roman empire, mathematical study was kept alive by the Islamic empires, and algebra was invented by them. (Interestingly, Spain--which would suffer a horrifying progrom that we know now as the Spanish Inquisition only a few hundred years later--was, under Moorish occupation, one of the leading centers for mathematical study in Europe!) Moslems also extended our knowledge of geometry. Info was shared also with other cultures, in particular Hindu and Buddhist nations, including the invention of the zero and positional notation (aka listing tens, hundreds, etc. as 10, 100, etc. rather than as X, C, etc.)

Around the 1500s--the time of the restoration--alchemists such as a certain Isaac Newton were doing their own researches on sacred geometry and figuring out how God built the universe in the purpose of their own spiritual refinement (often from substantially pre-Christian sources, and couched in a great deal of Christian imagery to avoid being burnt at the stake; quite a bit of alchemical knowledge actually dates back to at least Greek times (as evidenced by the references to Hermes Trigesimus) and sometimes even Egyptian sources). Mr. Newton incidentially invented calculus in this, and his work as well as those of other scientificially-minded folks helped spawn the birth of Deism as a philosophical movement.

(In other words, the very foundations of theoretical maths were set far in the past, and things like set theory are part of a progression that has gone pretty much uninterrupted for several thousand years--only having had to be rediscovered after the Dark Ages and carefully hidden away by other nations, monks and alchemists till then.)

And on set theory:

Quote:Set theory is, in essence, an extension of work in regards to discovering that there are an infinite number of real numbers--thus a way was needed to sort infinities. (Yes, it is possible to have multiple infinities in higher maths. This is one of the concepts that break people's brains.) A good example of set theory in action:

All real numbers (which is an infinity) are in set A. All real numbers that are divisible by 2 (which is also an infinitely long list of numbers) are in set B. All real numbers that are divisible by 3 (a separate, but also infinitely long list of numbers) are in set C.

Sets B and C will intersect; set A will include both set B and C, but there are numbers in set C that aren't in set B and vice versa.

You can even extend this--if number X is not divisible by 2, and is not divisible by 3, it goes in set A outside set B and set C.

One of the niftier things with this is that by doing this you can sort of "size" how big an infinity is. Set A is a huge infinity, B and C are smaller infinities. The intersection of B and C is a smaller infinity yet.

You can also sort stuff that's not infinite with this, too.

Set theory is, in essence, the basis of much modern logic and proofs, and is the foundation of a LOT of computer science (the whole and/or/not statements you learned in computer programming or that are on your scientific calculator are directly related to set theory).

Now that the explanation bits are out of the way, we get into how A Beka screws its math students over, and why they hate set theory. This is how it affects the students:

Quote:In pretty much ANY maths above the pre-algebra level (including geometry, parts of precal, pretty much ALL of calculus, even forms of business math like finite math and such) you are going to HAVE to know at least a little about set theory. Needless to say, if you don't know about set theory, you...aren't really going to do well at all in college course material.

There's also very important research going on where set theory is almost a prerequisite to understand things--especially (keeping this nice and Internet related!) in regards to the fields of cryptography. (One of the big questions in set theory now--which directly relates to things like your browser's security, the security of your GPG keys and so on--is whether "P" equals "NP" (in fact, it's THE biggest question in computer science, and the person or persons who successfully come up with a provable proof either way will likely become a millionaire). Whether P=NP or not has huge implications for things like the security of cryptography--if P does equal NP, theoretically it is MUCH easier to crack a secure cryptographic key (such as used in GPG or the SSL in your web-browser used for secure web pages); some people even think the NSA has a proof of this but isn't willing to give it out for fear of people knowing they can crack PGP keys! So yes, there ARE practical applications for set theory)

...

Needless to say, it is no shock whatsoever to me (again, as a maths geek and maths major) that California's state university system considers A-Beka's curriculum deficient as far as its maths go! Especially in such things as computer science, there is no way you can really go about it unless you've had at least pre-cal, and I ran into set theory in geometry and algebra in high school! Kids who are educated in A-Beka are essentially being sent to college with the equivalent of pre-algebra, if that. Most public school systems wouldn't allow you to graduate without at least learning algebra and geometry (both of which touch upon set theory, if in a basic form).

And here's the bit that pisses me off, another case of religious fanaticism corrupting education. Watch him explain why set theory doesn't set well:

Quote:Now, dominionists (to put it mildly!) probably don't like set theory and see it as absolutely evil for two reasons:

a) the whole concept of infinities within infinities (which doesn't set well, if you'll pardon the pun, with their concept of only one God, and two sets of humans that will never interact--the Saved and the Damned)

b) the entire concept of number theory and, for that matter, logical thought systems (in that entire systems of mathematics may be proven or disproven mathematically via logical proof; the entire idea of logic is frowned upon, partly because it encourages independent thinking and testing of a theorem (whereas dominionists are explicitly taught to trust their leaders and have faith, not thought) and in general doesn't work in a coercive society).


Another example of where A Beka's likely objections to set theory lie is in a particular part of advanced set theory known as the Axiom of Choice. What the Axiom of Choice is, in a sense, is it is an explicit acknowledgement that the particular system you use to work with a set is one's choice, and you're essentially choosing specific members of that set to work with. (A much better layman's description (well, if you've had high school maths, anyways) is at http://www.math.vanderbilt.edu/~schectex...hoice.html and sci.math has an entire FAQ on it.)

The Axiom of Choice has some interesting implications, and not just related to maths--much of it touches on the whole nature of how we construct numbers and maths at all (in some variants of the Axiom of Choice, it's acknowledged numerical systems are an artificial construct or "function", for instance).

The Axiom is important in computer science, for instance, because with some things it's acknowledged one must make compromises in certain calculations, and the Axiom of Choice is one of the factors used in, for instance, determining if one focuses on precision rather than speed in a calculation. (This is also one of those things where the whole P=NP/P!=NP debate comes up--if P=NP, by the Axiom of Choice one can optimise for searching out P problems in a set, which makes it much easier for, say, Three Letter Agencies to break PGP keys and such.)

The entire idea of mathematics being a construct is...to put it mildly...something that severely breaks the brain of your average dominionist. It doesn't set so well with mathematics being God's clockwork set. (I've actually seen similar objections to quantum mechanics in the dominionist community too--the whole "God Doesn't Throw Dice, Much Less Throw Them Where He Can't See Them" argument.)

After reading this over again, my faith in getting through college has dropped significantly. All because of a piss-poor curriculum designed by religious lunatics.

If you care enough to read Matt's relatively-huge take on A Beka in its entirety, see here: http://forums.school-survival.net/showth...?tid=35113

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brainiac3397 Offline
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A Beka's hatred for set theory

I thought mathematicians believed math was the logical language of the universe unburdened by the emotional subjectivity of humanity?

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(06-14-2013 08:02 AM)Potato Wrote:  watch the fuq out, we've got an "intellectual" over here.

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magikarp Offline
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RE: A Beka's hatred for set theory

(09-24-2015 03:09 PM)Hawkbit Wrote:  Set theory is, in essence, an extension of work in regards to discovering that there are an infinite number of real numbers--thus a way was needed to sort infinities. (Yes, it is possible to have multiple infinities in higher maths. This is one of the concepts that break people's brains.) A good example of set theory in action:

All real numbers (which is an infinity) are in set A. All real numbers that are divisible by 2 (which is also an infinitely long list of numbers) are in set B. All real numbers that are divisible by 3 (a separate, but also infinitely long list of numbers) are in set C.

Sets B and C will intersect; set A will include both set B and C, but there are numbers in set C that aren't in set B and vice versa.

You can even extend this--if number X is not divisible by 2, and is not divisible by 3, it goes in set A outside set B and set C.

One of the niftier things with this is that by doing this you can sort of "size" how big an infinity is. Set A is a huge infinity, B and C are smaller infinities. The intersection of B and C is a smaller infinity yet.
B and C, and the intersection of B and C have the same 'size' (cardinality).

Formally, you have:

B = {2n | n is a non-negative integer}
C = {3n | n is a non-negative integer}
B intersect C = {6n | n is a non-negative integer}

[I'm arbitrarily restriction this to non-negative numbers to keep it neat, but the same logic holds if you don't.]

Two sets have the same cardinality if you can definite a bijection between them. Informally, this means that for a set X and a set Y, they have the same cardinality if and only if there exists a function such that every element in X is paired with exactly one element in Y, and every element in Y is paired with exactly one element in X.

For example, for sets B and C:

2 -> 3
4 -> 6
6 -> 9
8 -> 12
...

Every element in B is paired with exactly one element in C, and vice versa, and the function can be extended for arbitrarily large elements of B and C.

The same idea can be used with B and the intersection of B and C:

2 -> 6
4 -> 12
6 -> 18
8 -> 24
...

The set of real numbers doesn't have the same cardinality as B or C, but the proof is kind of long so I'm not going to type it here.

Anyway, the point of this wasn't to like argue with a small point for no reason, but to point out that one of the weird things about set theory is that a set X may have the same cardinality as a strict subset of X.

edit: Also college math for non-math majors is kind of a joke unless you're trying to get into like Stanford or something, so I wouldn't worry about it.

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(This post was last modified: 09-25-2015 02:57 AM by magikarp.)
09-25-2015 02:54 AM
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