Scientists, particularly Physicists and others regard Mathematics as the final proof of anything. It is commonly held that once a proposition is able to find some mathematical argument in its support, it is deemed proved and settled for good. Physicists have even invented the term “Mathematical-logical derivation and elevated it to the level of gospel truth. The current cosmic view held by Physicists is a glaring example of where mathematical-logical derivation can lead to, if applied in interpretation of natural events, without due consideration of natural limitations of Mathematics. The current cosmic view in Physics has its origin in two fundamental hypothesis of Einstein’s Theory of Relativity, mathematically logically extrapolated to a theory of entire cosmos, directed towards reconciliation with quantum physics. As a result of unrestrained mathematical-logical derivation, the Physicist’s cosmic view has landed in to a cosmos which is beyond sensory perception and is therefore just not there. Strings of the string theory and M-Theory are beyond sensory perception – direct or indirect. These efforts have led to theoretical models which are quite similar to Ptolemy’s Model of Planetary System and Phlogiston Theory of Combustion.
This leads us to the question of natural limitations of Mathematics in understanding natural events, things and the cosmos as a whole.
1) Mathematics is not all real: For example Complex numbers and Boolean algebra are not grounded in natural events. These can be best regarded as mathematical fantasies, may be useful as tools and techniques in certain situations.
I once met Professor O. P. Mishra (Now retired) of Department of Physics. He claimed that by application of Schwartz Distribution Principle through dedicated software to the available data, it would be possible to predict incidence of earth quakes. He cited the reason for his belief that earth quakes can’t be predicted on the basis of available data only because of inaccuracy of available data which can be rectified by applying Schwartz Distribution Principle and thereafter accurate prediction of incidence of earth quakes can be made.
However, I found it farfetched and beyond comprehension. Mathematics is not like a magician’s hat that can apparently produce things out of nothing. Until and unless, one clearly knows what data is to be collected and how the relevant data is to be interpreted, how mere application of a mathematical tool or technique can ever lead to prediction of a natural event is clearly beyond comprehension.
Finally, these virtual components of Mathematics are relevant only as tools and techniques so far as they lead to results compatible with real world. If it is otherwise, then their use is clearly unwarranted.
2) Nature is hierarchically organized and rules change from one hierarchical level to another. Rules applicable to cosmic dust may not be the same as rules applicable to celestial bodies such as planets, stars etc. Therefore, unlimited mathematical extrapolation can’t be correct and proper. For example, by supplying heat to a block of ice, it is possible to convert it in to a liquid and then vaporize it to water vapor. But rules applicable to solid, liquid and gaseous state of matter vary and hence mathematical computation applicable to water in solid state can’t be valid when applied to water in liquid state and so on so forth. Natural events often involve change of state and change of phase with commensurate changes in applicable rules. Hence unlimited mathematico-logical extrapolations are wholly unwarranted until and unless it is shown that rules being so applied are valid across change of hierarchical level, change of phase, change of state etc.
3) Nature is not mathematically-logically perfect: Nature abounds in non-linearties. All acts of natural origin and creation are nonlinear in character. All phase changes and change of state are nonlinear in character. Some of these nonlinear acts are recognized as singularities by Physicists. A singularity in Physics means a strange point since Physicists fail to apply Laws of Physics across a singularity.
No unconstrained mathematical-logical extrapolation can be valid across a nonlinear act. All these nonlinear acts are empirically determinate but at the present level of human understanding are rationally indeterminate. For example decay of Uranium into lead is an empirically certain event but its occurrence can’t be predicted on the basis properties of Uranium and known laws of Physics.
Therefore, we have to be guided by Nature in application of mathematical tools and techniques towards interpretation of natural events. Application of mathematical tools and techniques is valid only so far as it corresponds to observations/ experimental results/ experience.
Notwithstanding the above said, mathematical tools and techniques are quite useful in discovering hidden conditions underlying things and events. Mathematical reconciliation of observed data indicates adequate and fair understanding of the given subject matter.
Hence in the book “Encounter of Science With Philosophy – A Synthetic View” Mathematics has been defined as science and art of computation of natural events.