Can someone please explain in simpler terms what does this:https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis, mean? Does this mean tegmark says for example: humans have corresponding math structure (e.g. humans can be described using some math objects, like the fact that they live some amount of time is a property of that mathematical object etc - but in that case that would also mean we don't know what kind of mathematical structures those are right? because there are some properties about humans which we may learn after some time.), as well as say lizards, and as well as anything in world?
Modern physics describes the physical world in purely mathematical terms, so when Tegmark talks about our universe being a mathematical object, he's talking about the most exhaustive possible physical description of our universe in some final, complete theory of fundamental physics (what physicists call a "theory of everything", one which would cover all particles and forces at all energy scales). So this mathematical structure would include the maximum physical information about the universe that could possibly be measured even in principle, dealing with every single particle (or other basic elements in fundamental physics) in every region of space and time. It wouldn't be any sort of simplified model that leaves out any quantitative information that would be measurable in principle (like a simplified mathematical model of a tennis ball that just treats it as a perfect sphere without modeling the shape of every fiber on its surface, or every molecule making up those fibers). One might think of such an all-encompassing mathematical structure as something like a perfect simulation of our own universe and its precise physical history.
The idea of such a maximally complete mathematical model of the universe takes for granted the physicist's view that every measurable event in our universe is the product of physical laws (whether deterministic or statistical) operating on prior physical conditions, that there are no exceptions to this; a believer in certain notions of non-compatibilist "free will" might disagree with such a premise, as would a believer in supernatural events like miracles. But if you do grant this premise, then if you also grant some form of mathematical platonism, such a maximally complete mathematical description or model of our universe should exist in the realm of mathematical forms (along with every other possible mathematically-describable world). And that maximally complete description would include descriptions of all the particles making up human beings and their surroundings, and their motions over time, including things like what pattern of sound waves they emit from their vocal cords and what keys they strike on their keyboards--meaning everything we might say or write about feeling confident we live in a "real, physical universe" would also be said by them within this detailed mathematical "simulation". So, if we grant the assumption that such a mathematical structure exists in the platonic world, why should we be confident that we are different from them? Tegmark's hypothesis, which assumes the premise of mathematical platonism along with the premise that we live in a universe governed by some ultimate set of physical laws, is that we are simply "self-aware substructures" of that mathematical structure, that there doesn't need to be an additional "physical universe" made of "matter" that is described by that structure.
The above is probably about the "simplest" way I can explain the idea, but I also think the idea can be understood more precisely in a philosophical context if we relate it to some ideas from philosophy of mind; if you're willing to read a much more long-winded discussion of how these ideas relate, see below.
A basic division in philosophy of mind is between those who accept some kind of eliminative materialism--roughly, the view that there are no truths about mental states which are anything more than just different ways of talking about physical brainstates--and those who reject it. Those who reject it commonly appeal to intuitions that suggest there are facts about first-person conscious experiences that go beyond the most complete possible set of facts about the universe as seen from a third-person perspective, like Thomas Nagel's famous paper What is it like to be a bat? which argues that even the most complete possible third-person description of a bat's brain functioning would leave out what its echolocation sense actually feels like from its own perspective, the qualia of echolocation similar to our own experience of the redness of red light. Some who reject eliminative materials endorse some kind of "interactive dualism" in which some physical events (like that of a human speaking) are directly caused by non-physical mental states. But others think that the model of all physical events being consequences of prior physical states and mathematical laws of nature is likely to be correct, so they instead endorse a view where there's a sort of one-way causality from physical brain states to mental states but no causality in the opposite direction: certain brain states "give rise to" certain experiences and qualia, but the physical world itself is "causally closed", a view known as epiphenomenalism.
Epiphenomenalism is usually coupled with some notion of mental properties supervening on physical ones, which means that if you have two possible world where at least some conscious experiences of the beings within them differ, this difference must be due to a physical difference between them (like a given life-form having different brain states in each of the two world which give rise to different mental states). Some philosophers go further and suggest there must be "psychophysical laws", analogous to the laws of physics, which determine the precise relationship between physical states and subjective experiences. Such laws might determine which physical systems or processes would give rise to conscious experiences and which would not, though some advocates of psychophysical laws like David Chalmers have also speculated that perhaps all physical states/processes would be linked to some form of experience by the psychophysical laws, a view known as panpsychism (see the section of Chalmer's article here on "type F monism", and he also discusses the issue at greater length in his book The Conscious Mind).
Some have also proposed that there might be laws determining a notion of the "degree of consciousness" of a physical system, with integrated information theory (IIT) being the only specific proposal I've seen discussed so far. Tegmark himself is interested in the notion of some kind of objective measure of the degree of consciousness in a system, probably based on IIT, see his paper Consciousness as a State of Matter. However, he is vague on the metaphysical question about whether he is actually arguing for a fully epiphenomenalist view which would include the idea of there being objective truths about the consciousness of different physical systems, or whether he is arguing for something more like a version of eliminative materialism where the "consciousness" of a system is simply defined in terms of integrated information, with there being no totally objective truth about whether such a definition is more correct than alternative possible definitions (I'm not sure if Tegmark has fully thought through the difference between these positions, though he was a co-author on this philosophical paper whose lead author was Piet Hut, who says on p. 4 that he is 'unenthusiastic' about the notion of 'a form of reductionism that states that the mental phenomena are “really just material processes”, as if the material were somehow more fundamental than the mental').
It is also often suggested that the psychophysical laws would determine whether or not two distinct physical states/processes (whether in a single universe or in different possible worlds) would give rise to the same subjective experience or not. One common hypothesis among epiphenomenalists is that mental states would obey a principle of multiple realizability, where physically distinct systems could give rise to the same subjective state, in much the same way that we say two computers with physically different hardware can "run the same program". Some epiphenomenalists do in fact believe that the psychophysical laws would involve a one-to-one relationship between distinct experiences and distinct computations, so that it is the computational structure produced by biological brains that determines the type of experience it has, and a sufficiently good simulation of an organism's brain would produce the same experience (the physicist David Deutsch has proved that according to quantum physics, 'every finitely realizable physical system can be perfectly simulated' by a sufficiently powerful computer, so that would include a biological organism). David Chalmers has given some arguments for this involving thought-experiments where the biological neurons in a brain are gradually replaced by computationally identical artificial neurons. He makes the point that if we don't accept the idea that the artificial brain at the end of this process would have the same types of experiences as the original biological one, this would imply the psychophysical laws would have to have some fairly ugly or inelegant properties, like laws where experience was unchanged when you replaced the first million neurons but shifted dramatically when you replaced a million and one neurons.
Whether or not the psychophysical laws would give a one-to-one relationship between physical computations and experiences, it at least seems conceivable that one way or another, the laws would be such that the details of a given conscious experience depend only on mathematical properties of physical systems, the type that the laws of physics deal with. We could imagine that whenever two brain-like (or computer-like) physical systems are completely identical in these measurable mathematical properties (like the exact spatial arrangement of the neurons and the timing of all the nerve impulses), they give rise to the same experiences, even if the two systems might differ in other physical properties we don't know about. For these other unknown physical properties to lie completely beyond the reach of any future mathematical theory of physics, we would have to assume they are properties that are completely irrelevant to predicting measurable changes like motion. Perhaps they could be something like the so-called "hidden variables" that are imagined in certain untestable interpretations of quantum mechanics like Bohmian mechanics (all these interpretations make identical predictions about all measurable variables so it's impossible in principle to distinguish them experimentally). Or perhaps the additional properties could be like Kant's noumena, which were by definition supposed to be beyond any of the mental categories we use to conceive of objects. Either way, the point is that even if you believe physical objects have additional properties beyond the quantitative and measurable ones that would be associated with them in a future complete theory of physics, the psychophysical laws might be such that these additional nonmeasurable properties wouldn't affect the contents of conscious experience.
If reality worked like this, then it would suggest there are at least three basic classes of facts about reality: facts about the platonic world of mathematics, facts about the physical world, and facts about conscious experiences/qualia. Facts about one domain could make reference to entities from one of the other three domains (for example, at least some facts about the physical world would be stated in mathematical form). The physicist Roger Penrose included in one of his books a memorable diagram of this idea, seen below, along with a discussion of the relationship between these three "worlds", how each seems to give rise to the other in certain circumstances; Penrose's discussion is the focus of the philosophical paper by Piet Hut, Mark Alford & Max Tegmark that I mentioned earlier. But this view would allow for the idea that out of all the possible descriptions of universes in the platonic realm, only one would be a description of a real physical world, and only physical states/processes within this real world will give rise to real conscious experiences.
You could then imagine that to create the real world, God looks out over the face of the platonic realm, and selects a unique mathematical form to serve as a template for the physical reality he creates, and then that particular mathematical form, combined with the psychophysical laws, determines what possible experiences are made real (mathematical descriptions of observers in different possible universes that are not granted physicality in this way would not correspond to any actual conscious experiences--these platonic observers would be p-zombies, in effect). Of course one doesn't have to literally believe in a God to take a view like this, but the idea is that there has to be some kind of metaphysical "selection" that determines which mathematical form will correspond to the real physical universe and the experiences it gives rise to, whether that selection is due to a choice by God or random chance or some unknown metaphysical principle.
We could equally well imagine a different scenario for how God might select from the platonic forms in making a unique reality--after choosing a unique mathematical form, He could decide to "cut out the middleman" and just directly cause the appropriate conscious experiences to become real, namely those that are linked to the chosen mathematical structure by the psychophysical laws (which, remember, were assumed to only depend on measurable mathematical properties of the universe). This scenario would be completely indistinguishable from the first scenario from an experiential point of view--rocks would still feel just as solid for example, even though by assumption no physical world has been created, reality consists only of mathematics and experiences. So, nothing in our experience can count as evidence in favor of the first scenario over the second scenario, though one might still have philosophical intuitions that favor one or the other.
From this scenario, it seems easy enough to make the leap to a third scenario where there doesn't have to be a "selection" at all, and all mathematical forms are by default equally real in the way they give rise to conscious experiences. Or if you prefer, this third scenario could be described as one where all possible mathematical structures are selected to have this sort of experiential reality, with the psychophysical laws just determining which experiences are associated with which mathematical structure. So this is how I like to think of Tegmark's proposal, as a way of showing it's a coherent metaphysical hypothesis and not just a category error as Pigliucci argued. Though I do think there are some additional difficulties with Tegmark's proposal related to the need to have some notion of relative frequencies or probabilities of different types of experiences in order to make predictions about the future (a computer scientist named Jurgen Schmidhuber has proposed a similar idea in which all computable universes exist, but a "speed prior" assigns higher probability to universes that are in some sense "easier" to compute).
(Hypnosifl's response is sufficient, I think, but in case I can help out...)
The first thing to consider is what Tegmark proposes the hypothesis in relation to. He offers it as an answer to the question regarding the "unreasonable effectiveness of mathematics in the natural sciences." What easier way for mathematical information to explain physical information than for the latter to collapse into the former? He therefore makes a more elliptical argument about how mathematical structures must objectively exist in order for the collapse to go through.
So the second thing is that Tegmark looks like a Platonic realist: his universe contains perfect triangles, perfect spheres, and so on. But it goes on to contain all "perfect" geometrical structures and sequences. Assuming that there is at least one sufficiently complex and exact structure/sequence that maps to the geometry/topology of our world (over time), then there is a Platonic Form of that geometry/topology. But instead of having the Platonic Forms existing in a separate world, Tegmark has it that they just are all the worlds, or rather that "a world" is an instance of a sufficiently complex and exact mathematical type-set, so if the type exists, it exists as its own world, in the "ultimate ensemble" of all possible type-worlds.*
The intuitive reason that this is questionable is the knowledge argument about qualia: is it possible to represent color information (by itself) as mathematical in the required sense? Are colors to be thought of as numbers, in the way we think of integers and reals and so on and on? Could they be "constructed" from sufficient iterations of the empty set, so to speak? But if the set-theoretic background for the mathematical universe contained urelements, we could escape this issue: colors, for example, or any irreducible qualia for that matter, could be treated as urelements, so that while they don't "look" mathematical (in the general numerical sense), they can be interpreted as mathematical. Certainly, colors are subject to interesting mathematical combinatorics and the Munsell color solid might be thought of as "what colors are" in the Tegmarkian sense (with the caveat that the overarching geometrical type would really be a conscious urelement juxtaposed with the color solid, say).
*This is why Level 3 is given as the Everett multiverse: think of reality like a graphing calculator, then assume that the calculator graphs all its possible outputs at any interval, and then assume that the quantum splitting of the graphs carves off one universe from another, until all the graphs are realized as an endlessly branching multiverse. But the parameters of this graphing calculator are a subset of all possible such parameters; so there is a Level 4 where all multiverses based on all such parameters occur.
Tegmark is channeling Pythagoras, so one could call him a neo-Pythagorean except that does a dis-service to real Pythagoras.
Plato did it much better, with much more breadth, width and heft. It's what his philosophy of forms was about. Plato acknowledged mathematics as a way of understanding the eternal and the timeless, the realm of necessity. But only as a very first and preliminary step.
On this basis Tegmark is still stuck in the doorway and hasn't begun to make his way out on the path to greater enlightment...