Lancelot Hogben, in the book Mathematics for the Million, states that

Transcription

1 Lancelot Hogben, in the book Mathematics for the Million, states that mathematics is the language of size, shape, and order. (p. 22, Mathematics for the Million) In the section on the development of algebra, Hogben lists some of the changes from rhetorical algebra where rules were written out in full to symbolic algebra expressed with the aid of letters and operative signs exclusively. (p. 259) This brief note adds to the vocabulary of mathematics a new verb, expressed by some symbol like ~ to mean develops or develop, and new adjectives in the form of a description of conditions written in parentheses. Development Strings, Development Patterns and String Theory Consider a description of action potentials, the sequence of electrical changes occurring when a nerve cell membrane is exposed to a stimulus that exceeds its threshold, in nerves and muscles. 1 action potential + 1 action potential = 2 action potentials is the conventional expression. The new expression I am arguing for would say: 1 action potential ~ (or develops) into 2 then 3 then 4 then 5 action potentials. In tachycardia, or rapid heartbeat, 1~ 2 then 3 then 4 with an increased rate. In fibrillation, a kind of arrhythmia or irregular heart beat, 1~ 2 action potentials, or electrical/chemical changes, then 3 then 4 then possibly 6 or 8 action potentials, or electrical/chemical changes. The muscle contractions are uncoordinated and irregular, so the action potential pattern is irregular. Fibrillation might have a unique development of action potentials, or electrical/chemical changes, 1

2 compared to a normal heart. We can call this new language a development string or development pattern and the impact might be a better description of phenomena than are currently described. A pattern to the development string could possibly be established leading to lessons on how fibrillation should be dealt with. A development string could also be used to describe photon emission from an atom and the phenomenon of non-locality. An atom induced to simultaneously emit two photons in opposite directions ~ entangled photons Measure an entangled photon on one side of lab ~ affect the other photon instantaneously on other side other lab. (p.62, Quantum Physics- A Beginner s Guide to the Subatomic World) Relevance to String Theory In String Theory, the String is argued to be the fundamental component of structures in the universe; the string vibrates and different vibrations are said to be different types of matter and objects. I will assert, perhaps too boldly, that the assertion that the String is the fundamental structure, rather than a vibrating box, sphere, circle, triangle, hyperbola, ellipse, or any other kind of shape or object, is an arbitrary assertion. Any type of object could vibrate and be a fundamental structure of the universe. Similarly, my use here of the term development string is just an arbitrary analogy. The term is used simply to conjure up images of a pattern of events that relate like how a string relates to two cats playing with the string. The string connects the two cats. A ball of string looks like a ball, but it is also a long string which could be laid out and form just a line of string. The idea of a development string is like this ball of string; we see a 2

3 ball, but a line of string is the component of the ball. The new mathematical word develops relates events to each other. The pattern might form a ball, but the component pieces might look like a line of string. Somehow, the line of string can develop into a ball of string; the development string concept relates phenomena to each other. Patterns can, of course, be misleading. The pattern ABABAB does not necessarily have the next letter as A, even though that is the pattern. Any letter or symbol could follow next. The key point here is that a verb ~ or develops gives another word in the language of math. Conditions Language Another linguistic change would be concentrations, temperatures, pressures, and other conditions included in expressions describing phenomena. 1 action potential (high concentration of K, potassium) ~ 0 action potentials. and 1 action potential (low K, potassium, concentration) ~ disordered and irregular potentials or a distinct pattern instead of 2 then 3 then 4 etc. Many equations could be described better with Conditions Language. 1. E=mc 2. With the parentheses language I have described, one could describe conditions like temperature, pressure, location near a black hole and time passage. At low temperatures, the energy, or capacity to do work, is vastly different than the capacity to do work at different temperatures. This language change would include variables like this in Einstein s famous equation. 3

4 mc 2 ~ E (temperature x outside a black hole) mc 2 ~ different value of E (temperature y inside a black hole) Additionally, E=mc 2 only describes certain mass energy relationships. For example, as argued in a Chicago Tribune Letter to the Editor called Einstein s Equation by this author on December 10, 2000, hydrogen fusing into helium is one set of phenomena that E=mc 2 correctly describes. There are many, many other relationships between mass and energy which have been proven to follow Einstein s famous equation. Popular discussion of Einstein s equation often misapplies it to all relationships of mass and energy. Only the mass lost in certain phenomena can be multiplied by c 2 to equal energy. Only certain mass energy relationships have been proven to follow the relationship. Dark energy has not been proven to follow the equation and dark matter has not been proven to follow the equation. It could be argued that applying conditions to equations would lead to more misunderstandings of equations, but some conditions would limit the number of some misinterpretations and ensure that only experimentally proven cases are learned. 2. Nuclear Annihilation in Political Science Hypotheses. Arguments have been made that lack of negotiation, excessive competition, lack of respect for others, or growth of weapons like nuclear weapons could lead to annihilation of mankind with nuclear weapons. One could express relationships like this with conditions language. 4

5 Growth of weapons + Excessive Competition (Lack of respect for others, lack of support, lack of food, lack of water and other resources, lack of negotiation) = Nuclear Annihilation Of course, this equation is just an example; the variables are just possible examples. The equation is not the result of any careful study or analysis of political science concepts. The point is however that a different set of conditions would give rise to a different result. Growth of weapons + Excessive Competition (Negotiation, support, resources and respect for others, arms control) = Avoidance of Annihilation. 3. Virtual particles. Virtual particles are another example where the language might be helpful. 1 virtual particle ~ one pattern of development 1, 2, 3 virtual particles ~ might develop into a unique pattern of development, because of the unique interaction of the probability functions of the particles. Different numbers of virtual particles might become real particles as the particles - and their probability functions- interact with each other. The conditions determine the relationship. With quantum mechanical interactions of small particles, one equation describes the relationship. Particles pop in and out of existence in different places. With large masses, like the Sun and Earth, the masses behave differently. The idea becomes a commentary on the struggle that Einstein faced unifying quantum mechanics with gravitational theory. With billions of particles in the Sun and Earth, the probability functions describing the particles interact 5

6 differently. The conditions of particles are critical to understanding particle interaction. If there are many particles, that is a different condition than if there are few particles surrounding each other particle. 4. Coordinate system of space-time. A set of coordinate axes describing space-time in which there is very little mass will develop into a different set of coordinate axes when more mass moves into the area. The curvature-gravity according to general relativity- of space-time will change. The conditions alter graphical relations as well as relationships between particles, numbers, behaviors and ideas. 5. Conditions language is critical to physics, engineering, medicine and many other problems. Einstein s Equation E=mc 2 (for most mass energy relationships not proven for dark matter or dark energy) Coulomb s Law F= k qq/r 2 (conditions x, y, b but not proven for conditions y, f, d) Newton s Law of Gravitation F= G m m E / r 2 (for r < infinity, not proven for all values of m, not proven for r = infinity) Equation for a Circle 2 Pi r = c Pi = c /d (for r>0) Angular Magnification of a Telescope 6

7 M= -f 1 / f 2 (not for very large arrays, only certain types of telescopes) Intensity Level of a Sound Wave B= 10 log I / I o (for I o corresponding to a sound for person with normal hearing) Multidimensional Systems Set of equations describes universe (4 space time dimensions) Not = Set of equations describes universe (higher dimension like 25 th dimension). Cardiac Muscle Contraction Equation describing cardiac muscle contraction (Strength of contraction determined partially by the extracellular calcium ion concentration and other factors). (p. 286, Human Anatomy and Physiology) Each condition helps describe the equation, so it is possibly better understood. These modifications to the language of mathematics might help describe more phenomena and possibly lead to a better understanding of phenomena. Resources Consulted Gribbin, John. (2002). Quantum Physics A Beginner s Guide to the Subatomic World. New York, NY: DK Publishing, Inc. Hogben, Lancelot. (1993) Mathematics for the Million-How to Master The Magic of Numbers. New York, N.Y.: W.W. Norton. 7

8 Hole, John W. Jr. (1993) Human Anatomy and Physiology. Dubuque, Iowa: Wm. C. Brown Publishers. Sears, Francis W. and Mark W. Zemansky and Hugh D. Young. (1990) College Physics. Seventh Edition. Reading, Mass: Addison-Wesley Publishing Company. (source of all physics equations without conditions). Struck, James Timothy. Einstein s Equation. (Letter to the Editor) Chicago Tribune. December 10 th, Acknowledgements Brian Green s Nova television show Elegant Universe helped me learn about string theory. Two mathematics professors at universities looked at a version of this submitted in March 2000 and One did not have time to help with it, and the other said it was not his field. I am still grateful for their time. Dr. Steve Cohen offered comments back in Dr. Silvio Levy also looked at an earlier version of this. Richard Garwin helped me learn about the threat of nuclear war and the importance of talking about the threat in a lecture at the Illinois Institute of Technology on 11/5/03. Kinko s, Roosevelt University, and IIT provided me with computers to get this done. The University of Chicago s John Crerar Library had a webpage that allowed me to learn about the CERN preprint server. Paula Koperniak offered comments about the article. Thanks to CERN for allowing this to be looked at by others. 8

9 Jane Goodall once wrote that only when we understand can we care, and that only when we care sufficiently will we help. ( Profile--Field Tested Lincoln Park Zoo For Members of Lincoln Park Zoo Fall 2003, p. 12) I include this only because it is a set of conditions describing human behavior; this could be described by this language change too. 9