Reasonable Effectiveness of Mathematics: A paper published to FQXi.
INTRODUCTORY
The validity of a mathematical statement is judged by its logical consistency. The validity of a physical statement is judged by its correspondence to reality. We collect too much data and without judging properly reject most (like at LHC). If we re-envision classical and quantum observations as macroscopic overlap of quantum effects, we may solve most problems. The physics community blindly accepts rigid, linear ideas about the nature of space, time, dimension, etc. These theories provide conceptual convenience and attractive simplicity for pattern analysis, but at the cost of ignoring equally-plausible alternative interpretations of observed phenomena that could possibly have explained the universe better. Modern theories do not give a precise definition of the technical terms used, but give an operational definition that can be manipulated according to convenience. Wigner1 defined mathematics as the science of skillful operations with concepts and rules invented just for this purpose. This is too open-ended. What is skillful operation? What are the concepts and Rules? Who invented them? What is the purpose? Do all concepts and rules have to be mathematical? Wigner says: The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible, but leaves out what is permissible and what is not; leaving scope for manipulation.
Relations between material objects must be expressed in a language compatible with the way in which objects in the real world actually interact – through the transmission/reception of mass/energy/information. Every object is a summation of the same fundamental stuff (quarks, leptons, etc) in varying orders. Events are energy rearranging fundamental particles. The space-time location makes intervals in both space and time dependent on where we measure them from. This implies space-time is related to the origin of the coordinates of the observer’s frame of reference. Measurement is carried out at here-now – thus, time variant (since now is the fleeting interface between past and future). Its quantitative description is mathematics – it describes the changing physical phenomena when the number or arrangement of any of the constituent parameters is changed. The changes are expressed as the result of measurement after comparison with a scaling constant (standard unit). These are always pure numbers, i.e., scalar quantities, because measurement is only the operation of scaling up or down the unit for an appropriate number of times. The results of measurement, which are time invariant, are frozen even though the object measured continues to evolve in time. Your 10 year old photo is not you.
Mathematics is the ordered accumulation and reduction in numbers of the same class (linear or vector) or partially similar class (non-linear or set) of objects. Number is one of the properties of all substances by which we differentiate between similars. If there is nothing similar at here-now, the number associated with the object is one. If there are similars, the number is many. Our sense organs and measuring instruments are capable of measuring only one at a time. Thus, many is a collection of successive one’s. Based on the sequence of perception of such one’s, many can be 2, 3, 4….n. In a fraction, the denominator represents the one’s, out of which some (numerator) are taken. Zero is the absence of something at here-now that is known to exist elsewhere (otherwise we will not perceive its absence at all).
Burrowing from M. Polanyi, Wigner says1: The principal point …. is that the mathematician could formulate only a handful of interesting theorems without defining concepts beyond those contained in the axioms and that the concepts outside those contained in the axioms are defined with a view of permitting ingenious logical operations which appeal to our aesthetic sense both as operations and also in their results of great generality and simplicity. Wigner admits not only the incompleteness of mathematics but also its manipulation according to the aesthetic sense of the operator. He gives the example of complex numbers and burrowing from Hilbert2, admits: Certainly, nothing in our experience suggests the introduction of these quantities. Indeed, if a mathematician is asked to justify his interest in complex numbers, he will point, with some indignation, to the many beautiful theorems in the theory of equations, of power series, and of analytic functions in general, which owe their origin to the introduction of complex numbers. The mathematician is not willing to give up his interest in these most beautiful accomplishments of his genius. A reverse self-fulfilling effect!
Negative numbers are related to mutually exclusive objects or events of a coupled system. For example, position (fixed coordinates) and momentum (mobile coordinates) are mutually exclusive. In two accelerating frames of reference, one who gains has positive value corresponding to the negative value of the other. Since one is without similars, it does not change the value in any operation except linear addition and subtraction (becoming many or zero). Thus, squaring or square-root of 1 is 1 (these involve field). Since negative numbers belong to mutually exclusive couplets and not exclusive one’s, complex numbers are neither physically nor mathematically valid. No computer algorithm is possible using complex numbers.
Infinity is like one: without similars. But whereas the dimensions of one are fully perceptible, the dimensions of infinity are not perceptible. There cannot be negative infinity to positive infinity through zero, as it will show one beginning or end of infinity at the zero point, which is non-existent at here-now. No mathematics is possible with infinity, as all operations involving it will have undefined dimensions – thus indistinguishable from each other. History shows that whenever infinity appears in any theoretical model, it points to some fundamentally different and novel phenomena. In aerodynamics formulas, as the velocity approached the velocity of sound in the medium where the aircraft moved, the resistance of the medium returned infinite figure. It was believed that supersonic motion is impossible. But when supersonic motion became obvious, the formulas were reviewed. It was noted that they described resistance only in a continuous medium without abrupt jumps in density and pressure. However, transition from subsonic to supersonic motion involves a shock wave in front of the body, leading to abrupt jumps in density and pressure. When these factors were taken into account, the infinity vanished.
The so-called irrational numbers are also perceived as the nearest fraction of integers. Otherwise, we cannot use them in programming. We may be as precise as we want to fix the value of a number tending to zero, but it will never be zero, as that will make it non-existent at here-now making the operation impossible. Similarly, a number tending to infinity will never become infinite, as the result of all such operations become indistinguishable from each other. Like energy, infinities coexist. Only, space, time, coordinates and consciousness are infinite.
Language is the transposition of information to another system’s CPU or mind by signals or sounds using energy (self communication is perception). The transposition may relate to a fixed object/information. It can be used in different domains and different contexts or require modifications in prescribed manner depending upon the context. Since mathematics follows these rules, it is also a language. Mathematics explains only how much one quantity, whether scalar or vector; accumulate or reduce linearly or non-linearly in interactions involving similar or partly similar quantities and not what, why, when, where, or with whom about the objects. These are subject matters of physics. The interactions are chemistry. There is no equation for Observer. The enchanting smile on the lips of the beloved is not the same as geometry of mouth or curvature of lips. Thus, mathematics is not the sole language of Nature.