Max Tegmark's doorstopper of a book takes aim at three great puzzles: how large is reality? What is everything made of? Why is our universe the way it is? Tegmark, a professor of physics at MIT, writes at the cutting edge of cosmology and quantum theory in friendly and relaxed prose, full of entertaining anecdotes and down-to-earth analogies. His book seeks to induct the reader into a wildly speculative cosmic vision of infinite time and space and infinite parallel universes. Close to his heart is an extreme Pythagorean/Platonic thesis: physical reality is ultimately nothing other than a giant mathematical totality.

The story goes as follows. Our observable universe is the result of an explosion, the big bang, some 13.7bn years ago. The explosion, now called inflation, is a solution to Einstein's theory of gravity in which a subatomic blob of non-diluting substance (dark matter might have this property) can double its size at regular time intervals while keeping its density constant. After inflation ends the expansion of space slows to roughly the present rate, gravity having coalesced matter into the familiar stars and planets we observe in space. The inflation solution explains many puzzles but includes a shocker: the process doesn't terminate – inflation is eternal. This means that just as our universe inflated and coalesced an infinite number of others, remote from us in space and time (but in principle accessible), were doing the same. And since, in an infinite space "everything that can happen according to the laws of physics does happen", some of these would contain planets and inhabitants virtually indistinguishable from ours. The assemblage of all these "Level 1 parallel universes" containing our observable universe is called the multiverse – that is, the Level 1 multiverse.

But there is more. Inflation, the "gift that keeps on giving", Tegmark quips, contains a second shocker. Starting from something smaller than an atom, inflation "can create an infinite space inside of it, containing infinitely many galaxies, without affecting the exterior space". This allows him – through an argument that involves the so-called fine-tuning of the fundamental constants of physics – to posit the existence of a multiverse Level 2, composed of an infinite number of Level 1 multiverses, each of which is inaccessible to us or any of the other universes in our Level 1 multiverse. What's more, unlike the situation with Level 1 parallel universes, many physical laws and fundamental constants may vary across a Level 2 multiverse, though little is known. Indeed, Tegmark admits that parallel universes themselves remain "highly controversial" and that by no means all physicists are convinced by inflation.

This cosmic "zooming out" makes up part one of the book. Part two zooms in to the ultimate building blocks of matter, from electrons to mesons, gluons and so on. The interpretation of quantum physics is notoriously contentious, with many competing versions. Dominant in most textbooks is the Copenhagen interpretation. According to this, Schrödinger's wave equation, which deterministically describes an electron as being in superposed states and spread out in space, breaks down under observation: the electron is found to be in a definite, but random position and state. Tegmark is made uneasy by physics' recourse to the intervention of an "observer" to explain experiments and opts for the so-called many-worlds interpretation of quantum phenomena, introduced by Hugh Everett in 1957. Dismissed at the time as bonkers, it now apparently has a sizable following.

According to Everett, the wave equation never collapses: it always applies. But not just to the particle; it applies to the combined system of particle and observing apparatus. This means an experimental observation "puts your mind into two states at once … so that at the end of it, there will be two different versions of you. Each subjectively feeling just as real as the other, but completely unaware of each other's existence." It's as if the universe repeatedly splits in two, resulting in innumerable copies of you in parallel worlds. As a bonus, the randomness of the universe that so troubled Einstein disappears, being simply "how it feels when you're cloned". Tegmark calls these parallel worlds Level 3 universes that together make up the Level 3 multiverse. He plays with the consequences of this mind-splitting picture of multiple realities, exploring such topics as quantum suicide and the possibility of quantum immortality, and demonstrating why your brain and its consciousness can't be a quantum computer.

So far, in parts one and two mathematics appears as no more than physics' indispensible tool kit for describing the world. The final part ups the ante: "Our reality isn't just described by mathematics – it is mathematics … Not just aspects of it, but all of it, including you." In other words, "our external physical reality is a mathematical structure". He calls this the Mathematical Universe Hypothesis (MUH). Admitting that the idea sounds far-fetched, he proposes to demonstrate it in the time-honoured manner of Euclid by logically proving it from a self-evident truth, namely the External Reality Hypothesis (ERH): "There exists an external physical reality completely independent of us humans." Tegmark's demonstration that ERH implies MUH is complicated, difficult to follow, and resists summary. It requires, among other things, a certain (disputable) characterisation of mathematics; an insistence that a "complete" description of external reality must be defined in a form "devoid of any human baggage like 'particle', 'observation' or other words"; and it is obliged to argue – given that mathematical structures are static objects – that time and physical motion are illusory. Thus everything physical is ultimately mathematical, including you and me, "making us self-aware parts of a giant mathematical object". He calls this vast nebulous entity the Level 4 multiverse, an omnium eternally "there" stacked above or beyond or behind the other multiverses than contain us. To the objection we don't feel such a truth, Tegmark counters that we don't feel like we're travelling in space at thousands of kilometres a second, but we are.

Perhaps physicists really do understand themselves to be part of a giant mathematical object. For them Tegmark's conclusion that we are living inside a Platonic, Level 4 all-encompassing mathematical multiverse will be appealing: the ultimate guarantor and resting place of their quest for a theory of everything. (After all, physics excludes, in principle, anything in physical reality not measurable or describable in mathematical language, so what other ultimate truth can it think.) But as a mathematician, convinced that mathematics is a human construction, that numbers don't exist in the world before humans put them there, I don't buy it. And the claim that our minds are ultimately self-aware mathemes – insofar as that assertion makes sense – seems like saying that Shakespeare's plays are ultimately patterns of words: it tells you next to nothing about its object.

• Brian Rotman's books include Ad Infinitum: The Ghost in Turing's Machine.