This is pretty good, and it will exercise your mind for a minute.
Source: Saturday Morning Breakfast Cereal
The distinction between mathematical notation and its underlying reality is a crucial one. The first is an invention of humans, the second is universal. In fact, I’ve increasingly become convinced that the second actually is the universe, and mathematics is just us recognizing reality’s fundamental patterns, and devising mechanisms to describe and to model, to extrapolate, to make predictions, based on those patterns.
Of course, many of those predictions have no correlation in observed reality, at least none that has been observed yet. Many mathematicians take delight in pointing out how useless many of their endeavors are. Yet, despite this, many mathematical structures initially thought to be purely abstract do eventually end up being useful to model some aspect of nature. The ones that don’t could be thought of as either untested or falsified scientific theories.
Another way to describe what I’m saying is that mathematics is the universe. This is similar to but the reverse of the Mathematical Universe Hypothesis, which posits that the universe is a part of mathematics. Both of these ideas see an equivalence between underlying mathematical realities and the universe, but with opposite ideas of which is the more primal reality.
Which one is true? Like all metaphysical conundrums, I can’t see any way to know for sure. But my personal judgment is that mathematics being the universe is simpler. The universe being a subset of mathematics requires us to assume a trans-universe reality that we can’t observe, an assumption mathematics being the universe doesn’t require.
Of course, depending on exactly what we mean by “mathematics”, even if there is no trans-universe reality, the universe could still be thought of as a part of mathematics, but only in the same sense that it is a subset of all scientific theories, including both true and false ones.
Unless I’m missing something?
SelfAwarePatterns 
Mathematics is a model of the universe. For example, the integers can be used to model (count) sheep. But sheep are not math. And math is not sheep.
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And yet, there is an ontological difference, an actual difference in reality, not just in modeling, between three sheep and four sheep.
It seems like you’re positing these layers:
1. the universe
2. mathematics
3. mathematical notation symbolizing 2
What would you say the difference is between 1 and 2?
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I am partly just teasing, of course, in the spirit of the SMBC comic.
I am open to there being a deep and fundamental reality to mathematics, and I am open to it being merely a set of symbols that we invented as a game. I think that the correlation between mathematical reality and mathematical notation is stronger than some people think (i.e. it’s just notation and a set of rules, nothing more.) Conversely, I think that the correlation between mathematical reality and physical reality may be weaker than some people think (i.e. the reason why mathematics has predictive power is less to do with physics and more to do with maths itself.)
I say this as a physicist who understands the deep symbiosis between math and physical reality at the most fundamental level. One of the most intriguing features of mathematics is that it has predictive power even over processes that are disordered and random. Statistical mechanics is like this. In a different sense, so is quantum mechanics. So even if the sheep randomly transmute into ducks or goats, we can still use mathematics to model their behaviour at some level.
So what I’m saying, in a roundabout way, is that mathematics is a fundamental abstract synthetic language that can be used to describe not just this universe but all possible universes (and therefore places constraints on those universes), but that simultaneously it is a series of learned patterns inspired by our experience of living in this universe, and is therefore derived from it.
Rather ambiguous, but this is a work in progress for me (as well as you), and this is the best I can do right now, and in any case, as I say, I am open to suggestion on this topic, and may be hopelessly wrong.
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Well put Steve. This is definitely still a work in progress for me. It may always be. One of the reasons I post on it periodically (like just about any topic) is to see if anyone points out something I’ve missed.
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More evidence would be useful. By that I mean a really decisive move forward in physics. If we ever arrive at a set of fundamental theories, with no inconsistencies or bits missing, then we might find that we can finally understand the relationship of math to the universe. Or maybe we won’t. We will have to see, if we ever arrive at that position.
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What kind of new evidence (if any) would affect your position?
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I think evidence for other universes, particularly ones that appeared to follow different physical laws but still fit in with existing mathematics and logic, would sway me toward mathematical platonism.
Any evidence for anything that could never be described mathematically might sway me toward mathematical nominalism. (It would have to be something that relentlessly defeated any attempts to expand mathematics to take it into account.) I sometimes wonder if the quantum wave function collapse / decoherence / whatever, might qualify, although it’s been pointed out to me that even though we can’t mathematically predict what happens there, we can mathematically describe it.
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Something I’m not clear on: On the one hand mathematics is the universe. On the other, that the universe is a “part” or “subset” of mathematics. Those aren’t matching relationships. The first is an equality, the second is a subset. You were consistent in maintaining that distinction, so it’s clearly meaningful. I’d like to hear more about that. How it fits in.
There’s a definition question here: What exactly do we mean ontologically by math “is” the universe?
For me, and perhaps for you, ’cause I think we’re on the same page here, it kind of goes like this:
The physical universe has regular, isotropic, observable laws and relationships we can discover and describe. The best, most precise, most useful, language we’ve discovered for expressing those laws is mathematics. It’s so good that it allows us to predict laws and relationships we later observe.
As you say, the notation is particular to culture, but the underlying laws and relationships are not (the whole point is how isotropic they are).
Any description of those laws necessarily converges on mathematics because math is inherent in the relationships and patterns of the physical reality. As you said to Steve, there is a fundamental ontological reality about a given number of sheep.
Recognizing them as “sheep” at all leads directly to mathematics!
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The phrase “mathematics is the universe” is, admittedly, a bit ambiguous. I use it in the same way that Max Tegmark uses “the universe is mathematics”, flipping the labels to make the point of what I perceive to be the more primal reality. But both phrases require clarification that we’re talking about the underlying mathematical structures and not the notation.
The subset part is difficult to discuss without using language that implies a value judgment I don’t want to signal. I think the fact that there are mathematical objects without any known physical correlates is what tempts mathematical platonists to conclude that math is a superset of the universe. But I think these abstract objects exist in the same place that rigorous but incorrect scientific theories exist.
In both cases, there is some factor, some complication, that prevents them from matching reality. Both can be useful for certain purposes. (String theory has reportedly led to advances in mathematics, so even if it turns out to be false, it’s still been useful.) Again, I don’t want to imply that the things like Mandelbrot set is “false” or “wrong”, just that it exists in the same mental space as theories without correlates in the universe. Another way to say it is that they are “right” mathematically but “wrong” scientifically, or perhaps a better way is that they are useful mathematically but not scientifically.
Totally agree with your other remarks.
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We do seem on the same page, so this is just thinking out loud, pondering points. As you and Steve said, a work in progress…
Tegmark is saying the universe is made of math — that if you go down far enough, somehow all you find is numbers. I get stuck on what that could even look like ontologically, but maybe I need to get more details on Tegmark’s view.
There is a view that if you go down far enough, “everything is just relationships” (although surely there must also be things that have those relationships). We definitely do see information encoded in relationships, and relationships are quantified by numbers and math. So maybe there’s some sort of Tegmarkian connection there.
If we reverse that, math is made of the universe, it still works and, I think, perhaps a bit elegantly. Go down far enough and you find… the universe (and the mathematical relationships it creates).
I’m not a Platonist in the common sense of believing in some other perfect realm of forms (but I’m not clear that even Plato actually thought that). I am a Platonist in believing that the abstractions we recognize are based on physical reality and have a level of reality themselves (and that many are a priori given a concept of physical space and objects).
Unicorns are real as demonstrated by how you know what I mean by “unicorn.” 🙂
I think I follow on the subset aspect. I wonder if Tegmark’s reply would be that, since we can conceive of unusual mathematics in this world, that mathematics is part of this world (in the same sense that unicorns are).
Perhaps another way to look at it is that physical reality is the basis of our imagination. Horses lead us to imagine unicorns just as Euclidean geometry leads us to imagine hyberbolic geometry.
It gets interesting trying to decide if a perfect circle is a horse or a unicorn. Circles do exist. But it’s like imagining a perfect horse. Presumably there is no such real animal.
In a sense, all math is an idealization — a subset of it idealizes physical things — but it seems an abstraction that moves towards deeper truth. In the real world, things fall. The idealization lets us see the deeper truth behind how they fall.
I know what you mean about things like the Mandelbrot not being “false” or “wrong”… That string theory has resulted in advances in mathematics shows we can’t really even say “unscientific” (math, of course, is the purest science there is). About the best word I can think of is “abstract.”
The real debate, perhaps, is about the ontological reality of an abstraction.
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Based on my readings of Tegmark, he expects the final theory of everything to be entirely relational, entirely mathematical, in other words, with no things in it. But he also posits that we may never be able to derive that theory because it might require observation of aspects of reality we will never have access to.
On abstraction…yeah. Abstract objects certainly exist as patterns in our brains. I think we can communicate them because they’re derived from patterns of sensory inputs that we’ve both experienced. As you said, knowledge of a unicorn requires having seen a horse and a horn before, at least in pictures, and the ability to simulate combining them in a certain way.
Of course, most of us have seen an artist’s impression of that model, which helps. But I can imagine a pink singing horse by taking the pattern of a horse in my mind and modeling modifications, but to model those modifications, I must have seen the color pink and someone singing. On the other hand, I personally can’t visualize a 12 dimensional horse.
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“Abstract objects certainly exist as patterns in our brains.”
Yes, and those patterns are physical instances of the abstraction (like source code is a physical instance of an abstract algorithm). It comes back, I think, to the ontological weight you give an abstraction.
As you mentioned, pink singing horses can be imagined but not so much 12-dimensional ones. Except that you just did imagine a 12-dimensional horse! 🙂 You couldn’t picture it, but you did imagine it.
(♩ ♪ A 12-dimensional horse is a horse, of course, unless it’s the famous Kaluza-Klein. ♩ ♪)
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“Except that you just did imagine a 12-dimensional horse! 🙂 You couldn’t picture it, but you did imagine it.”
Well, that depends on what we mean by “imagine.” I strung the words, “12”, “dimensional”, and “horse” together, and did visualize (unbidden) a Picaso-like version of a horse, but I’m not sure if it’s honest to say I actually imagined it. If I say “the sound of a square root”, am I actually imagining anything coherent?
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“Well, that depends on what we mean by ‘imagine.'”
Imagine! It’s easy if you try; it isn’t hard to do. 🙂
You may say I’m a dreamer, but I take ‘imagine’ to be more inclusive than ‘visualize’ — I can easily imagine things I can’t visualize (often before I’ve even had breakfast).
When you strung those words together, you also strung the concepts behind those words together to create a new concept. That’s quintessential imagination! It’s storytelling!
“If I say ‘the sound of a square root’, am I actually imagining anything coherent?”
Who said imagination has to be coherent? 🙂
(And who says that’s not coherent anyway? If you go to the Online Encyclopedia of Integer Sequences you can listen to MIDI music versions of those sequences. The digits of a square root might indeed be used to generate sound.)
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“those patterns are physical instances of the abstraction”
I think this is exactly it. The cornerstone of Tegmark’s argument is that an abstraction has ontological validity. It could equally be argued, as Wyrd hinted, that only physical instances of an abstraction exist.
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@SteveMorris: Yep. How real is an abstraction? (I say very; your mileage may vary. 🙂 )
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Hi,
Since I’m a Tegmarkian I can probably answer some of your questions.
I wouldn’t say that’s quite right. There’s more to math than numbers. It’s true that he is fond of making the point that the properties of an electron (for instance) are all numeric… charge, spin and so on, but there is more to these properties than their values. There is also how these values affect the interactions of the electron with other entities. Having a mass of “1” is not the same as having a charge of “1” (you can’t even compare the two since they use different units). For this reason I don’t much like this way of explaining the view.
I think what he means instead is that there is in principle a mathematical model of the universe and that the universe is identical to that mathematical model. That all there is are relationships and so on and there is no “stuff”, or rather that stuff is just formed from these relationships.
Pretty much.
OK, but those things can themselves be abstract nodes which have no meaning outside the context of their relationships. So even the “things” are made of their relationships, in essence.
For instance, I can describe a graph of nodes A,B,C and D, and that A is connected to B and C while C is connected to D. I have defined a simple mathematical system composed of relationships between things which have no meaning outside the context of that system. The system defines what D means even as D is used to help define the system. We see this kind of circularity in math all the time, and it isn’t a problem.
I’m not sure that you can model the universe itself as a simple graph like that, although you could probably encode the laws of physics in such a way if you wanted to. I am a programmer so instead I think in terms of programs. There is an algorithm (which is just a mathematical object) that would simulate the universe, and the universe is that algorithm (or any other mathematical object which is isomorphic to that algorithm in respect of the features that we observe). Like the nodes in the graph, the algorithm uses variables which have no meaning outside the context of that algorithm. The algorithm defines the variables as much as the relationships between the variables define it. This is what it means to say everything is just relationships.
I don’t think so, since that rather goes against what he is trying to say. We can conceive of mathematical objects that are not this universe. We can consider those objects to be part of this universe if you like (since they are reflected in it within our brains) but that’s straying from the crux of the issue.
Tegmark and I believe that all possible mathematical objects exist. Some of them we can conceive of and some of them perhaps we cannot. But the crucial point is that this universe (i.e. the laws of physics and everything in the universe) is just a mathematical object like any other. Some mathematical objects we would be inclined to call universes (if, say, they are big, complex, have the potential to support self aware observers and especially if they resemble our universe) but this is just a convenient label and not a true distinction between what is physically real and what isn’t.
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“There’s more to math than numbers.”
Oh, absolutely! I was just speaking poetically there. 🙂
“I think what he means instead is that there is in principle a mathematical model of the universe and that the universe is identical to that mathematical model.”
That does seem different than my impression (which is no doubt ignorant). I had understood there wasn’t much “in principle” to it and that the mathematical model had a higher ontological weight (was “more real”) than physical reality.
I would think most physicists think there is in principle some mathematical model that represents the universe (the universe’s Schrödinger wave equation, at least) and can be said to be mathematically identical to it. I had understood Tegmark to be making a much stronger statement.
“I can describe a graph of nodes A,B,C and D,…”
But your graph is both the relationships and the nodes A–D. If those are completely unbound variables, what is the value of the graph?
What I was going to reply to Mike about relationships “all the way down” is that it does remind me of infinite summation or product series in math. There are frameworks involving recursive functions that infinitely recurse.
To quote an old saying, “There’s no there there.” It’s all journey, no destination.
“There is an algorithm (which is just a mathematical object)…”
A point I’ve been making in my recent series of posts about using computers to model minds!
“Like the nodes in the graph, the algorithm uses variables which have no meaning outside the context of that algorithm. The algorithm defines the variables as much as the relationships between the variables define it. This is what it means to say everything is just relationships.”
Surely the variables of any algorithm are meaningfully bound at run time? If I write:
def some_function (a, b): return a+bThere is an abstract relationship: (a,b) => (a+b)
But it’s not meaningful until run time when ‘a’ and ‘b’ are bound to real objects (values).
“I don’t think so, since that rather goes against what he is trying to say.”
Okay, I think I see the distinction you’re making. (So, in a sense, perhaps yes, but mostly no. 🙂 )
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Hi,
I’m saying “in principle” because we don’t currently know and may never know a complete mathematical description of the universe. I’m just saying that such a description exists in the abstract.
I don’t know if I would put it like that. Physical reality is just a subset of the mathematical universe, but being such a subset it is just as real as any other part of it. But we only perceive it as physically real because we are in it — what is physically real to me is just that which I can physically interact with in some way, perhaps indirectly. What is physically real to an observer in some other universe is entirely different. So, on the MUH the property of physicality is not objective but subjective and observer-dependent. I hope that explains what we are driving at.
Right. But without the relationships the nodes have no context, no meaning, no properties. If I replace these nodes with nodes E-H, nothing has really changed, only our notation. We’re still describing the same mathematical object. So I’m going to say that these particular nodes are in fact defined by their relationships (just as the relationships are defined by reference to the nodes). Now, I have no major problem if you want to insist that there is more to the graph than the relationships and that these nodes also exist. I’m just explaining to you what I take “it’s all relationships” to mean. Since the nodes are also abstract, even if I take your point I’m still not presupposing the existence of any concrete objects.
I don’t know what you mean by the “value” of the graph. The graph is what it is. It’s just an abstract mathematical object, not necessarily mapping or referring to anything in the physical world.
He is. He is not saying it is just mathematically identical to it, he is saying it is identical to it. It shares the same identity. It is the same thing. That it has no properties that are not mathematical (such as the property of being physically real).
A Platonist would disagree. From a Platonist perspective, this relationship meaningfully exists even when the algorithm is not instantiated in a running computer process or even a programmer’s mind.
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As the late, great musician Jaco Pastorius said about musical composition – “It’s just maths ‘n shit.”
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And Sinatra said, “Doobee-Doobee-Doo!” XD
What would Bach have done had he actually had a computer and a Moog?
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‘Sactly! I still have that vinyl around here somewhere. (If only I had a turntable.)
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I loved watching the score, and what a neat notation. I’m pretty sure the colors are bound to specific notes.
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Mesmerising isn’t it? And it helps you hear the music as well I think – e.g. check out the close interval bass at 4’06” (green on red dots) . Yes, the colours are pitch-specific I’m sure, and the octaves just appear vertically stacked. Anyway, pretty cool as you American chaps say.
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i do not know if this has any relevance, but just now i’m reading…
Thinking of space and time together as a sort of fabric which knits the Universe together is very useful way of approaching things on the mathematical level. But does it tell us very much about what time actually is?
It is not entirely clear to what degree the elegant world described by the quantum physicists represents reality, or conversely, a mathematical model of it. Modern physics is full of wonderful models and descriptions of reality, and this is a problem.
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Excellent point. Scientific models strive to match reality, and we generally judge them by how well they fare in experiments to test that match. But there’s never been a perfect theory. We’ve always found anomalies that eventually led to the theory being replaced by a closer understanding. There’s no reason to think our current theories are immune. (Quite the contrary.)
That being said, established scientific theories are generally established because they’ve survived mountains of experiments testing their soundness, which means they give us at least a close approximation of reality. We’ll ultimately find cracks in them, just as we did in Newton’s laws, but like Newton’s understandings, they’ll remain useful in many cases, and any new theory will have to account for all the things the current theories account for.
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Some thoughts have been lurking at the back of my mind re maths for a while that may be apropos here. I hope i am not completely off base.
I see what appears to be ‘mathematical’ calculation in animals. Playing with a less than 1 year old puppy, I noticed how it jumped off the couch repeatedly, spiraling down to land facing in the opposite direction. This complex move required a calculation of how much force to apply with each hind leg, timing it perfectly at the right moment during the descent.
Similar but greater displays of coordination are regularly seen in sports: a forty yard pass over the heads of defenders hits the wide-receiver in full stride. There is a lot of intuitive maths going on.
“Primitive quantitative abilities play a role in how modern humans learn culture-specific, formal mathematical concepts (1). Preverbal children and nonhuman animals possess a primitive ability to appreciate quantities, such as the approximate number of objects in a set, without counting them verbally. Instead of counting, children and animals can mentally represent quantities approximately, in an analog format.” Cantion, JF 2012. It is also suggested that children learn basic maths concepts between the ages of 2 and 8 years that took humankind perhaps 20,000 years to acquire.
Primitive human activity was probably loaded with intuitive mathematical activities; throwing, catching, jumping, diving, etc. Our ability to reduce these activities to symbolic equations has been the great advance, allowing us to describe formally what we have always been doing. We can also imagine new possibilities and reduce them to symbolic formulae. Perhaps we should not be that surprised that these new symbolic relations find application in the real world.
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Well said Liam. I think the fact that animals have to experiment with movements before they get it right also increases my sense that this is all, ultimately, an empirical activity. (Or an instinctive one, but evolutionary instincts arise from the same realities as empirical sensations.)
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Hi Mike,
I would say that the universe being uniquely physically real requires us to assume that there is a distinction between this one particular mathematical object we call the universe and all the others, an assumption the universe being just another mathematical object doesn’t require.
The trans-universe reality you’re talking about exists in any case, at least on Platonism. And Platonism is in my view just a way of looking at things, not something which could be true or false.
From where I’m standing, the usual assumption that there is not a vast multiverse arises from false parsimony. It’s not the simpler view, it’s the more intuitive, and the unintuitive parsimony of the MUH is thus mistaken for extravagant complexity when it is anything but.
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Ha! Interesting way to look at it!
I knew a guy once who was clearly Tegmarkian. He explained that the waveform collapse thing that MWI seeks to solve isn’t a problem mathematically. It’s as simple as how √4 is both +2 and -2. Multiple answers come from math problems with no problem! That was a long time ago, but it always stuck with me. (And makes me somewhat sympathetic towards Tegmarkian views.)
(I don’t care for the energy-mass requirements of either MWI or MUH. The explanations in all cases sound like wishful hand-waving to me. This is one place were I tend to be more of a Constructionalist.)
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Hi DM,
That’s one of the problems with the usual description of parsimony as being what’s most simple. What actually feels most simple to someone is what’s closest to their existing beliefs.
What I try to do is ask, what are we required to accept? What do we have evidence for? That’s the first tier. Every theory has to at least account for the things we are required to accept. Parsimony has no role at this tier, because if we use it here, we end up rejecting inconvenient data.
The second tier is what we’re prepared to assume. I define parsimony as the theory with the fewest number of these assumptions beyond what we have to accept.
So, we’re required to accept the universe. Even if it’s an illusion, it appears to be an illusion that exacts painful penalties for not taking it seriously.
But what about outside of the universe? I think assuming there even is an outside is an assumption. Granted, assuming there is not an outside is also an assumption. Each is an assumption. (Both are big ones.)
I’m agnostic on whether there is an outside (and the definition of “universe” is always a factor here), but if we assume there is an outside of our time and space, and then furthermore assume it follows the same mathematical and logical rules of this universe, I perceive that to be at least one other assumption. (It might be a multitude.) My thinking is that, if there is an outside, a trans-universe reality, it may be unimaginably different from anything we observe in this universe, including mathematics and logic.
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Hi Mike,
That’s not a bad definition as far as it goes, but I think it leads you astray a little. I don’t think you can really get by with just counting assumptions, as I’ll try to explain.
Perhaps. It seems to me that you’re questioning induction. We’re required to accept that the sun has risen every day in the past. But what about tomorrow? Even assuming there is a tomorrow is an assumption.
That’s true as far as it goes. But is it a big assumption? I wouldn’t have thought so. You’re asserting that it’s a big assumption either way to say there is more to existence than the universe or that there isn’t. But in my view one of these assumptions is far more parsimonious than the other.
Here you seem to be departing from me on another front. I am pretty convinced that the laws of physics have no power to constrain mathematics or logic. I firmly believe that the same mathematics will work in all possible universes. There can be no universe where 2+2=5 because this is false by definition, not due to some contingent aspect of the world we live in. Again, you may say this is an unnecessary assumption, but I think it’s far more plausible than the assumption that logic is contingent. Unfortunately this is probably not possible to argue further because I can’t use logic to prove the validity of logic. Logic is just one of those things that has to be taken as a given. For me this means it is a given in all possible worlds.
The trans-universe reality is not really following any particular mathematical or logical rules. It doesn’t evolve with time for one thing, so “following” probably isn’t quite the right word, or at least there is the risk that it could lead to the wrong picture (not that you meant this). Rather it is just the set of all mathematical objects, including those not conceived of or not conceivable to human beings. Yes, you are right, it obeys the rules of logic in the sense that only logically possible or coherent objects can exist, but this seems to me to be self-evident if not outright tautological.
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Sorry, I left out the most important point!
Which is that nobody is assuming that there is more stuff outside the universe. This is a conclusion, not an assumption, which arises from assuming as little as possible. In particular, I don’t assume that physical reality is an objective observer-independent property, and I don’t assume that this is the only world. As I’ve outlined on my blog, my assumptions are only:
that Platonism is a reasonable stance to take
that naturalism is true (no magic: everything that happens in this universe is governed by laws expressible in the language of mathematics)
functionalism/computationalism is true in philosophy of mind.
As I’ve argued, it seems to me that the MUH follows from these assumptions. So it’s not correct to simply say I’m assuming there is a reality outside the universe.
Again, the assumption I’m very explicitly not making is the intuitive assumption that there is an objective ontological difference between a physical object and an abstract object. That’s one less assumption than most people make and that’s what makes the MUH more parsimonious.
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Great discussion you guys are having!
DM, I’m not sure how you see me as questioning induction. But induction does have its limits. The sun will rise tomorrow, but it won’t rise forever (unless maybe the earth survives the red giant phase and we want to consider a white (and later black) dwarf “the sun”). The further beyond observation we go, the less certainty we can have in our extrapolations.
On logic and mathematics, I think you’re right that further debate is unlikely to clear anything up. I’ve come to see mathematics and logic as our most basic theories about reality, which means I do think they are constrained by physical laws. But I can’t prove that’s all they are, and I see no way to prove that they’re more than that.
On assumption versus conclusion about the multiverse, I can see that for quilted and bubble universes, which still exist in our space continuity, but beyond that, the logic for other universes becomes much more speculative to me. Again, this is not something we’re likely to resolve here 🙂
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“Again, the assumption I’m very explicitly not making is the intuitive assumption that there is an objective ontological difference between a physical object and an abstract object.”
Disagreeable Me, I’m glad you brought this up. I think it’s important to tackle the question of what constitutes reality head-on. In Platonism (by that, I mean Plato), the abstract object would have “more reality” than physical objects (at least on my interpretation of Plato). I think you’re saying here that you make no such ontological claims, but are pointing out that we often assume the opposite, that physical objects are “more real.” Hence the intuition that mathematics is “real” only insofar as it applies (or can apply) to the observable world…the extreme version of which is that math is only “in our heads.”
Also, curious to know whether you would assign ontological priority to abstract mathematical objects, or is that going too far?
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Hi rung2diotimasladder,
I don’t think I would go along with classical Platonism at all really. I certainly don’t see real objects as pale reflections of Platonic ideals. I don’t think there is an ideal horse, for instance.
I don’t think the question you’ve asked is meaningful on my view, since I think everything there is is a mathematical object. There are no physical objects to compare to the mathematical objects, or rather physical objects are just mathematical substructures we are causally connected to by virtue of being embedded in the same larger mathematical object (our universe) as us and so we perceive them as physical.
You could say that this means mathematical objects have ontological priority, because they are all that exists, but at the same time I’m not saying that the concept of a circle is any more real than a drawing of a circle on a page.
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Linking to Disagreeable Me’s blog post on this, because I think it’s relevant, and politeness probably inhibits him from doing it.
http://disagreeableme.blogspot.com/2013/12/the-universe-is-made-of-mathematics.html
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@ SAP,
Sorry, I couldn’t figure out how to reply in order. Thanks for the link. That did clarify things quite a bit.
@ Disagreeable Me,
I checked out the blog post and will comment there too.
Are you saying the physical world is an illusion? Is it time to coin the phrase “mathematical Parmenidean-ism?” (Or maybe not…I certainly wouldn’t want to hear myself say that out loud). 🙂
“You could say that this means mathematical objects have ontological priority, because they are all that exists, but at the same time I’m not saying that the concept of a circle is any more real than a drawing of a circle on a page.”
I’m confused. That last bit comparing the concept of a circle to a drawing of a circle seems to point to the classic ideal/physical dichotomy. But then you’re saying mathematical objects are all that exist, and the physical is nothing more than mathematical. How can the circle drawing qua physical exist then? Or was your point to dismiss this dichotomy altogether?
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Meh. I tried to leave a comment on your blog, DisagreeableMe, but it wasn’t agreeable at all.
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rung2diotimasladder ,
I don’t know why you wouldn’t be able to comment on my blog. If there’s some issue I’ll try to fix it.
No. Although I could call an illusion the perception that the physical world is any more real than any other mathematical object. All mathematical objects exist. The physical world just seems more real to us because we are a part of it. It is only due to our point of view that it seems especially real. That’s not to say it isn’t real, but to say that it is no more real than other mathematical objects, which are all real.
“Physical” just means “part of the same universe as the speaker”. The drawing of a circle exists in at least a couple of of ways. Firstly as an arrangement of atoms within the universe of the person who drew it — again, being just another mathematical object that universe is just as real as the ideal circle and so are constituent parts of it such as the drawing. Some things are said to exist by virtue of the role they play as recognised by observers. For instance, money and organisations and so on only exist because people agree that they do. The drawing also exists in this way — it exists qua a drawing of a circle by virtue of its being recognised as such by observers within that universe, e.g. the person who drew it.
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“money and organisations and so on only exist because people agree that they do”
How can you say this, but also say that mathematical objects exist independently of us?
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I don’t really see a contradiction. Money isn’t really a mathematical object, is it? Neither are organisations. Although they could be seen as patterns that can be superimposed on the goings on in a mathematical object (the universe).
For money to exist, there have to be people. So it isn’t independent of us. The idea of money without people doesn’t seem to me to be viable because money is more or less defined as a medium of exchange used by people. Circles and numbers and so on are independent of us because they can be defined without reference to people.
Maybe there is a problem there and maybe you can spell it out. Perhaps I might retreat from my view that money only exists because people agree that it does. I guess money exists as an abstract solution to the abstract problems of having a medium of exchange, store of value and unit of account, but to say “I have 5 pounds” is only true if society and other people agree that I have 5 pounds.
But none of this is really to say money doesn’t really exist. It really does exist, because people (who really exist) say that it does. It’s just that for it to exist, there have to be people using it, because those people form part of its definition.
Ultimately, you could say it is just another mathematical object, but it is a mathematical object that is made of patterns in the structures of the minds of people (who are also just mathematical objects). Viewed as a mathematical object in this way (as an entity distributed in the minds of all people), money is ridiculously complex, so it is not helpful to view it this way and I usually wouldn’t.
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Well I think this might be the nub of our different views. Money and organisational structures are abstract concepts. But they were clearly invented by people. Why is this different to a mathematical object?
Can you define (or even give an example of) a mathematical object? Is a circle a mathematical object? Is an equation? How are they different to the abstract objects we create, or are they not? Is there for example, a mathematical universe made of money, or one that has the same organisational structure as Microsoft?
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Hi Steve,
For me, it’s not a question of whether some concept was invented by people (incidentally, I would personally say mathematical concepts are discovered, not invented), but whether it can be completely well defined by reference only to other well-defined entitites with absolutely no ambiguity. The well-defined entities are just the mathematical entities.
Money is not a well-defined entity in this sense, because it cannot be defined without reference to people and goods and exchange other mushy concepts of that sort.
A circle and an equation are perfectly well defined. There is no ambiguity there. They have objective properties that can be explored. Independent explorers will find the same results. We would not be too surprised if alien civilisations had independently discovered the same concepts.
Now, we might not be too surprised if alien civilisations had discovered the abstract concept of money, but we would be surprised if they used British Pounds Sterling or American Dollars as their currency, and we would be surprised to find they had a local branch of the Catholic church. So I’m happy to say that there is something real about the concept of generic money or a generic organisation as a solution to a problem, but they’re still a bit fuzzier than mathematical objects and they can still only be defined with reference to people (or rational agents of some sort at least) and so I guess it’s a secondary kind of existence for me.
How could it be made of money if there were no people to treat it as money? Money is only money if there are people exchanging goods and services for it. It’s baked into the definition. The idea of money in the abstract without such people is meaningless.
If you treat the organisational structure of Microsoft as a directed graph, say, then that graph exists as a mathematical object. But that graph is not Microsoft, because (for instance) it doesn’t publish software for people to use. So there are aspects of organisations which are mathematical objects but those are not the organisations themselves. An organisation is a structure that supervenes on people, and you can’t have an organisation without having people for it to supervene on.
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Hi Disagreeable,
I figured out what I did wrong when I tried to comment on your blog. I wrote “rung2diotimasladder” instead of “philosophyandfiction”…sorry about that! Unfortunately I didn’t save the first comment and now I’ve forgotten my point.
On the physical, thanks for the clarification. I think I get what you’re saying now.
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” I am pretty convinced that the laws of physics have no power to constrain mathematics or logic.”
Can you be sure? If I spin a ball, it can rotate clockwise (A) or counter-clockwise (B). Classical logic states that A and B are mutually exclusive states. But if the ball is a subatomic particle, the laws of physics allow A and B simultaneously.
“There can be no universe where 2+2=5 because this is false by definition.”
If it is a definition, how can it act as a constraint on the universe?
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Hi Steve,
No it doesn’t, any more than classical logic states that being orange (A) and spherical (B) are mutually exclusive states.
What you want to say is that classical logic holds that being in state A and state not A are mutually exclusive. But the actual state of a subatomic particle is not spin up (A) or spin down (not A). Rather the actual state is a superposition of the two.
So quantum mechanics is not contrary to logic. It couldn’t be. The apparent contradiction arises from an incorrect mathematical or logical modelling of the state. You’re modelling things as black and white while in reality there are shades of grey and there is no logical inconsistency in saying something is, say 75% white and 25% black.
A universe is a mathematical object, and mathematical objects are basically just things that can be rigorously defined. Any universe you can define can and does exist. But it doesn’t make sense for that definition to include “and in this universe 2+2=5”, because that is nonsense.
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“there is no logical inconsistency in saying something is, say 75% white and 25% black”
Indeed, but quantum mechanics requires something to be 75% A and 25% NOT A, which is in complete violation of classical logic.
“it doesn’t make sense for that definition to include “and in this universe 2+2=5”, because that is nonsense”
I’m not a mathematician, so I’m going out on a limb here, but I’m pretty sure you could define such a universe. I could certainly define a computer algorithm that obeys this law. It’s simple. Every time a set contains 4 objects, there is a rule that says it immediately spawns another object, so there are 5. I could write code to ensure this. It would be a rigorously-defined system, so it would be a mathematical object according to your definition.
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Hi Steve,
So what happens if you have two sets of two objects? How many objects do you have then?
Or if a set has 5 objects, then it has two disjoint subsets, one of 4 objects and one of 1 object. But the former subset must actually have 5 objects, so the set has 6 objects, not 5. But we stipulated it had 5 objects.
If you do actually manage to create a system or a simulation where we can make sense of this kind of thing, then that is a real mathematical object where “2” “+” “2” “=” “5”, but every term needs to be put in scare quotes because what we’re actually doing is not addition of integers but something completely different and just reusing the same notation to muddy the waters.
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“So what happens if you have two sets of two objects? How many objects do you have then?”
It would depend how I counted them. The answer would depend on the order of operations.
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Hi Steve,
Missed this point because it was italicised and looked like a quote from me.
No it isn’t. white and black are opposites too. There is no logical violation in something being 75% white and 25% black and there is no logical violation in something being 75% A and 25% not A as long as the model we are using allows for this. If the logical model you are using for quantum systems yields conclusions that differ from experiment, what you should take away from this is not that logic doesn’t apply but that your model (consisting of logical premises) is incorrect.
You can model and simulate quantum systems on a computer, which are devices built on classical logic. But you can’t do so by modelling a quantum system as a system where a particle has a defined mutually exclusive state of either spin up or spin down. You need to model the state as a superposition. And once you do so you no longer have a logical inconsistency.
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Well then, let us say that the MUH is 50% correct and 50% not correct. 🙂
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I would say at the heart of the idea of numbers is that it doesn’t matter how we count objects. There are either four objects or five objects, and if you’re saying that the number of objects depends on how you are counting them then you are not counting them but doing something else. As long as you can define without contradiction what you are doing (e.g. by writing a computer program) then that’s all perfectly legitimate but you are not making a system where 2+2=5. You are making a system where those symbols have been repurposed to have entirely different meanings.
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So, on Tegmark, at one level it’s a theory that’s impossible to falsify and has no predictive power.
And yet, there is one prediction you can make from the Tegmarkian world view. If all mathematical objects are “real” and constitute “universes”, and the universe we inhabit is just one of these mathematical objects, then one prediction we could make is that, if all mathematical objects are equally likely, then we would expect to be living in a very complex one, with many relationships, types of objects, interactions between objects, etc. And by many, I mean something like infinite. And yet, our universe appears to have just a few dozen fundamental types of particle, merely 4 forces, a number of dimensions that you can count on 1, 2 or at most 3 hands, and a whole bucketload of symmetries. Given the multitude of weird mathematical objects that we could have found ourselves living in, this one seems exceptionally boring, don’t you think?
The anthropic principle argues for a universe not less complex than this one, but it’s hard to justify why a more complex universe couldn’t support life. So, this apparent simplicity of our universe seems like an anomaly to me, that the Tegmark model needs to explain.
Paradoxically, it’s the apparent simplicity of our universe, and all its regularities that presumably led Tegmark to his theory.
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I suppose one reply might be that, if there are a plenitude of universes, there is at least some odds we just happened to end up with a really boring one. Or one could argue that too complex a universe leads to too much chaos for life to arise. Maybe it takes a little boredom for life to get a chance to bang two cells together.
I’m not entirely convinced there’s anything beyond the Oort cloud other than a painted backdrop, so you can just imagine how I stand when it comes to multiple realities. XD
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Hi Steve,
That is a good argument, and one that has bothered me in the past. I have two answers.
Firstly, we don’t know that this universe isn’t infinitely complex. The further we probe into the fundamental building blocks, the more detail we seem to find. From molecules to atoms to fermions to nucleons to quarks to strings(?) to who knows what. Not to mention the possibility that there are all kinds of particles and forces out there which simply do not interact with ordinary matter. The dark matter universe could be a whole complex system in its own right, and we only know about that because it happens to interact via gravitation. There could be even more stuff that doesn’t even do that.
In other words, it could be that in an arbitrary mathematical object containing observers, it is perhaps not unreasonable to suppose that most of the detail is irrelevant or hidden and just a few rules tend to dominate. It could also be that if you add a great mass of rules together you tend to find that on aggregate they tend to be equivalent to a simpler set of rules. I mean, there is a lot of complexity and really impossibly difficult quantum mechanical calculation determining the physics of the phase transitions of water, but in the end we are left with some pretty simple rules such as that water freezes at 0 and boils at 100 degrees C at standard pressure. Perhaps it is not that easy to make a truly complex byzantine set of fundamental laws accessible to the observer.
But, if you could design such a byzantine world, I would tend to agree with Smythe’s answer. If the laws of nature were really infinitely complex on a scale we could appreciate, it would make it very hard to predict what would happen next, and this effectively makes the world so chaotic it’s hard to see how self-replicators could arise. Self-replication relies on a stable and relatively homogenous environment and an ability to “predict” what will work in future. My feeling is that those worlds which look like you would expect a typical world to look like on the MUH are not host to any observers and so are filtered out by the anthropic principle.
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In that case, the MUH seems to be predicting that the laws of physics are far more complicated than we currently believe and that there are many particles and forces that we have yet to observe. Is that a fair assessment of your stance?
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Well, if there are particles which we can in principle never interact with, that’s pretty much the same as them not existing.
What I’m saying is that if you try to invent a complex universe on paper, you might find either that you end up with observers closed off from a lot of the complexity or that the complexity produces too much chaos so you end up with no observers.
Now, I could be wrong on either of those counts. You might indeed find that the typical result is something very different from what we see. I offer these as plausible explanations for why the universe seems so simple, not as hard refutations of your argument.
I’m not smart enough to give an answer to the question of what we should expect a typical observer in the MUH to see, and I’m not sure any body else is either, so I’m not really that comfortable with trying to make predictions from the MUH (and this includes your argument predicting more complexity than we see).
Tegmark wants to make predictions because he wants to claim it as a falsifiable scientific hypothesis, but to me it’s just a philosophical conclusion and I don’t make any attempt to defend it as science.
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“if there are particles which we can in principle never interact with, that’s pretty much the same as them not existing”
What if there were universes like that? Perhaps they wouldn’t exist either. 🙂 Seriously, I do think that the MUH (and mathematical platonism) basically tries to redefine what is normally meant by “exist” or “real”. Nothing wrong with that, I’m all for free thinking, but it makes me queasy. That way of thinking seems to lead to a kind of nihilism, where everything is real, and therefore nothing is.
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Hi Steve,
Yeah. I should have said “pretty much the same as them not physically existing (at least from our perspective)”.
This is kind of an important point to me, in that I think it is wrong to suppose there is a fact of the matter about whether such particles exist in our particular universe. There are some mathematical models of our universe in which they and we co-exist, and some in which they do not exist. I don’t think it is correct to say we are in either the former or the latter. We are in both. If there are two universes which contain identical copies of me and the differences between those universes are closed off to me then I think it is reasonable to identify with each copy of me equally and to deny that there is a fact of the matter regarding which mathematical model describes my particular universe (or, to put it another way, there is no fact of the matter regarding which of these universes I am in).
In the case where a previously closed of facet of the physical universe becomes available, say through a new discovery at the LHC, I prefer to think of it as my identity forking much as in the MWI. One copy continues in the universe where the fine detail turns out to be one way, and another copy continues in the universe where the fine detail turns out to be another way.
Sure, I suppose it does, but that doesn’t mean it’s wrong, or even a bad thing. I think you can still be a positive, productive member of society even if you do subscribe to that kind of “nihilism”, without the negativity, pessimism and cynicism usually associated with that label.
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But now you’re suggesting that you inhabit multiple universes at once. In one of those universes, another version of you is denying this. In yet another universe, another version of you has never even heard of the MUH. Very quickly you are spiralling into nonsensical statements where everything and nothing is a valid statement about reality.
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Hi Steve,
What I actually said was insofar as the two copies of me are perfectly identical, I think it is problematic to identify with one of them in particular and not the other. Another “version of ” me is not me because it is not identical.
I disagree that I’m spiralling into nonsensical statements. Perhaps I’m spiralling into statements you regard as nonsensical, but that’s not necessarily the same thing.
But you are right that the core of the MUH is to call into question the assumption that reality is a concept we should take for granted.
And I think that’s right, because I think it is problematic. It’s fine in everyday discourse when we’re talking about tables and chairs and so on, but not when we’re talking about universes or String theory or mathematical objects in general.
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Interesting discussion, even as it is viewed from my very different sub-universe.
Not much attention has been paid, so far, to the reality of ‘time’ or, more precisely, the various clocks that constitute time as we experience it. The way I see it, time eliminates all other possible universes from the possibility of being real. There are an almost infinite number of real or imaginary sub-universes, but all of these are founded on, and dependent on, our one real universe.
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Hi Liam, I think most physicists would say that time, like space, and more precisely space-time, is a feature or property of our universe. So if another hypothetical universe existed independently of our own, it would not exist on the same spacetime continuum as us. It would not be meaningful to say that the other universe existed in our past, future or present, just as it would not be meaningful to say that it existed within or outside our own space. If it did exist in our own spacetime, it would be a part of our universe, or our universe would be part of it (or both.)
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I think that’s right.
Liam, you could consider other universes to have their own independent timelines, much like two different DVDs sitting alongside each other on a shelf. The events of The Fellowship of the Ring do not take place before or after the events of Harry Potter and the Sorceror/Philosopher’s Stone.
Time only really flows from a perspective within those worlds. From an outside perspective we can view them as complete works, not with a past, present and future but with a beginning, middle and end.
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Hi Steve Morris and DM,
I basically agree with both of you, but still come to a different conclusion:
“So if another hypothetical universe existed independently of our own, it would not exist on the same spacetime continuum as us.” Agreed, and so there would be no interaction whatsoever between these universes; they would forever be ‘ignorant’ of each other.
“.. you could consider other universes to have their own independent timelines, much like two different DVDs sitting alongside each other on a shelf.” Yes, the two DVD’s, left to their own devices would never interact with each other. However, I could watch (interact with) them via my player, so they are sub-universes in my universe.
Any hypothetical universe as posited via philosophy, maths or religion is absolutely dependent on a functioning brain, and so would be sub-universes of that consciousness, itself a member of our real universe (reality as it is). Any truly parallel, independent, self contained universe on another space-time continuum would never interact with our universe.
There could, alternatively, be multiple completely independent sub-universes, each on its own space-time continuum. A god-like entity would then be the only one that could interact with all such trans-cosmic sub-universes of god.
We have no option but to interact with our universe, the only one that there is, as far as we will be able to tell.
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Hi Liam,
I hope you realise that’s a pretty controversial, if not radical statement. The usual idea is that these other universes are not at all dependent on a brain. The brain is not sustaining these universes but is deducing their existence. The universes exist regardless.
Agreed. But a brain here in this universe deducing the existence of other universes is not interacting with them. The brain does not affect those universes in any way. I would also say those universes are not affecting the brain imagining them. What’s going on the brain can be explained without appeal to the causal powers of foreign universes — it’s just neurons firing because other neurons fired and so on. This activity reflects in some way the idea that these universes exist, but those universes did not cause that idea to be born and that idea would exist even if those universes did not.
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Hi DM,
Thanks, I did not realize that my opinion was that radical! I would have thought that most realist (neuro-)scientists would find the statement unremarkable. Sort of like Santa Claus who exists but is not real – a distinction that I recognize, but some don’t. Without a human mind to conceive of Santa, he ceases to exist. Similarly, by my definition, mathematical universes outside our space-time continuum are not real, even the thought of them is a real thought existing in the mind of a mathematician. If such a mathematical object is communicated to other mathematicians it becomes part of their culture, but it is still not real, just as Santa is not real even though millions think of him at times.
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Hi Liam,
Perhaps I shouldn’t have said radical. I think we may be talking at cross purposes, because it rather seems to me that you are conflating thoughts and beliefs about other universes with the universes themselves.
You should not interpret most multiverse proponents as claiming that other universes exist, but only as ideas in our heads (like Santa Claus). Instead they are claiming that they really do have an independent existence.
So when I claim another mathematical universe exists outside of our spacetime, I am claiming that it is real, just as real as ours, and independent of any mental intentions we may have towards it.
You’re saying it is only in our heads, and indeed this is not at all radical but perhaps even mainstream because many people deny the idea of a multiverse. But then all you’re basically saying is that you don’t buy the idea of a multiverse.
But since you’re not presenting yourself as simply assuming that the multiverse idea doesn’t hold water, it comes across a little like a radical reinterpretation or misunderstanding of what the multiverse idea means.
In particular, since multiverse-proponents don’t buy your premise that “all of these are founded on, and dependent on, our one real universe”, your claim that “time eliminates all other possible universes from the possibility of being real” doesn’t follow.
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Great discussion! I’ve bookmarked this page for future reference.
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This, to me, at least somewhat implies that the universe is some sort of simulation. I might have this backwards, but in the simulations we build, we define parameters and “patterns” mathematically. It could be coincidence that our universe seems to follow a similar pattern, but based on how many simulations our civilization produces, one would expect vastly more simulated realities than “real” realities.
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We can never know for sure that we’re not living in a simulation. But if this is a simulation, it appears to enforce painful consequences for not taking it seriously, and we appear to have little choice but to play the game. Indeed, if it is a simulation, then the simulation is our reality, at least until or unless the outer reality intercedes.
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Very true. The reason I bring this up is that, it seems to me, thinking of our existence as a simulation both rationalizes the mathematical nature of our universe and explains the ways mathematics in general and determinism in particular break down.
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me, as a fool pole, only one wonders. with such a great tool (math) which offers our great mind and such a wonderful description of this universe provided by math – “Mathematical Universe”. (of course if we assume that mathematics is such a great tool as we think). that’s why our knowledge about Universe is so fragile that even a small detail can overthrow it.
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Stan, not sure if I’m following your point here. I do think the usefulness of mathematics is more than just an assumption. Most of science and technology collapses if we see math as an optional facility. It seems forced on us by the universe.
But I do agree that details that don’t conform to a theory (such as the anomalies in Mercury’s orbit that eventually led to the superseding of Newtonian dynamics) can be crucial.
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there are many such small details, but it requires a detailed description. i also don’t want it to oppress science fruitless quest actual evidence of the existence of dark matter, energy. it isn’t too politely in order to dig lying*. but there is something that seems to us that we understand perfectly. perhaps we know it in great detail, we have it investigated and described. Mathematics captured it perfectly. but we absolutely don’t understand what gravity is. and there is even scientific evidence. of course according to my belief.
*idiom needed here, but i don’t know. (how to express that to oppress someone who can’t defend themselves).
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I think it’s important to remember that dark matter and dark energy are basically bookmarks, placeholder terms for phenomena whose effects we observed, so we know they exist, but that we basically know little about, which is why they start with “dark”. I’ve long thought that “dark matter” was a name that assumed too much, that a better one would have been “dark mass”.
Indeed, we don’t know what gravity is, or energy for that matter. We know a lot about what they do (warp space, etc), but we don’t fundamentally know what they are. (Newton was reportedly pretty bothered by this.)
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but probably it not change the fact that they are products of Mathematical Universe and we may find that this doesn’t exist in reality. because it is that we are seeing some effects may confirm the existence of dark energy,matter but we may find out that our mind has the next illusion, very significant qualities of our minds!
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This discussion seems to me to be an excellent example of how implicit differences of ontology and metaphysics end up garbling a conversation. This really reinforces to me the importance of those disciplines, somewhat neglected nowadays.
I agree that ‘mathematics’ is a fundamental process but it seems a little glib to conclude that it is what constitutes the universe. If ‘mathematics’ is defined as information processing, then such information would be even more basic for an understanding of the universe.
The true nature of information is a mystery, so we are left with our best efforts which still seem quite meager. Approaches that leave out swaths of information should be questioned. The crucial instrument in all of this is our brain, and nobody knows what is really going on inside of it – but we are beginning to make progress.
I have tried to survey all domains of information, including some of the underlying processes. Needless to say, it is way too complicated, but my alter ego has summarized all of it in plain language as a Perspective on Everything. (It already seems a bit dated.)
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and besides, someone wise said … “math is a tool you use to describe things you understand. It’s not a substitute for understanding”
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I like sometimes to think of the universe as the unfolding of a mathematical proof..What is universe we see today but a set of fundamental behaviors repeating themselves again and again..each iteration applying to the result of the previous iteration.. the distinction between substance/matter and abstract ‘laws’ applying on it sounds like nonsense to me.. reality as we experience it and interact with it is patterns of behaviors..
I guess what i m trying to say is that i agree with your view that mathematics IS the universe.. 🙂
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“usefulness of mathematics is more than just an assumption”.
Michael. it isn’t true i don’t appreciate mathematics. because this is a mathematics underlies the development of our mind. but unfortunately it has created so many universes where there is no room for reason.
But let’s get back to reality. it is the year 1997. we know that the universe is expanding. we don’t know yet that this expansion is accelerating. maybe someone knows what led scientists to conclude (before 1998) that at some point the Universe will begin to shrink under the influence of gravity. because according to me is one of the greatest scientific idiocy of the last century (XX). and the problem is really serious. because if it turns out it was my fault i don’t understand this problem, it will mean i don’t understand virtually nothing!
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Your point, if I’m understanding correctly, is the chief reason I’m primarily an empiricist rather than a rationalist. Logic and mathematics can only carry us so far. Their based on what we currently have evidence for in the universe. They can’t take into account aspects of reality we don’t know about yet and haven’t imagined.
This is why I find the desire of some theoretical physicists to have their theories judged by mathematical elegance, rather than their success in passing empirical tests, to be so disturbing.
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Wow. I’m sorry I missed out on this. An abstract system is what it describes. An interesting proposition. Are not models easily confused for what they model if they are good enough. It is hard to imagine that an equation is a thing though. Is description the same as what is describes?
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