In the following sections, I will briefly describe some of my work published in 2019 and 2022 in explaining the role of mathematics in science, and introduce my theory pertaining to the origin and evolution of mathematics in particular and cognition in general. 

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Mathematics has played an undeniable role in the progress of science. Such a significant role calls for special attention to the origin, evolution and function of mathematical reasoning. Mathematics is not only so effective in formulating the laws of nature, its effectiveness in doing so reaches beyond present into the future. I was reminded of such quality following the unfortunate early passing of a brilliant mind in mathematics, Maryam Mirzakhani (1977 – 2017), the first woman in the world to receive the most prestigious award in mathematics, the Field Medal, in 2014. The extraordinary discoveries made by mathematical geniuses such as Emmy Noether, Evariste Galios, Maryam Mirzakhani and others have contributed significantly to our understanding of natural phenomena beyond the boundaries of time many times and is a reminder of the notion attributed to the Polymath Avicenna that it is the depth of life that matters the most, not its length. The long reach of mathematics into the future is a testament to the vast horizon of possibilities a mathematician brings before scientists, an idea that motivated me to complete the manuscript in understanding such effectiveness that I had almost forgotten to finish, and pursue this line with the investigation of the role of mathematics in studying perception. 

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Mathematics and Physical Sciences:

 

Physics is considered by many scientists and philosophers of science a gold standard for evaluating how close other branches of human knowledge is to a (physical) science. Many other fields such as psychology and neuroscience have been inclined to follow the path that physics has taken in order to stay in the domain of science. Nevertheless, there are areas in which physics, as arguably the longest established discipline in modern science, has shown clear differences when compared with a field such as psychology. One of these areas of difference is the extraordinary effectiveness of applying mathematics in physics. Such a feature has contributed to the growth of other differences at more superficial levels. The trace of this trait can be observed in the structural formation of scientific approach in physics according to which the realization of distinct areas of studies such as theoretical physics versus experimental physics has become possible. When it comes to a field such as psychology or neuroscience, there is no such clear distinction, not at least very common in practice. This distinction between the two groups of physicists is so well accepted that it may even appear in daily conversations among physicists. For instance, in a discussion among distinguished scientists in a panel including the late Nobel laurate Steven Weinberg and other prominent theorists and experimentalists, Weinberg expresses his view that a theoretical physicist's work is more trustworthy, and while he is making this comment perhaps in a somewhat humorous context (see a short segment of the panel by clicking here), it is implicitly confirming such well accepted distinction between theorists and experimentalists in physics. Another witness to such a difference is the fact that in physics, a theoretical physicist can develop theories without running any experiments to produce any direct supporting data. String theory whose development relies heavily on mathematics is one of such examples. In addition, the history of physics is a witness to the theories that fundamentally were products of some strong logical arguments equipped with mathematical reasoning. Einstein's theory of relativity or Higgs's theory to explain the mass of subatomic particles are two examples of such theories. Nevertheless, thanks to the effectiveness of mathematics in formulating the laws of nature, such theories have been supported following the creative work of experimentalists, years after their publication. 

 

In sum, even though mathematics does not serve the function that experimental data does, the notion that mathematics is a valid and reliable language in physics is agreed upon by virtually all physicists. This does not seem to be the case in fields such as psychology. First, psychological principles can not be expressed in mathematical language in many cases yet, and second, not all psychologists believe that such a translation is possible in psychology at all. Nevertheless, it seems that among those who see psychology and neuroscience (or at least many areas of such fields) as physical sciences, there are those who have developed theories describable in mathematical language and have recognized the importance of studying the role of mathematics in science, its origin and its evolution as well as outlining their suitability for the investigation of perceptual and cognitive phenomena. This is part of what I spend my time doing. Working on these subjects within the context of the philosophy of mathematics, biology and cognitive sciences, I have published two articles in peer-reviewed journals, one aiming at providing a theoretical solution to the problem of the (seemingly) unreasonable effectiveness of mathematics in natural sciences as a matter of cognitive evolution in 2019 and the other on the suitability of topology for the investigation of geometric-perceptual phenomena (i.e., the Müller-Lyer Illusion) in 2022 as follows:     

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A Theoretical contribution in 2019: 

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My 2019 article encompasses a theory that even though it has provided a framework for understanding cognitive evolution in general as has been briefly described in the Cognitive Evolution section, it particularly addresses the problem of unreasonable effectiveness of mathematics in the natural sciences as follows:  

 

In a series of my previous studies, I set the goal of analysing the behaviour comparatively in order to understand the essential distinction between humans and non-human animals. Over the course of my investigation, I became more and more inclined to see such distinction as a product of qualitatively distinct brains that have somehow taken advantage of the laws now known to be fundamental to nature and to the understanding of natural phenomena. Following some analyses of behaviour, which I published in 2015 and 2017, the puzzling problem of unreasonable effectiveness of mathematics in natural sciences that was first formally formulated in 1960 by Nobel laureate Eugene P. Wigner, grabbed my attention, especially that he had found the process of natural selection inadequate in explaining the perfection of  reasoning (i.e., mathematical reasoning). In order to provide a solution to this problem, I finally found it easier to advance a theory that fills some of the gaps that exist in explaining cognitive evolution in general and evolution of mathematical reasoning in particular. This could be very well achieved based on a biological approach to this problem. The theory is explained in the following article:

 

Nemati, F. (2019). The role of nature and brain in demystifying the “unreasonable effectiveness of mathematics”, Philosophical Psychology, 32:8, 1221-1245, DOI: 10.1080/09515089.2019.1646895

 

* For access to a free pdf of the above article offered by the journal please click here. 

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The theory is formulated based on my comparative analysis of behaviour with respect to the concept of symmetry embedded in the mathematical axioms of group theory. I have proposed a mathematical framework to outline the transitional steps from behavioural to cognitive mathematical structures over the course of evolution. The article provides a theoretical solution to the problem formulated by Wigner (1960) and I have argued that the epistemological implications of the symmetry principle following Einstein's theory of relativity is consistent with the proposed theoretical framework in guiding the construction of mathematical structures and the invention of mathematical constructs where physics, mathematics and biology meet.