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How do people understand algebra if they never encounter it?

File:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg

early Arabic treatise on algebra, 8th c AD

In “Geometric Principles Appear Universal in Our Minds” (Wired Science, May 24, 2011) , Bruce Bower reflects on the fact that research among peoples who do not even count suggests that abstract geometric principles are probably innate in humans:

If geometry relies on an innate brain mechanism, it’s unclear how such a neural system generates abstract notions about phenomena such as infinite surfaces and why this system doesn’t fully kick in until age 7. If geometry depends on years of spatial learning, it’s not known how people transform real-world experience into abstract geometric concepts — such as lines that extend forever or perfect right angles — that a forest dweller never encounters in the natural world.

As always, we needn’t wait long for a Darwin answer:

Whatever the case, the Mundurucú’s keen grip on abstract geometry contrasts with past evidence from Izard’s group that these Amazonian villagers cannot add or otherwise manipulate numbers larger than five. Geometry may have a firmer evolutionary basis in the brain than arithmetic, comments cognitive neuropsychologist Brian Butterworth of University College London.“If so, this would support recent findings that people who fail to learn arithmetic, or ‘dyscalculics,’ can still be good at geometry,” Butterworth says.

Some find the leap here puzzling: That “dyscalculics” can learn geometry better than arithmetic sheds no light in particular on evolution: Both are branches of mathematics are abstract arts.

It’s one thing to say that evolution explains why wolves are better at sniffing than at the times tables; another to simply plunk an assertion about “evolution” in the midst of discussion of variable human groups’ performance with abstractions, providing no evidence apart from the humble assent of millions that “evolution is true.” But Butterworth is on safe ground. Hard to imagine Bower asking him to explain.

These data are interesting in the light of the fact that the historical development of mathematics puts geometry about a millennium and a half before the development of algebra, implying that the latter follows the former in a hierarchy of concepts. Thoughts?

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5 Responses to How do people understand algebra if they never encounter it?

  1. But alas, besides the profound mystery of us innately understanding such ‘abstract mathematical constructs, why should reality itself be reducible to, and governed by such ‘abstract’ mathematical constructs??? The only answer is that the ‘abstract mathematical constructs’ are not abstract, in the sense of being separate from concrete physical reality, but that concrete physical reality does in fact derive its very being from the ‘immaterial’, abstract, realm of ideas in the first place!!! Moreover, this ‘anomaly’ is very strong evidence that man was indeed made in the ‘image of God’ to have a intimate relationship with the Creator, and ‘sustainer’, of this universe!!!

    =======================

    notes:

    Finely Tuned Big Bang, Elvis In The Multiverse, and the Schroedinger Equation – Granville Sewell – audio
    http://www.metacafe.com/watch/4233012

    At the 4:00 minute mark of the preceding audio, Dr. Sewell comments on the ‘transcendent’ and ‘constant’ Schroedinger’s Equation;

    ‘In chapter 2, I talk at some length on the Schroedinger Equation which is called the fundamental equation of chemistry. It’s the equation that governs the behavior of the basic atomic particles subject to the basic forces of physics. This equation is a partial differential equation with a complex valued solution. By complex valued I don’t mean complicated, I mean involving solutions that are complex numbers, a+b^i, which is extraordinary that the governing equation, basic equation, of physics, of chemistry, is a partial differential equation with complex valued solutions. There is absolutely no reason why the basic particles should obey such a equation that I can think of except that it results in elements and chemical compounds with extremely rich and useful chemical properties. In fact I don’t think anyone familiar with quantum mechanics would believe that we’re ever going to find a reason why it should obey such an equation, they just do! So we have this basic, really elegant mathematical equation, partial differential equation, which is my field of expertise, that governs the most basic particles of nature and there is absolutely no reason why, anyone knows of, why it does, it just does. British physicist Sir James Jeans said “From the intrinsic evidence of His creation, the great architect of the universe begins to appear as a pure mathematician”, so God is a mathematician to’.

    i.e. the Materialist is at a complete loss to explain why this should be so, whereas the Christian Theist presupposes such ‘transcendent’ control,,,

    John 1:1
    In the beginning was the Word, and the Word was with God, and the Word was God.

    of note; ‘the Word’ is translated from the Greek word ‘Logos’. Logos happens to be the word from which we derive our modern word ‘Logic’.

    To solidify Dr. Sewell’s observation that transcendent ‘math’ is found to be foundational to reality, I note this equation:

    0 = 1 + e ^(i*pi) — Euler

    Believe it or not, the five most important numbers in mathematics are tied together, through the complex domain in Euler’s number, And that points, ever so subtly but strongly, to a world of reality beyond the immediately physical. Many people resist the implications, but there the compass needle points to a transcendent reality that governs our 3D ‘physical’ reality.

    God by the Numbers – Connecting the constants
    Excerpt: The final number comes from theoretical mathematics. It is Euler’s (pronounced “Oiler’s”) number: e*pi*i. This number is equal to -1, so when the formula is written e*pi*i+1 = 0, it connects the five most important constants in mathematics (e, pi, i, 0, and 1) along with three of the most important mathematical operations (addition, multiplication, and exponentiation). These five constants symbolize the four major branches of classical mathematics: arithmetic, represented by 1 and 0; algebra, by i; geometry, by pi; and analysis, by e, the base of the natural log. e*pi*i+1 = 0 has been called “the most famous of all formulas,” because, as one textbook says, “It appeals equally to the mystic, the scientist, the philosopher, and the mathematician.”
    http://www.christianitytoday.c.....ml?start=3

    (of note; Euler’s Number (equation) is more properly called Euler’s Identity in math circles.)

    Moreover Euler’s Identity, rather than just being the most enigmatic equation in math, finds striking correlation to how our 3D reality is actually structured,,,

    The following picture, Bible verse, and video are very interesting since, with the discovery of the Cosmic Microwave Background Radiation (CMBR), the universe is found to actually be a circular sphere which ‘coincidentally’ corresponds to the circle of pi within Euler’s identity:

    Picture of CMBR
    https://webspace.utexas.edu/reyesr/SolarSystem/cmbr.jpg

    Proverbs 8:26-27
    While as yet He had not made the earth or the fields, or the primeval dust of the world. When He prepared the heavens, I was there, when He drew a circle on the face of the deep,

    The Known Universe by AMNH – video – (please note the ‘centrality’ of the Earth in the universe in the video)
    http://www.youtube.com/watch?v=17jymDn0W6U

    The flatness of the ‘entire’ universe, which ‘coincidentally’ corresponds to the diameter of pi in Euler’s identity, is found on this following site; (of note this flatness of the universe is an extremely finely tuned condition for the universe that could have, in reality, been a multitude of different values than ‘flat’):

    Did the Universe Hyperinflate? – Hugh Ross – April 2010
    Excerpt: Perfect geometric flatness is where the space-time surface of the universe exhibits zero curvature (see figure 3). Two meaningful measurements of the universe’s curvature parameter, ½k, exist. Analysis of the 5-year database from WMAP establishes that -0.0170 < ½k < 0.0068.4 Weak gravitational lensing of distant quasars by intervening galaxies places -0.031 < ½k < 0.009.5 Both measurements confirm the universe indeed manifests zero or very close to zero geometric curvature,,,
    http://www.reasons.org/did-universe-hyperinflate

    This following video shows that the universe also has a primary characteristic of expanding/growing equally in all places,, which 'coincidentally' strongly corresponds to e in Euler's identity. e is the constant used in all sorts of equations of math for finding what the true rates of growth and decay are for any given problem trying to find as such:

    Every 3D Place Is Center In This Universe – 4D space/time – video
    http://www.metacafe.com/watch/3991873/

    Towards the end of the following video, Michael Denton speaks of the square root of negative 1 being necessary to understand the foundational quantum behavior of this universe. The square root of -1 is 'coincidentally' found in Euler's identity:

    Michael Denton – Mathematical Truths Are Transcendent And Beautiful – Square root of -1 is built into the fabric of reality – video
    http://www.metacafe.com/watch/4003918&quot;

    I find it extremely strange that the enigmatic Euler's identity would find such striking correlation to reality. In pi we have correlation to the 'sphere of the universe' as revealed by the Cosmic Background radiation, as well pi correlates to the finely-tuned 'geometric flatness' within the 'sphere of the universe' that has now been found. In e we have the fundamental constant that is used for ascertaining exponential growth in math that strongly correlates to the fact that space-time is 'expanding/growing equally' in all places of the universe. In the square root of -1 we have what is termed a 'imaginary number', which was first proposed to help solve equations like x2+ 1 = 0 back in the 17th century, yet now, as Michael Denton pointed out in the preceding video, it is found that the square root of -1 is required to explain the behavior of quantum mechanics in this universe. The correlation of Euler's identity, to the foundational characteristics of how this universe is constructed and operates, points overwhelmingly to a transcendent Intelligence, with a capital I, which created this universe! It should also be noted that these universal constants, pi,e, and square root -1, were at first thought by many to be completely transcendent of any material basis, to find that these transcendent constants of Euler's identity in fact 'govern' material reality, in such a foundational way, should be enough to send shivers down any mathematicians spine. Further discussion can be found here relating Euler's identity to General Relativity and Quantum Mechanics:
    http://www.uncommondescent.com.....ent-364379

    Here is a very well done video, showing the stringent 'mathematical proofs' of Euler's Identity:

    Euler's identity – video
    http://www.youtube.com/watch?v=zApx1UlkpNs

  2. f/n; The mystery doesn’t stop there, this following video shows how pi and e are found in Genesis 1:1 and John 1:1

    Euler’s Identity – God Created Mathematics – video
    http://www.metacafe.com/watch/4003905
    This following website, and video, has the complete working out of the math of Pi and e in the Bible, in the Hebrew and Greek languages respectively, for Genesis 1:1 and John 1:1:
    http://www.biblemaths.com/pag03_pie/

    Fascinating Bible code – Pi and natural log – Amazing – video (of note: correct exponent for base of Nat Log found in John 1:1 is 10^40, not 10^65 as stated in the video)
    http://www.youtube.com/watch?v=Wg9LiiSVae

    Another transcendent mathematical structure that is found imbedded throughout our reality is Fibonacci’s Number;

    Fibonacci Numbers – Euler’s Identity – The Fingerprint of God – video – (See video description for a look at Euler’s Identity)
    http://www.metacafe.com/watch/5988843/

    ========================

    Researchers Succeed in Quantum Teleportation of Light Waves – April 2011
    Excerpt: In this experiment, researchers in Australia and Japan were able to transfer quantum information from one place to another without having to physically move it. It was destroyed in one place and instantly resurrected in another, “alive” again and unchanged. This is a major advance, as previous teleportation experiments were either very slow or caused some information to be lost.
    http://www.popsci.com/technolo.....-computing

    Explaining Information Transfer in Quantum Teleportation: Armond Duwell †‡ University of Pittsburgh
    Excerpt: In contrast to a classical bit, the description of a (photon) qubit requires an infinite amount of information. The amount of information is infinite because two real numbers are required in the expansion of the state vector of a two state quantum system (Jozsa 1997, 1) — Concept 2. is used by Bennett, et al. Recall that they infer that since an infinite amount of information is required to specify a (photon) qubit, an infinite amount of information must be transferred to teleport.
    http://www.cas.umt.edu/phil/fa.....lPSA2K.pdf

    Quantum no-hiding theorem experimentally confirmed for first time
    Excerpt: In the classical world, information can be copied and deleted at will. In the quantum world, however, the conservation of quantum information means that information cannot be created nor destroyed. This concept stems from two fundamental theorems of quantum mechanics: the no-cloning theorem and the no-deleting theorem. A third and related theorem, called the no-hiding theorem, addresses information loss in the quantum world. According to the no-hiding theorem, if information is missing from one system (which may happen when the system interacts with the environment), then the information is simply residing somewhere else in the Universe; in other words, the missing information cannot be hidden in the correlations between a system and its environment. (This experiment provides experimental proof that the teleportation of quantum information in this universe must be complete and instantaneous.)
    http://www.physorg.com/news/20.....tally.html

    The following articles show that even atoms (Ions) are subject to teleportation:

    Of note: An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge.

    Ions have been teleported successfully for the first time by two independent research groups
    Excerpt: In fact, copying isn’t quite the right word for it. In order to reproduce the quantum state of one atom in a second atom, the original has to be destroyed. This is unavoidable – it is enforced by the laws of quantum mechanics, which stipulate that you can’t ‘clone’ a quantum state. In principle, however, the ‘copy’ can be indistinguishable from the original (that was destroyed),,,
    http://www.rsc.org/chemistrywo.....ammeup.asp

    Atom takes a quantum leap – 2009
    Excerpt: Ytterbium ions have been ‘teleported’ over a distance of a metre.,,,
    “What you’re moving is information, not the actual atoms,” says Chris Monroe, from the Joint Quantum Institute at the University of Maryland in College Park and an author of the paper. But as two particles of the same type differ only in their quantum states, the transfer of quantum information is equivalent to moving the first particle to the location of the second.
    http://www.freerepublic.com/fo.....1769/posts

  3. ba77 I think you run a cut-and-paste database! I don’t know how you have time to write all that stuff for every blog!

    Here’s my attempt to answer Denyse’s question. In Gen 1:26 we see the creation of male & female, which I have previously identified with the CroMagnon appearance about 60,000 BC.
    http://procrustes.blogtownhall.....port.thtml
    DNA analysis reveals that CroMagnon is 100% modern DNA, which is to say, “in the image of God”.

    Gen 2:7 is the creation of Adam, whom all agree is homo sapiens, and I identify with the Neolithic Revolution about 10,000 BC. It is also associated with the development of language, and you will note that everything after Gen 2:7 has a name, whereas nothing before that was named.

    But it is CroMagnon cave paintings that astonished the world. So here we have a dilemma. Before Adam, before language, there was no moral injunction, no original sin, no Fall. Somehow, sin is related to language. But language is not related to art, since CroMagnon could certainly paint.

    Therefore sin is a word that didn’t exist before language, and algebra is a language of math. But art is in our genes, and like geometry, requires no words.

  4. Dr. Sheldon, interesting ‘spiritual’ insight as to the ancient art.,,, as to my ‘cut-and-paste database’,, Yes, I have collected quite a few references over the years in defense of ID, Theism in general, as well as Christianity,,,,

  5. I’ve encountered algebra and I still don’t understand it.

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