The author holds an unredeemed Platonic conception of mathematics and I will state at the outset of this review that I have a Kantian conception of mathematics. For Max Tegmark, the physical world is not only described mathematically, “…but that it is mathematics.” This is to claim too much. I will concede that mathematics occupies a privileged intermediate space between physical science and metaphysical speculation but I cannot conceded that existence is mathematical. To do so actually reaches back to the Pythagorean perspective of reality upon which Plato built. The mathematical structure that Tegmark sees as the instantiation of the physical world is one that has been found to be riddled with paradoxes and proven to be incomplete. I do not think that mathematics can be regarded as an a explanation in itself of anything. Physical theories are not what they are because of mathematics. Mathematics are the language in which we state our theories about the physical universe. That is, nothing is the way it is because of a mathematical principle. However, even if reality can be described mathematically, it does not follow that the final or ultimate ontology of realty is mathematical. If the ultimate nature of reality is mathematical it follows that mathematics is the cause of ultimate reality. This is, I am sorry, prima facie absurd. For example, we can describe everything we take to be reality with words, but it does not follow that the final or ultimate ontology of realty is linguistic. What is most interesting is that, from either or any philosophical perspective, paradoxes and incompleteness notwithstanding, the practical application mathematics proceeds and works as the author more than adequately demonstrates in this book. Mathematics still provides the most precise manner in which to express our theories, but it is also possible that we have a cognitive bias to seek theories to explain the universe that can be expressed mathematically. My concern here is more philosophical than practical.

Thinkers working in the philosophy of mathematics since the time of Plato are traditionally separated between those who say that mathematical statements are true about the physical world (empiricists view), those who feel that this does not do justice to the inexorability of mathematics and claim an eternal truth status for mathematics (Platonic view) and a third view offered by Kant which is that mathematical statements are true for the 'form of our intuition'; that we bring them to the world to organize our experience of exigence to better navigate the world (intuitionist view). For Wittgenstein (twentieth-century), the whole idea that mathematics is concerned with the discovery of truth is a mistake. This mistake arose from the treatment of pure mathematics as an area of study apart from application to the physical sciences. When mathematics is treated as a tool, or a series of techniques for calculating, measuring, analyzing etc, philosophical questions about the nature mathematics simply do not arise. The philosophical nature of mathematics is so mush resolved as it dissolved by Wittgenstein.

From a purely utilitarian point of view, what matters most is that mathematics works and produces results. But since all mathematical models of the physical world break down at some point, combined with many inconstancies, paradoxes and unproven assumptions found at the heart of mathematics, I come down on the side of it being a human invention like chess, but a great invention it is!!!! Mathematics is not the language in which the universe is written, it is a selective tool which we use to explore the universe. Both chess and mathematics are highly useful systems, but constitutive of their rules, rules that are subjectively imposed by humans. In this frame of mind, we can take the “shackles” off our thinking – we need not wait around for a new discovery. Instead, we can be creative and free to invent more, better and new mathematics as needed. What the author does not see is that mathematics is a human creation, a tool that we use to explain physical reality to ourselves, not a feature of nature to be discovered.

In author’s defense, this book is not intended purely as a history of mathematics, but I think Russell’s paradox (shockingly not mentioned in this book) demonstrates that mathematics is a human invention. Russell’s paradox shows that mathematics is not rooted in logic as both Russell and Frege had originally set out to demonstrate. It is not objective. Mathematics is not the product of logic and objectivity separate or apart from the sensible subjective world. Mathematics, and logic, is more as Kant described it after all, its origins lie not in objective knowledge but in our own a priori subjective intuitions about space and time. Mathematics is not fundamentally an objective science - the product our discoveries about reality. Instead, at its foundations, it is a synthetic enterprise and its findings are based on our ability to use our imagination and harness our intuition. Mathematics is not a body of immutable absolute truth as Pythagoras, and later Plato, tells us and with which the author agrees. Rather, it is a collection of useful problem-solving techniques constructed upon, and built up from, the most banal tautologies.

The fact that we constantly fall into paradoxes combined with our ability to construct logical contradictions and traps demonstrates that the basis of logic itself is flawed or contains fundamental contradictions, e.g., again Russell’s paradox which is the result of a logical contradiction in the use of classifications to explain numbers and organize reality, number is a mathematical notion and class is a logical notion. The relationship between the internal reality of the human mind and external physical reality is a cacophony of concatenated asymmetrical subjective approximations that we invent and impose. Mathematics is the human way of imposing organization and determination onto an underlying reality of randomness and indeterminacy. To know mathematics, to know something about math & logic shows more about how human beings think, perceive and reason; not an objective truth about reality. Tegmark recognizes the subjectivism in a field such as economics, but not in mathematics, the only difference is the degree of subjectivity. Mathematics is not a body of metaphysical truths out there to be found. This is to succumb to a seductive ontological temptation. There is nothing there to be found that we did not put there ourselves. The author tells us that mathematical equations offer us a window into the working of nature. Maybe he is correct for reasons he does not realize, nature as we describe it with our mathematics is our subjective imposition so of course nature is full paradox and contraction because we put them there. Nature does not create paradoxes and contractions, humans do this. We then make the mistake of looking back on our subjective impositions upon the physical world and please ourselves by calling them objective.

Mathematics is an abstraction of the human mind, it does not have a separate existence to be discovered. There is no unified origin of everything, call it truth, math, science or God. The meaning is not in the mathematics, we do the mathematics and create the meaning. Mathematics is a discursive exercise. Instead, the author treats us to a curious mathematical fantasy whereby the path to ultimate knowledge is opened.

All knowledge cannot ultimately be squeezed into a form of pure mathematics. Knowledge cannot be narrowly reduced to necessary truths.

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