I challenge the student of mathematics 
I challenge the student of mathematics 
coberst 
Sep 08, 2009, 02:30 PM
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#1

DemiGod Group: Basic Member Posts: 721 Joined: May 21, 2007 Member No.: 11167 
I challenge the student of mathematics
It appears to me that most people look on math as something with supernatural qualities. I challenge the student of math to develop and post short essays on Internet discussion forums about those fundamental aspects of math that you think people can and should comprehend. What follows is something that I have posted regarding my idea of what ordinary citizens should know abut this very fundamental domain of knowledge. Arithmetic is object collection It is a hypothesis of SGCS (Second Generation Cognitive Science) that the sensorimotor activity of collecting objects by a child constitute a conceptual metaphor at the neural level leading to a primary metaphor that ‘arithmetic is object collection’. The arithmetic teacher attempting to teach the child at a later time depends upon this already accumulated knowledge. Of course, all of this is known to the child without the symbolization or the conscious awareness of the child. The pile of objects became ‘bigger’ when the child added more objects and became ‘smaller’ when objects were removed. The child easily recognizes while being taught arithmetic that 5 is bigger than 3 and 3 is littler than 7. The child knows many entailments, many ‘truths’, resulting from playing with objects. The teacher has little difficulty convincing the child that two collections A and B are increased when another collection C is added, or that if A is bigger than B then A+C is bigger than B+C. At birth an infant has a minimal innate arithmetic ability. This ability to add and subtract small numbers is called subitizing. (I am speaking of a cardinal number—a number that specifies how many objects there are in a collection, don’t confuse this with numeral—a symbol). Many animals display this subitizing ability. In addition to subitizing the child, while playing with objects, develops other cognitive capacities such as grouping, ordering, pairing, memory, exhaustiondetection, cardinalnumber assignment, and independent order. Subitizing ability is limited to quantities 1 to 4. As a child grows s/he learns to count beyond 4 objects. This capacity is dependent upon 1) Combinatorialgrouping—a cognitive mechanism that allows you to put together perceived or imagined groups to form larger groups. 2) Symbolizing capacity—capacity to associate physical symbols or words with numbers (quantities). “Metaphorizing capacity: You need to be able to conceptualize cardinal numbers and arithmetic operations in terms of your experience of various kinds—experiences with groups of objects, with the partwhole structure of objects, with distances, with movement and location, and so on.” “Conceptualblending capacity. You need to be able to form correspondences across conceptual domains (e.g., combining subitizing with counting) and put together different conceptual metaphors to form complex metaphors.” Primary metaphors function somewhat like atoms that can be joined into molecules and these into a compound neural network. On the back cover of “Where Mathematics Comes From” is written “In this acclaimed study of cognitive science of mathematical ideas, renowned linguist George Lakoff pairs with psychologist Rafael Nunez to offer a new understanding of how we conceive and understand mathematical concepts.” “Abstract ideas, for the most part, arise via conceptual metaphor—a cognitive mechanism that derives abstract thinking from the way we function in the everyday physical world. Conceptual metaphor plays a central and defining role in the formation of mathematical ideas within the cognitive unconscious—from arithmetic and algebra to sets and logic to infinity in all of its forms. The brains mathematics [b]is mathematics, the only mathematics we know or can know.”[/b] We are acculturated to recognize that a useful life is a life with purpose. The complex metaphor ‘A Purposeful Life Is a Journey’ is constructed from primary metaphors: ‘purpose is destination’ and ‘action is motion’; and a cultural belief that ‘people should have a purpose’. A Purposeful Life Is A Journey Metaphor A purposeful life is a journey. A person living a life is a traveler. Life goals are destinations A life plan is an itinerary. This metaphor has strong influence on how we conduct our lives. This influence arises from the complex metaphor’s entailments: A journey, with its accompanying complications, requires planning, and the necessary means. Primary metaphors ‘ground’ concepts to sensorimotor experience. Is this grounding lost in a complex metaphor? ‘Not by the hair of your chineychinchin’. Complex metaphors are composed of primary metaphors and the whole is grounded by its parts. “The grounding of A Purposeful Life Is A Journey is given by individual groundings of each component primary metaphor.” The ideas for this post come from Philosophy in the Flesh. The quotes are from Where Mathematics Comes From by Lakoff and Nunez 
Flex 
Sep 14, 2009, 02:42 PM
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#2

God Group: Basic Member Posts: 1954 Joined: Oct 17, 2006 From: Bay area CA Member No.: 5877 
What people should know about math:
It is a human creation, and is based solely on observations. This being said, mathematics is every bit as fallible as the rest of scientific discovery. It may be practical, but has no real meaning, and for the common folk will be of very little use. Your intuition will do pretty much all of the computing you need i.e. spatial relationships via sight an sound. 
code buttons 
Sep 14, 2009, 05:07 PM
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#3

Supreme God Group: Basic Member Posts: 2451 Joined: Oct 05, 2005 Member No.: 4556 

Flex 
Sep 15, 2009, 01:37 PM
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#4

God Group: Basic Member Posts: 1954 Joined: Oct 17, 2006 From: Bay area CA Member No.: 5877 
Math is an observation itself. Most mathematical observations deal with rates. We create an arbitrary distance, and an arbitrary conception of time, and observe how nature acts. Nature exists before math. I repeat math is an OBSERVATIONcomputations are a way to somewhat reliably predict how the natural world will function in the future. The better our observations, the better our predictive powers.
I can see an apple is red. A spectrometer can see that the apple reflects light with around a 700nm wavelength. No need to quantify the results for the layman. 
doodoobrain 
Jul 20, 2013, 12:03 PM
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#5

Newbie Group: Basic Member Posts: 1 Joined: Jul 20, 2013 Member No.: 35244 
Math is an observation itself. Most mathematical observations deal with rates. We create an arbitrary distance, and an arbitrary conception of time, and observe how nature acts. Nature exists before math. I repeat math is an OBSERVATIONcomputations are a way to somewhat reliably predict how the natural world will function in the future. The better our observations, the better our predictive powers. I can see an apple is red. A spectrometer can see that the apple reflects light with around a 700nm wavelength. No need to quantify the results for the layman. WOW. You seriously misunderstand the nature of mathematics. Or at the very least you are conflating mathematics and either physics or computation. Math is MUCH more than computation. Perhaps your idea of math is limited to elementary arithmetic or algebra, or perhaps the only math you know is that used for the purpose of solving engineering or physics problems. This type of math is just an extremely small portion of applied math. Math is also not physics. Math is not based on observations of the physical world. The entire philosophy behind math is using nothing but logic in order to solve problems. It is easily one of the highest endeavors in human thought to determine what problems may be stated and solved wholly within the realms of logic. Pure math does not closely resemble applied math. Pure math has grown into a staggeringly large body of knowledge containing many distinct fields. Math does not require any knowledge from the physical world in order to be developed, but it is indeed true that it has enormous amounts of application to things in the physical world. Here are some examples: 1. The concept of a Fourier transform is a rather abstract notion from the field of analysis, which you may think of as being similar to calculus, and this has given rise to algorithms that achieve an impressive compression ratio for data, and without which cellphone technology and other data transmissions technology would be severely retarded. It is just not possible to transmit that much raw data in real time. 2. The field of algebraic topology is even more abstract than analysis. Topology has its roots in geometry, but it deals with studying space itself, and not shapes. Using topology, mathematicians have been able to learn a mindboggling amount about space: how can it be compressed or stretched into other configurations. For example, did you know if you take two solid 4dimensional balls and glue their surfaces together in a certain way that you will obtain the boundary of a solid 5dimensional ball? Or did you know that it is impossible to deform a 2dimensional sphere smoothly into the shape of a 1dimensional sphere, but it is possible to deform a 3dimensional sphere smoothly into the shape of a 2dimensional sphere? If you were an algebraic topologist you would know not only these facts but all the unique ways of performing these operations. I would like to emphasize that these statements have absolutely nothing to do with the physical properties of materials, but only whether certain shapes or configurations of space are theoretically possible to exist. All of this was figured out on a chalkboard, with no laboratories. And yet, the knowledged obtained from the fields of algebraic and differential topology have manifold applications to theoretical physics (pun intended). 3. The field of algebraic number theory is arguably more abstract even than algebraic topology. It incorporates many concepts from abstract algebra and field theory, which originated over a century ago and are each still areas of ongoing research. There are several entirely distinct fields of number theory, and algebraic number theory is one which has many applications to cryptography. A large portion of the people doing classified cryptographical work in government agencies have a good understanding of algebraic number theory. ANT is central in developing a sound enough understanding of the subtle relations between numbers, especially prime numbers, to create the modern encryption algorithms (RSA, etc.). While I might not disagree that prime numbers are observable in nature, I don't think that the intricate structure existing within prime numbers is consistent with your claims of observation. In order for any of the above concepts to be developed, brilliant, hardworking humans have had to spend centuries thinking and thinking, but not experimenting. Often physical observations will influence mathematicians by giving them ideas of what types of math to develop, but this does not affect the veracity of the resulting math. It is in this way that mathematics distinguishes itself from any other field of study. If you take the body of knowledge of sociology developed in America, it may or may not be applicable when you direct your attention to a foreign country, but this is not the case for biology. Humans (and animals) will have the same fundamental biology no matter what continent you are on. However, if life were to exist on another planet, there would be absolutely no guarantee that any biology you know from Earth will have any use in understanding the aliens. Still, physics will be the same on that other planet as it is on ours (presumably), as it will be at any location in this universe. However, as many physicists believe, there could be many other universes in existence, either now or in the past or future, and in these universes none of the physics we think we know from our studies on Earth would have any guarantee to hold. The point I want to make is that no matter what continent, planet, or universe you are in, every bit of math will still be just as true, because it is formulated from pure logic, nothing more. 
Flex 
Jul 21, 2013, 06:20 PM
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#6

God Group: Basic Member Posts: 1954 Joined: Oct 17, 2006 From: Bay area CA Member No.: 5877 
You are absolutely correctI was limited in my comprehension of math. Let me put it another way:
Say you have two magnets. They feel each others presence very strongly when in close proximity. As you move away the magnets apart they become less aware of on another, they less conscious of one another presence. Since the strength of the magnetic field decays exponentially, magnetic bodies far apart will be much less conscious; however, they still feel some force, since the function has an asymptote at 0, the magnets can be arbitrarily far apart yes still feel the presence of one another. In quantum mechanical terms, an electron orbital is defined as 95% probability that the electron will be found in that particular locationbut may still exist anywhere in the universe. The super positioning of all of the orbitals in the universe gives rise to the shape of all things in the universe. We perceive ourselves to be separate entities due to the proximity of out atoms. In reality, using the timedependent Schrodinger equation, while applying the concept of super positioning, life itself is like a quantum holographic display. We perceiver space to be void of matter (loval minimums) while we perceive local maximums as something which is tangible. 
Flex 
Jul 22, 2013, 05:27 PM
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#7

God Group: Basic Member Posts: 1954 Joined: Oct 17, 2006 From: Bay area CA Member No.: 5877 
While we are at it, here is my take on quantum consciousness:
A rock has consciousness in that it is aware of its surroundings (i.e. iron works great for compasses). As the biological system becomes more complex, we perceive it to be more conscious (i.e. a virus or bacteria that responds to the environment). Then we develop even more complexity till you get to the level of a cuddle fish, dolphin, elephant, etc. So why then do we perceive ourselves as conscious entities separate entities apart from everything in the Universe? I believe it has to do with the proximity of atoms in our bodies and how closely. Why? Asymmetric crystalline structures are pretty rate in the Universe; save for "living" conscious creatures. In biological systems, there are plenty of asymmetric crystalline structures, such as DNA. When in crystalline for it DNA possesses a property called birefringence, which means it has a property called birefringence. DNA polarizes light in two directions due to the asymmetry of the crystal. Proteins also have regions of asymmetric crystalline structure such as beta sheets or alpha helices. These regions protein structure are highly conserved evolutionarily because they are the sites where the actual chemistry goes down, where as nonessential biological sites such as random coils are much less essential to the function of the protein, and consequently these regions mutate at a much faster rate. Asymmetric crystals can be used to "entangle" two photons if passed through an asymmetric crystal. Once entangled, the two photons will instantaneously respond to one another faster than the speed of light, disobeying Einstein's theory of relativity, in which traveling faster than the speed of light is not possible. I theorize that if during a chemical reaction, such as a ligand binding to a protein, two photons are given off and thus they are entangled. To prove quantum consciousness, one only need 2 asymmetric crystals such as a prism: 2 photons pass through crystal A one proton is left as is, while the other photon passes through crystal B with another photon. If the changing the spin of one of the three photons causes the other two to react, then quantum consciousness is real. Another Idea of mine is to loop tie the two end of DNA together, making it a Mobius loop (an inorientable structure that bas volume in 3 dimensions, but no volume in 4 dimensions (time dependent Shrodinger's equation). Doing so would mean that telomeres would never become shorter since the 3' and 5' ends of the DNA could be continuous. A way to study epigenetic would be to pass a photon through the middle of a crystalized DNA helix, a vacuum. It ought to move at the speed of light according to Einstein. The spin I theorize would trace the backbone of DNA, spinning at different rates depending on how the bases are stacked. This data could then be used to do epigenetic studies, by creating a Bessel function. One could transfer this wave data to generate a song should they so desire Each genome would be unique even identical twins. Since their environment, mood, food, etc. directly impact our genomes and are heritable from one generation to another. It is sort of like how the US constitution is set firm, yet judicial review and amendments are passed down from one generation to the next. Tons more ideas where that came from, but I will save that for another time. My puppy misses me. 
Omen 
Dec 17, 2013, 04:03 AM
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#8

Newbie Group: Basic Member Posts: 2 Joined: Dec 14, 2013 Member No.: 36729 
Everybody should know about set theory and discrete mathematics. In extension of this, I also think that everybody should learn what logical fallacies are.
To answer the poised question fully, I should add that I believe the ordinary Joe will appreciate the greater understanding of the world around him, that learning these concepts will bring. 
LoFi Version  Time is now: 22nd July 2018  11:15 PM 