Oh, no! Not math again!
Among those who try to imagine alien first contact, many believe that mathematics will be the basis of initial communication. This is based on the perceived universality and inevitability of mathematics. They see math as so fundamental any intelligence must not only discover it, but must discover the same things we’ve discovered.
There is even a belief that math is more real than the physical universe, that it may be the actual basis of reality. The other end of that spectrum is a belief that mathematics is an invented game of symbol manipulation with no deep meaning.
So today: the idea that math is universal and inevitable.
A while back I wrote about how we may have discovered the basics of mathematics in our need to count things (from sheep to ships). That progression assumed humans on Earth (with sheep and ships).
Count us!
This time I want to try starting from first principles with as few assumptions as possible. I’m going to start by assuming physical existence, intelligent self-awareness, and minimal sensation.
Let’s see where that takes us.[1]
The One True Fact that we can know regarding intelligent self-awareness comes from René Descartes’ famous statement: Cogito ergo sum — “I think, therefore I am.” From this fact, and our self-aware thought process, what might we derive?
We can notice that we seem to experience the passage of time.[2] There are thoughts we had earlier. There are the thoughts we are having now. We may even realize there are thoughts we haven’t had yet.[3]
A sense of time passing leads to the need to measure its passage. We might do that by noting heart beats or breaths; for us time’s basic units are on the order of seconds or minutes.
I sense… I’m late again!
A being capable of sensing chemical processes might measure time on a much smaller scale.
A being capable of sensing only the passing of days (or seasons) might measure it on a very large scale.
But sensing the passing of time leads to the need of a metric — a means of measuring, and also recording, the passage of time.
“Yesterday” and “an hour ago” are relative measures that reference the current moment. Dates (and times) are more absolute measures, although even those are relative to some “mile marker” moment.
Time implies the concepts of before and after — relationships between moments in time.
The sense of units of time, whether vibrations of electrons, heartbeats, mere thoughts, or the passing of seasons, seems to lead directly to the need to count those units, to the idea of the natural numbers.[4]
Merely thinking thoughts, let alone noticing the passing of time, introduces the key ideas of objects and — crucially — classes of objects.[5]
Not only do there exist things, but some things are similar enough to be grouped together.
Cogito ergo maths
The passing heartbeats, or the sensation of passing days, each pulse is like another.
To recognize multiples of a type of thing is to recognize the natural numbers.
But what if there are beings who sense time only as a smooth continuum? It’s hard to see how a mind would never discover periodicity or multiple objects, but that could be a failure of my imagination.
Perhaps such a being might see past events as magnitudes of time from its current moment. Even so, there are still relationships between events and the idea of intervals of time between them.
We have the idea of a continuum of time, and that gives us before and after as well as shorter and longer.
If we consider two past events, even if we sense the time spans as, say, an electrical potentials in our minds — if all we can say is that event A happened longer ago than event B — we still have the two events!
If we can remember specific moments, or specific thoughts, then some form of counting seems inevitable. Remember that the natural numbers are really set size cardinals. Counting is nothing more than considering how many objects are in the set.
§
S’real!
So far we haven’t opened our eyes. We’re just inside our own heads noticing that our thoughts occur in time. And sensing how our bodies feel passing time.
Yet we have almost certainly recognized objects of thought — ideas and memories.
It would seem hard for intelligence to progress from that without the natural numbers. All it takes is the recognition of objects (which seems impossible not to have) and the recognition of classes (which might be an indicator of intelligence — it is something of a distinction between humans and animals).
Now we have sets, and sets have cardinality, and — voila — out pops the natural numbers. (Given the obvious question of what’s between two natural numbers, it’s likely we’re also onto the rational numbers or even the real numbers.)
We have also discovered relationships, which, with a little thought, lead to the idea of operators.[6]
All without opening our eyes or experiencing the physical world around us.
Let’s open our eyes now and experience the physical world.
§
Closer < Farther!
Just as we realized we exist in time, now we see that we exist in space. If we have any experience of space at all, what might be basic to that experience is a sense of distance (of spatial extent).
Exactly as time separates points in time, spatial extent separates points in space.
Recognizing spatial extent leads us to the need for a spatial metric, a way of measuring space.
As with time, we can ponder the idea of a mind that doesn’t see distances as having units, but as magnitudes.
Perhaps they can only be compared as greater or lesser (but again comparing magnitudes implies distinct, thus countable, objects: the magnitudes themselves).
These metrics of time and space allow more than measuring. They allow us to label, and thus remember, think about (calculate with!), and discuss, these concepts.[7]
Even if an alien race consisted of a single vast individual or mind-sharing collective (think hive), remembering and thinking about time and space requires labels.
So it almost seems impossible to think about these things without discovering (some might say “inventing”) mathematics — the language necessary for those thoughts!
§
The German mathematician, Leopold Kronecker, famously said: “God made the integers, all else is the work of man.”
“Lee” Kronecker
This recognizes how fundamental the natural numbers are (they are called “natural” for good reason).
In a universe of discrete objects, shared properties create classes (sets) of all objects having that property (e.g. “redness” or “spaceshipness”).
Sets of objects not only can be counted, they demand to be counted. The one thing all sets have is a size, a cardinality!
The natural numbers, either through the division operator, or through pondering natural steps between integers, lead fairly directly to the rational numbers. The world of countable objects seems intimately part of a universe of recognizable objects.
Arriving at the real numbers — the world of uncountable objects — is another milestone along the journey, and there are multiple ways to get there. A species who thinks naturally in magnitudes might even get there instinctively.
The same “what’s in between” logic that leads from the naturals to the rationals can lead to the reals. A growing ability to measure magnitudes with precision can also lead to the reals — or if not, at least to the rationals (which just adds to their inevitability).
…and so on!
A self-imposed word limit prohibits getting too deep into the inevitability (and universality) of geometry now, but careful consideration of a circle tells us something weird is going on — something beyond the discrete world of countable objects.
The universe’s duality turns up once again. It’s filled with both countable and uncountable things. As such, experiencing the universe in any meaningful and intelligent way seems to lead naturally — inevitably — to mathematics.
The other question involves the universality of math. I’ve written before about the difference between how we spell numbers versus what those numbers actually mean. As words are just local handles to ideas, physical numbers are just handles on mathematical ideas.
These ideas, at the very least, have the reality of ideas and in many cases represent real-world objects. Taking the metaphor a step further, mathematics is the language that uses those words.
Still just “1” “2” “3”
But just as different languages express the same idea different ways, different mathematics can only express the same mathematical ideas in unfamiliar ways.
Put simply, 2+2=4 regardless of how it’s expressed!
What might happen — what does happen in language — is that one may express ideas unknown to a culture that hasn’t encountered them (so far).
Certainly an advanced culture might have advanced mathematics, but they wouldn’t expect to use that in first contact (any more than we would).
Further, they still have to balance their alien expense reports, so ordinary mathematics won’t have gone away (any more than it has for us).
For me, the question isn’t whether mathematics is inevitable and universal. It’s how can it not be?
Stay numerate, my friends!
[1] WP blogger Steve Morris recently did a similar post. This isn’t a response — I already had notes for this — but it does elaborate on comments I made there.
[2] Note that we are adding another assumption: That all intelligent minds experience time in the same serial fashion that we do. Could there exist beings who experience time all at once or that have no (or just a different) sense of past or future?
Yet time seems to be a universal aspect of reality. Chemistry, and thus biology, is a process that occurs in time. Motion also occurs in time (in fact, you can’t define it without time).
The idea of thought without the experience of time seems incoherent to me, so I’m going to claim that time is inherent to thought. (If anyone disagrees, we can consider it Debate Point #1!)
[3] Yet another implicit assumption: That intelligence comes with memory, but I think this is also a reasonable assumption. Memory is necessary to learning (and judging comparisons) and therefore, surely, intelligence. (Alternately: Debate Point #2!)
[4] Maybe intelligent trees (who communicate via chemicals wafted on the wind) sense only the passing days and the longer rhythm of seasons. But these are distinct, very similar, objects of perception, so even intelligent trees with no language (written or verbal) might still discover counting and, thus, mathematics.
[5] If you are disinclined towards “objects of thought” consider that, really, everything is an object of thought (even your sensations).
[6] It starts by realizing we can add two numbers together. That leads directly to subtraction. Multiplication is just serial addition. Division is a bit more of a leap, but it follows from multiplication (and division gets you to the rational numbers.). Exponents are just fancy multiplication.
We’ve already recognized some relational operators: greater-than, less-than, and all the others. These lead to the idea of logical operators: not, and, or, xor. At this point you’ve discovered arithmetic.
[7] We might ponder whether a species necessarily discovers the idea of three-dimensional space. Could a physical being exist without a notion of volume? Might a being only be aware of two-dimensional space (surfaces) or even one-dimensional space (lines)?
That seems unlikely to me, but I won’t rule it out entirely. It does seem that any sense of extent, or of movement in space, must lead to the recognition of dimensions. After all, we ponder 4+ dimensional space. (I’ve written about dimensionality before. See Dimensional Coordinates.)
Logos con carne 
October 12th, 2015 at 6:23 pm
This post adds up for me Smitty! ❤
Diana xo
October 12th, 2015 at 6:51 pm
I’m glad you weren’t divided by it! XD
October 12th, 2015 at 8:32 pm
Funny guy! 😀
October 12th, 2015 at 8:43 pm
Thanks! I feel it multiplies my value.
October 14th, 2015 at 4:32 am
Very interesting post. I used to be at the “math is more real than the physical universe” end of the spectrum, but now I am closer to the “mathematics is an invented game of symbol manipulation” end. Is it possible to believe that mathematics is invented, but also a powerful way of thinking about the universe? I think so.
I agree with you that time is a core requirement for thought. As I see it, thought, or consciousness, or processing, is the dynamic manipulation of data, which requires that data changes. Theoretically, data could change in space, rather than in time, but I confess that I can’t imagine how that would work. In any case, that wouldn’t affect your argument.
Numbers pop into existence as soon as we define classes, but I can’t get away from the problem that defining classes is not a real activity, but simply a matter of manipulating symbols and making definitions. Therefore, it seems to me that the integers are simply symbols that we invented, and that arithmetical operators are rules that we think might be useful.
Here, we seem to be zeroing in on the core problem – that classification itself is a tool of the way we think, and one that we find helpful, and has no further significance. Very young babies seem not to do this, although they learn to very quickly. I remember watching my kids as toddlers repeatedly trying to fit large cups inside smaller ones – it became obvious through these observations that ideas of magnitude are empirical. Mathematics, like logic, seems to be a useful tool for modelling the physical universe and not something fundamental.
Adult humans are exceptionally good at manipulating abstract symbols, and we should be concerned about bias in our way of thinking. Other paradigms may be possible.
The fact that mathematics breaks and ties itself in paradox – as per Godel and Turing – isn’t a reassuring fact for those who think the universe is built from math.
My view is that it is highly likely that intelligent aliens would understand mathematics in much the same way that we do – especially if they manage to pull off engineering feats such as space travel – but I think it is something that we will have to verify empirically if we ever meet any aliens. I wouldn’t be surprised if they had different mathematics, although I can’t conceive what it might look like.
October 14th, 2015 at 1:17 pm
Thanks!
“Is it possible to believe that mathematics is invented, but also a powerful way of thinking about the universe? I think so.”
We use metaphor as a tool for understanding, so perhaps you’re saying math is a metaphor we’ve invented.
“As I see it, thought, or consciousness, or processing, is the dynamic manipulation of data,..”
But then why do we experience things? Processing data implies no requirement for experience. That’s the vexing point about consciousness. (But let’s not go down that road here.)
It’s a good point about change. Even “just” changes in space wouldn’t be just that — they’d also be changes in time. The passage of time is fundamental to the concept of “change”! (I’m inclined to agree with you that time is a fundamental property of reality, just like space.)
“I can’t get away from the problem that defining classes is not a real activity, but simply a matter of manipulating symbols and making definitions.”
Or is it a matter of recognizing that some objects share properties? In a handful of small rocks, each rock has a unique shape and color, but they all share a basic shape, a basic size, and a basic substance. Is it really “made up” that we think those rocks all belong in a class of objects?
“Therefore, it seems to me that the integers are simply symbols that we invented, and that arithmetical operators are rules that we think might be useful.”
Do you deny the world of objects? If we ignore property classes, we still have a world of discrete objects.
If I have an object in my right hand and another object in my left hand, and I put both objects on a table, how do I describe that? Did I not just add one and one to get two? If I remove one, have I not subtracted one from two?
We do make up a language to describe this, but as far as I can tell we’re describing a real thing. Objects have real distances between them, for instance, and those distances have relationships. Describing those real things is mathematics. Addition and subtraction are real processes we carry out every day.
To me it’s like saying cats, dogs, and elephants, aren’t real because we made up the words to describe them and how they behave. Of course we did. But we needed that language to describe what we were seeing.
“Here, we seem to be zeroing in on the core problem – that classification itself is a tool of the way we think, and one that we find helpful, and has no further significance.”
I’m struck by the “no further significance” part of that. I agree classification is a tool, but it again seems more a recognition that objects can share properties rather than something imposed.
Try it this way: We see no real signs of justice in physics or the animal kingdom. Justice is indeed a concept we’ve invented to make society work better. (Some animals do seem to have a sense of fairness, and it’s possible justice springs from this impulse.) We don’t look around and see it in the fabric of reality.
But math we do merely in virtue of existing in a world of discrete objects and relationships.
“Very young babies… [demonstrate] that ideas of magnitude are empirical.”
Indeed. Built into the fabric of reality as it were. Your babies are discovering the world of objects and the relationships between those objects (not unlike how babies discover pretty much everything about the world).
“Mathematics, like logic, seems to be a useful tool for modelling the physical universe and not something fundamental.”
I wonder if we’re running aground on a distinction over what fundamental really means here… A world of objects seems (to me, anyway) to come with the idea of natural numbers and (at least) addition and subtraction. To me that’s extremely fundamental.
We do invent a language to describe these objects and their relationships, but as with cats, dogs, and elephants, I don’t see how we’re not describing something real. I think it’s compelling that mathematicians often make up mathematical structures “just to see where the math takes them,” but those turn out to model some real-world phenomenon after all.
Among mathematicians, there is what is known as the “unreasonable effectiveness of math.” (See this famous paper.) Yeah, it does seem made up sometimes, especially some of the weirder parts, but time and again math turns out to fit some aspect of the real-world.
That’s really, really eerie for a metaphor we just made up!
“Adult humans are exceptionally good at manipulating abstract symbols, and we should be concerned about bias in our way of thinking.”
Perhaps. A nice thing about math is how pure it is intellectually.
OTOH, consider what bias being 3D beings might cause. If beings exist in other dimensionalities (assuming that’s even possible), they might have very different biases.
I read a short story once about an alien life form that existed as large “sheets” in the ocean. They lacked obvious senses and weren’t spotted as intelligent immediately. Basically, they turned out to be a kind of “cellular automatons” — a version of Conway’s game of Life brought to life. Lacking senses they were beings of pure (as it turned out) mathematical thought.
But, of course, that’s science fiction. 😀
“The fact that mathematics breaks and ties itself in paradox – as per Godel and Turing” – isn’t a reassuring fact for those who think the universe is built from math.
(As an aside, I’m working on a post about Turing’s Halting Problem — I almost posted it last night.)
This reply is getting long for getting into Gödel and Turing in detail (maybe next time), but it could also be viewed as a comfort. If the universe really was based on math (FTR: I’m not in Tegmark’s camp), this implies it must be fully deterministic (there’s no Heisenberg in math).
What Gödel showed (and it is a subtle point) is that, if math is consistent (which it seems to be), then there are statements within that system that cannot be verified as true or false. You might call that “breaking” or “paradoxical” but I call that “wonderful.”
It means math is so rich and complex that it can surprise us. So if reality were math based, I’d be happy math has that kind of richness and complexity.
(Keep in mind Gödel only demonstrated that a given system of math cannot prove all true statements within that system of axioms. A “higher” system can prove a lower system is both complete and consistent (but then the higher system cannot prove itself to be both). His second, stronger, theorem amounts to saying that any system that can prove completeness cannot be consistent.)
“My view is that it is highly likely that intelligent aliens would understand mathematics in much the same way that we do – especially if they manage to pull off engineering feats such as space travel…”
I agree, obviously, and that’s a good point about engineering (and I think it supports what I’m saying). There are physical realities, and any sciences designed to deal with those is going to end up being similar. Many languages describe elephants, but they all describe the same object (big floppy ears, trunk, etc.).
“I wouldn’t be surprised if they had different mathematics, although I can’t conceive what it might look like.”
I’m also working on a companion post to this called Weird Math. 😀
It’s an attempt to describe some weird ways of doing math. Of course, I’m limited in being human and living in 3D space, so it’s really more of a hand-wave. I’m not sure that post will see the light of day.
October 14th, 2015 at 1:50 pm
Thanks for the detailed response. I am of course, not disagreeing with you, just raising points that I find interesting. To put the ball back in your court, a few points about your points:
“In a handful of small rocks, each rock has a unique shape and color, but they all share a basic shape, a basic size, and a basic substance”
Actually, I think that what we are really doing when we invent classes is that we are creating a simplified model of reality. The details of the different rocks are so many, and so hard for us to remember, that instead of carrying round a model of each particular rock, we dump 99% of the detail and say, “it’s a rock.” A key component of any model is that it drops most of the information about objects and systems, retaining only the most essential elements. In a way, our intelligence depends on the ability to ignore everything we think doesn’t matter. The smarter we are, the less we know 🙂 Could there be other types of intelligence that don’t work in this way? I wouldn’t rule it out.
“Do you deny the world of objects?”
I don’t deny physical reality, but an object is a concept that we invented. For example, to say that a rock is an object is an arbitrary division between the rock and everything else adjacent to it. It’s a simplified model of a complex reality that’s too hard for us to think about directly. So, rocks are real, but their identity as rocks is invented by us.
“If I have an object in my right hand and another object in my left hand, and I put both objects on a table, how do I describe that?”
Anyway I choose, or not at all. It doesn’t affect what’s on the table (which itself is an invented concept – a metaphor, if you like.) But really, I think that what we are doing here is mapping our world of abstract symbols onto the real world, and doing a lot of simplification and modelling as we go. First, we abstract the idea of the table, and the notion of a surface (ignoring the details of this specific table, which is unlike any other table we have encountered, and doesn’t even have a flat or level surface.) Then we examine what’s on the table and simplify what we see by grouping the physical stuff there into discrete objects. Then we classify the objects on the table as rocks. Then we map our abstract model of numbers onto our conceptual model of rocks, and now we can apply the rules we invented to manipulate these symbols, so that 1 + 1 = 2, and we conclude that there are 2 rocks. This is an intellectual exercise, surely?
“’m also working on a companion post to this called Weird Math.”
I’m hoping you will call it Wyrd Math!
October 14th, 2015 at 3:38 pm
“Thanks for the detailed response. I am of course, not disagreeing with you,…”
Well, now don’t be coy, I think you are (and that’s a-okay). You asserted before (and again here) that math was a useful tool and not fundamental. That is disagreeing with my premise (and belief). But that’s fine; ideas should be challenged and examined. That’s always been something I hoped would happen here and I quite value my small group of participants — none of you cut me a break, and that’s exactly as it should be!
And ultimately it’s a philosophical point anyway, one that’s been long-debated, and we’re not going to solve it here. (But the disagreement is key to bringing out the points! I realized I often can’t fit everything into a post — my max limit is about 2000 words with 1200-1500 for preference — so I depend on the comment section for finer-grained discussion.)
“I think that what we are really doing when we invent classes is that we are creating a simplified model of reality.”
Sure, that’s a big part of it. We recognize shared properties that allow us to say that this one is like that one, and that is a reduction or abstraction. And it strikes me this may be the core of where our paths diverge.
You see the abstraction as a fabrication that moves away from a messy reality, whereas I’m seeing it as recognition that moves towards fundamental truths about that reality.
The vexing thing about the fabrication idea — which is entirely reasonable and could turn out to be right — is the astonishing effectiveness of that fabrication. OTOH, if math reflects deeper existential truths, then that effectiveness isn’t surprising.
“Could there be other types of intelligence that don’t work [by abstracting, filtering and reducing information]?”
Can’t be ruled out. My dog had a sense (an imprecise, unreliable one) of the difference between the class “ball” and the class “toy”. I’ve often wondered to what extent male dogs differentiate between trees and phone poles — something we were taught in early grade school. Does the location and scents of each tall thin vertical thing make each one a distinct (unclassed) object for dogs? I’ve gotten the impression it does, but I’m not an impartial observer.
As you pointed out, babies learn to classify things, we’re taught that all through our education. Our abstraction ability seems part of what sets us above the animals. And at least sometimes abstraction does reveal underlying truths. Classifying all falling objects leads to the underlying truth about gravity, for example.
Computer programs are all based on abstractions upon abstractions (upon abstractions upon…). They’re key to at least this kind of information processing. If the world consists of discrete unrelated objects, there isn’t much to be said about groups of them.
It seems like abstraction is basic to intelligence, but that could be a failure of imagination.
“[A]n object is a concept that we invented.”
Sure not! There is nothing “arbitrary” about the division between the rock and what surrounds it. There is the simple fact it can be removed from whatever surrounds it that absolutely distinguishes it. It is clearly a discrete object in virtue of how it can be put into many surroundings.
“So, rocks are real, but their identity as rocks is invented by us.”
Do you mean the terminology we use to label them? If so, yes, agreed. But there is nothing arbitrary or invented about recognizing the sovereignty (so to speak) of the object itself. Nor is there anything arbitrary about recognizing rocks share properties not shared in other objects.
“Anyway I choose, or not at all.”
LOL! Well, you’re certainly free to walk away and chose not to describe it. We’re assuming you do want to describe it. 😀
But I’m not sure the example was clear, since I never mentioned I was holding rocks (I wasn’t) and the table has nothing to do with it (I could put them in my pocket or on the ground).
What I was trying to say is that, if there’s an object over there and another object over here, then how do we describe (assuming we do want to) the situation of putting them next to each other?
As you’re suggesting, one way is to continue to see them as an object and another object, to have no sense of grouping them, even just as distinct objects. But I don’t see how you develop any kind of technology with a view like that. As Mike pointed out, all of science, and thus technology, is built on math.
And math starts with recognizing an object and another object is two objects.
“Then we map our abstract model of numbers onto our conceptual model of rocks, and now we can apply the rules we invented to manipulate these symbols, so that 1 + 1 = 2, and we conclude that there are 2 rocks.”
I was actually holding a couple of elephant treats, but no matter. 🙂
Here’s the question: where did those conceptual models come from (if not from the initial recognition that a thing and another thing is two things)?
“I’m hoping you will call it Wyrd Math!”
HA! Then I’d have to make up a bunch of wyrd problems, and no one likes wyrd problems! XD
October 15th, 2015 at 3:39 am
“Our abstraction ability seems part of what sets us above the animals”
It seems to be the key to intelligence, in fact, or at least our intelligence. But the very notion of abstraction indicates a separation between the physical world we inhabit and the mental models we use to describe it.
“But there is nothing arbitrary or invented about recognizing the sovereignty (so to speak) of the object itself”
Really? How many objects are needed to make a human being? How many objects are needed to make a hydrogen atom? Are they even objects?
“Here’s the question: where did those conceptual models come from (if not from the initial recognition that a thing and another thing is two things)?”
Exactly. Mathematics is an empirical science based on how we observe objects to behave. I agree with Mike when he says, “math is an invented system built to represent the most fundamental relationships in the universe.”
If you put two rocks on the table and they coalesced, then we would have invented different mathematics to 1 + 1 = 2. Of course, we could imagine a world where objects behaved differently, and mathematicians do this. Could an alien species imagine a different kind of mathematics?
October 15th, 2015 at 2:02 pm
“But the very notion of abstraction indicates a separation between the physical world we inhabit and the mental models we use to describe it.”
Agreed, but isn’t everything we perceive an abstraction — a mental model — generated by our sensory input and how our minds function? Mental models separated from the physical world is all we have!
As I mentioned before, I think a fundamental split here seems to be how we view abstraction. You (if I’m following correctly) see it as separation and fabrication; I see abstraction as (potentially) fundamental truth about real relationships.
FWIW, I also think there is something more than abstraction going on. There is observation and recognition of regularities, and it is those regularities that lead to abstraction.
Perhaps a question for you is: What do those regularities mean?
“How many objects are needed to make a human being?”
Two! A mommy object and a daddy object! (Ya walked right into that one! XD )
Seriously, though, there’s no law saying objects can’t be made of other objects. Large rocks, if you hit them with a hammer, turn out to be made of smaller rocks. Humans, if you cut them open, turn out to be made of smaller objects.
“Mathematics is an empirical science based on how we observe objects to behave.”
Yet a big point of my post was showing how it can arise a priori. At the same time, even so, math does spring from analysis (at a minimum) and I’ve argued all along that recognition of objects is key to math theory.
So I can agree that math is (or can be) empirical, but I don’t see how that makes it not real. It almost seems like it would make it more real given that we consider empirical evidence a really good kind of evidence.
“I agree with Mike when he says, ‘math is an invented system built to represent the most fundamental relationships in the universe.'”
So do I! 🙂
But those fundamental relationships we all agree exist. What — if anything — governs their behavior? Are there physical laws built into the universe that do that?
“If you put two rocks on the table and they coalesced, then we would have invented different mathematics to 1 + 1 = 2.”
I’m not sure that follows. We’d have different physics, that’s for sure, but if we have, as you say, two rocks then we’ve already invented the natural numbers.
As you point out, mathematicians already play with alternate forms of mathematics (check out rings and group theory in math, for instance, or hyperbolic geometry).
So even a really weird species with a weird environment — I believe — has to stumble on mathematics we’d recognize, although it’s possible they might consider it an “alternate” (or even “weird”) form. That said, I also believe the foundations of math follow so directly from a world of objects that I’d be astonished any intelligent species wouldn’t develop them.
Especially, as you pointed out, if they get into engineering.
October 14th, 2015 at 1:25 pm
My quote for the day: “If math is a metaphor, it’s a metaphor with teeth!” XD
October 14th, 2015 at 11:38 am
I think math is an invented system built to represent the most fundamental relationships in the universe. So, I would definitely come down on it being broadly universal. It’s our most fundamental theory, heavily verified, about how reality works. All other scientific theories are built on top of it.
However, because it is a theory, our version could have species level biases in it. What would those biases be? Being in the effected species, and a member not particularly proficient at mathematics, I don’t have a clue. Steve’s mentions of paradoxes strikes me as possibly pointing to potential areas.
October 14th, 2015 at 2:22 pm
“I think math is an invented system built to represent the most fundamental relationships in the universe.”
I think we’re on the same page here? The relationships of objects are real and fundamental to reality, and math is a system we invent to describe those.
Like I said to Steve, different languages have different terminologies for elephants, but they’re all describing something real. (The story of the three blind men and the elephant might be apropos here. A given math system might only describe what it has encountered so far.)
And maybe it’s important to make clear that when I talk about the reality of math, I’m talking about the fundamental properties of existence (the elephants) and not the language we use. Any culture with elephants will invent a language to describe them, and those will differ.
“All other scientific theories are built on top of [math].”
Which says something, doesn’t it? Math is a language used by pretty much every other science. (Is there any form of science that doesn’t use it?) And it has a purity that doesn’t admit to much in the way of bias in application.
“However, because it is a theory, our version could have species level biases in it.”
It’s possible, sure. I think the purity and “unreasonable effectiveness” are compelling arguments towards the underlying truth of math, of the reality of those objects and relationships.
Going back to elephants, a very small species might paint elephants as giants, but an elephant-sized species would paint them as normal (and asteroid-sized space whales would wonder what those teeny tiny organic bits on those large rocks were).
But both species (and the space whales if they looked closely enough) would agree about the big floppy ears, the trunk, the four legs, and so forth. Elephants are real (as real as anything, anyway), so any language describing them has to converge on certain properties.
The approach and tone will certainly vary! Our language, obviously, is filled with bias.
But consider the mathematical properties of elephants: length, width, height, mass, frequency of reflected light. There are myriad smaller measurements, the many characteristics of various parts. About the only thing different species can differ on here is the units.
If a species experiences time and space anything like we do, then the only difference in the basic perceptions of those things is the units. How far apart are the markings on your ruler, and what do you call them.
And the thing is, given c (a universal constant), time and distance become defined in terms of each other. Distance is how far light travels in a given time. Time is how long it takes light to travel a given distance. We consider time more fundamental, so we define it based on vibrations of a cesium atom. Length is then defined by how far light travels in a given time.
Part of the science of an elephant is the measurement of its properties, and if these are ultimately based on universal standards, then there isn’t much room for species bias to stand. (The whole point being, since math is the purest of the sciences, it is the least prone to bias. Not immune, perhaps, but pretty well-protected.)
“Steve’s mentions of paradoxes strikes me as possibly pointing to potential areas.”
I’m not sure “paradox” is quite the right word (let alone “broken”)…
There is a point of view that the universe should be fully understandable. But physics shows that view is false. Heisenberg prevents us from fully understanding the physical world on a micro level. Quantum physics adds other weirdness that blinds us. That Gödel showed math is equally weird — just like reality — seems almost an expected result to me.
I see a striking connection between Heisenberg, Turing, Gödel, and Cantor. In particular how the last three all used the diagonal method in proving roughly the same thing in three different domains: You can’t know everything!
I think that’s fascinating! And heart-warming. It means that the most fundamental aspects of reality are rich and complex and surprising. That’s cool! 😀
October 14th, 2015 at 5:35 pm
I do think we’re on the same page. Sorry, didn’t mean to imply I was disagreeing. (I know from past discussions that we do have disagreements in this area, but I don’t think it enters into this post.)
The universe has definitely shown us that it has no obligation to meet our expectations, or to necessarily be understandable, although it has proven to be far more understandable than a 15th century monk might have expected.
The point I got from Steve’s comment is that mathematics describes our universe, but it occasionally runs into limitations (or whatever word you want to use). Are those limitations in mathematics only? Or does that limitation reflect a deeper reality? (It does in quantum mechanics, but not sure about the rest.)
October 15th, 2015 at 1:15 pm
“I don’t think it enters into this post.”
I think you’re right (so long as we focus on the inevitability and universality of math rather than its ontology, ’cause I think that’s where we probably see it differently).
“The universe […] has proven to be far more understandable than a 15th century monk might have expected.”
And paradoxically also less so. It’s much larger and older than any monk could have dreamed, plus there’s the whole quantum weirdness thing. It certainly didn’t turn out to have the clockwork simplicity that came from Newton and the Scientific Revolution. The view that reality can be fully understood starts in that era.
“Are those limitations in mathematics only? Or does that limitation reflect a deeper reality?”
Right. And my point is I do think those limitations (a fine word) reflect similar, deeper, physical realities exactly as math reflects other deeper realities.
October 15th, 2015 at 3:40 am
The thing about trying to deduce what aliens might think is this. I am constantly astonished by what other humans think, so I would hesitate to guess anything about aliens, no matter how logical the guesses seem to be.
October 15th, 2015 at 2:13 pm
True, our speculation is just that! (Still, it’s fun and good mental exercise.)
October 15th, 2015 at 4:03 am
More thoughts. The unreasonable effectiveness of math. The power of math lies in its abstraction. It not only describes our universe, it describes any possible universe. This is not a Tegmark statement – I’m not saying the universe is math.
Example. Thermodynamics. The point here is that it doesn’t matter what the properties of the billions of particles of a gas actually are. Statistics tells us how the gas will behave at a macro scale, and why we even have a concept of “gas”. Similarly, we understand the conditions for which normal distributions will arise (and other kinds of distributions.) Mathematics helps us understand and predict systems that are so complex we can’t describe them in detail. It tells us things that we ought not to know.
So mathematics tells us that a lot of things about our world are inevitable (because of the law of large numbers.) I sometimes have a thought that we will dig deeper and deeper into the fabric of the universe, and reach a point where, instead of everything becoming simple, it will become extraordinarily complex, and we will be left with chaos at the fundamental level. But that will be OK, because mathematics explains how the absence of deterministic laws at a sub-microscopic scale could still give rise to deterministic laws at a larger scale.
This is of course an argument for the universality of mathematics 🙂
October 15th, 2015 at 2:30 pm
“The power of math lies in its abstraction.”
To the rocket-builder, the power of math lies in its ability to predict what her design will do. To the electrical engineer, its power lies in its ability to predict what his design will do.
Maybe we’re saying the same thing in different ways. To me those abstractions, because they reflect fundamental realities, give us the power to predict the future. Successfully.
“It not only describes our universe, it describes any possible universe.”
(Which is a big problem in string theory right now.) Isn’t that an argument for math being deeply fundamental? That it can describe other realities?
“Mathematics helps us understand and predict systems that are so complex we can’t describe them in detail.”
Indeed. And gets it right. Why do you think that is? Is the behavior behind thermodynamics real behavior successfully described by the statistical laws of thermodynamics?
“I sometimes have a thought that we will dig deeper and deeper into the fabric of the universe, and reach a point where, instead of everything becoming simple, it will become extraordinarily complex, and we will be left with chaos at the fundamental level.”
Chaos at a fundamental level would be comforting. I don’t want to live in a clockwork universe! And, if we’re talking “quantum foam” then there is a great deal of chaos going on down at the Planck level.
We know that, between GR and QFT, one — if not both — must be incomplete. There definitely is some big discovery other there waiting for us. I think it would be awesome if it turned out to be another duality like the wave-particle thing. Maybe the universe does have two faces… a macro face and a micro face, and wave-particle-like they are mutually exclusive.
That disconnect between GR and QFT might also be evidence this is a virtual reality — one that uses one “program” for macro level stuff and another for micro level stuff. That matter-energy is quantized is also evidence supporting the VR view. 😀
(Personally, I want Einstein to have gotten it right. Matter-energy quantized, sure, they obviously are. But maybe, just maybe, space and time are not.)
“This is of course an argument for the universality of mathematics :)”
Why, it is, isn’t it! XD
October 16th, 2015 at 5:36 pm
Well you know how I am with actually doing math, so perhaps I’m not the one to have an opinion. Also, disclosure…I didn’t read the comment thread, although I might come back to it later.
I wonder if more can be gleaned from the quote: “God made the integers, all else is the work of man.”
Does it make sense to say that perhaps some of the more advanced math, or certain kinds, could be a sort of “manipulation of data,” but the more basic stuff is not? Or is this an inconsistent position?
The true mathematicians will probably bawl me out, but suppose we say there’s two different kinds of math—math that’s real and math that’s only functional? (The latter being the paradoxical stuff I would never be able to wrap my mind around…starting with imaginary numbers.)
BTW, I love this post. Great thought experiment. Reminds me of reading Kant, actually!
October 16th, 2015 at 10:33 pm
Thank you! Some of it is definitely informed by Kant. I didn’t take to Kant right away, but the more I dig into him, the more I go, whoa!
“I didn’t read the comment thread, although I might come back to it later.”
I think it’s worth reading (although — as usual — I got a bit long-winded).
“I wonder if more can be gleaned from the quote: ‘God made the integers, all else is the work of man.'”
The problem is something like π — which is just a ratio between parts of a circle. Or e, or many other values that show up in natural things. And these aren’t just real numbers, they’re transcendental irrational numbers! They’re pretty far away from the integers!
OTOH, that we live in a quantized reality makes one wonder if the real number realm could be entirely a unicorn. No perfect circles exist in the physical world, and one way to read that is that the weird numbers one gets in the ideal mathematical world are an idealized dream. Their weirdness is because they’re made up.
“Or is this an inconsistent position?”
To the extent most advanced math is built on the foundation of “natural” math, yes, probably. But something like, say, hyperbolic geometry doesn’t really have any counterpart in the physical world. But it does seem like some math is horses and other math is unicorns.
The confounding thing is how often the unicorns turn out to be horses wearing fake horns. Time and again, some mathematical unicorn suddenly turns out to model some real-world phenomenon and becomes a horse.
This seems to argue against even this realm of math being pure imagination. The success record is pretty astonishing for something just made up.
October 18th, 2015 at 9:53 am
Actually, I DO need to come back to the comment thread. I have this debate in my novel. My “Thrasymachus” character is on the side of “math is an invention, a game” (which seems to go along with “justice is the interest of the stronger”), and my math student character is unsure. Since they’re both students, not experts, I have some wiggle room. And they aren’t likely to be as knowledgeable about the debate as you and certain other folks I’ve been reading. Still, I’ve been remiss in not bookmarking all the talks on blogs, which are a wonderful resource. I’m obviously on the side that math is discovered, but I lack a lot of knowledge in the specifics of this area myself. So the point is, even if my characters don’t need to be all that knowledgeable, I probably should be slightly ahead of their game.
Plus, you guys speak informally and naturally, but articulately. It’s helpful to read these exchanges, although I’d have to make it more college student level. Hope you don’t mind…I promise to include acknowledgements to you all if I ever publish!
Good point about pi. (Shit, I almost wrote “pie” thanks to the Penzey’s catalog from “pi day”.) Don’t know how I could’ve missed that one, except that I hardly think of pi when I think of circles (the standard example in these discussions, of course).
Speaking of natural discussion, I love your metaphor of horses and unicorns. What a great way to cut through the verbiage!
On my inconsistency, yeah, I felt that was the case, but I also can’t help but think inconsistently on the matter. Certain examples make me go, “Yeah, of course math is a horse.” Other examples conjure up unicorns. Which makes me want to say, “Why can’t we have both?” But you’re right, the so-called unicorns are built on a so-called horse foundation, which makes the position problematic.
On math’s success, that’s a rough road to go down, but it makes some sense, at least in intuition. Those folks who think it’s invented will have plenty of arguments against using math’s efficacy as proof for its reality, but I think it’s astonishing too.
October 18th, 2015 at 12:46 pm
“Since they’re both students, not experts, I have some wiggle room.”
Ah, that is helpful. Plus it gives you the excuse for exposition.
“I probably should be slightly ahead of [my characters’] game.”
I suppose you have a choice of your characters just being at whatever level you are, but often with performance (which writing sort of is) it takes grace to pretend to be clumsy and smarts to pretend to be dumb. In that mode you have more control over your characters, I’d think, be able to steer them better.
“Plus, you guys speak informally and naturally, but articulately.”
I really do cherish the discussions I’ve gotten on this blog! (It may be just as well the group is small. I probably couldn’t keep up with a really active blog of folks like you guys.)
“It’s helpful to read these exchanges, although I’d have to make it more college student level.”
Should we start saying “far out” and “groovy” more often? (That is what college say, isn’t it, or am I out of touch?)
“Hope you don’t mind…I promise to include acknowledgements to you all if I ever publish!”
Aw, shucks. [blush][stammer] Ah don’t need no credit! (I want percentage points. Of the gross. I’ll have my people call your people.)
“Good point about pi.”
It’s one of the superstars in math. Couldn’t be more simple in conception, but what a number it turns out to be. (They Freshly Pressed my post about just how magical π is!)
“I love your metaphor of horses and unicorns.”
Thanks! It did turn out to be a useful way to look at it. Steve’s most recent comment is based on it!
“But you’re right, the so-called unicorns are built on a so-called horse foundation…”
No matter how fantastic, there’s a horse in there somewhere! And it’s funny how, sometimes many years later, someone discovers that an obvious unicorn is a horse after all. There is even a deliberate unicorn — math the author bragged would never have real purpose — that turned out to be hugely useful in cryptography.
“On math’s success, that’s a rough road to go down,…”
You might like the first two videos in those I posted today. There is an analogy that, without viewing the Bible as accurate or even true, you can still view it as a useful set of (moral) tools. Those on the fictionalism side of the math spectrum view math like that; it’s not true, but it’s useful.
Of course, on the opposite end are those who view math as fundamentally true. 😀
In a way, it almost does go back to Kronecker’s God-invented integers, although the divide might not be between integers and reals. Per the fourth of today’s posted videos, perhaps the real divide is between the transcendentals and the others.
Which would exclude π and e and other such numbers.
And yet,… the circle. If an idealized circle is real (and perhaps it isn’t — perhaps perfect circles are transcendental and therefore true unicorns) then π is real.
In a sense, perhaps it can be asked this way: Do you believe in perfect circles?
October 19th, 2015 at 7:49 pm
I believe in perfect circles!
I did see your post on pi…awesome that you got such attention! (I mean, groovy, like far out man.)
Well, I’m pretty pleased with myself for getting the Pythagorean theorem into a drug deal chapter. It took an insane amount of googling, but it works. And I didn’t have to do too much math to figure it out, but it seemed clever enough given the context.
“Those on the fictionalism side of the math spectrum view math like that; it’s not true, but it’s useful. Of course, on the opposite end are those who view math as fundamentally true.”
Yeah, exactly. It comes to an impasse usually.
As for percentage of profit from my book, sure. I’ll be lucky if I earn enough to buy a cup of coffee, which leaves you with maybe enough to ride the little horse in front of the grocery store. If you save up, you might be able to get something out of the temporary tattoo dispenser…maybe you’ll score a glittery unicorn. 🙂
October 19th, 2015 at 11:44 pm
Good one! XD
Maybe you can work Euler’s Identity into your story! It both the mysterious superstars of math: π and e. Plus it’s got that real trouble-maker, i. It really is one of the most singularly cool things in math.
October 22nd, 2015 at 6:24 pm
I have no idea what Euler’s Identity is. Maybe you can give me a Euler for Dummies version?
October 22nd, 2015 at 6:53 pm
Ah, you just haven’t gotten to that post, yet! 😀
Beautiful Math
October 22nd, 2015 at 8:17 pm
I think you’ll like some of the videos in the post after that. Numberphile is one of my favorite YouTube channels! You might find a lot of good material there for your book.
Moar Math!
October 23rd, 2015 at 4:51 pm
Well, I tried! But hey, it’s good to know. It might make it into my novel anyhow…even if I don’t get it.
October 23rd, 2015 at 5:44 pm
It was once said to be (one of) the most famous equation(s) in math. It’d be interesting to see if that plays out among average math students. Maybe it’s more for the serious student of mathematics.
But: Stunning! Amazing! Beautiful! (etc.)
October 17th, 2015 at 8:59 am
Wyrd, I have reached my maximum indent level for comments, and am now just confused. I think we’ve basically said what we’ve got to say anyway, and are in danger of entering an infinite loop. 🙂
October 17th, 2015 at 11:38 am
Okay, that’s fine. If you like, I think I can boil it down to two questions:
1. Is abstraction a “unicorn” or a “horse” — is it more of an arbitrary made up fantasy or a deeper truth about nature’s universal behavior?
2. Does the universe work according to isotropic laws (regularities we can observe), and — if so — what describes those?
October 17th, 2015 at 11:57 am
1. My vote is for unicorn-horse. It’s not arbitrary, as it’s clearly inspired by empirical experience, but is certainly a made-up fantasy. At the same time, it expresses a deep truth about fundamental reality, if applied in the right way.
2. It appears to. Mathematics appears to describe those laws, although there are plenty of things in the universe that it doesn’t describe, such as why a particular song makes me cry. Maybe it does, but the math is unbelievably complicated.
October 17th, 2015 at 12:12 pm
1. Okay. I suffer from a failure of imagination here. I don’t grasp how something can both express a deep truth about fundamental reality and be a made-up fantasy. Those seem exclusive to me.
2. I believe that, unless you subscribe to some form of dualism, you almost have to believe math does describe why you cry. (It probably is complex math, but there is, in fact, a single equation that is said to embody everything science understands about reality. I’ll be posting about it soon, but in the meantime here’s a blog post someone did about it.)
October 18th, 2015 at 12:26 pm
“Okay. I suffer from a failure of imagination here. I don’t grasp how something can both express a deep truth about fundamental reality and be a made-up fantasy. Those seem exclusive to me.”
I’m a simple-minded guy. I can’t understand complex arguments, so I have to take things step by step. Here is my step-by-step simple-minded answer 🙂
The universe is full of patterns and regularities. Consequently, humans have evolved to be excellent at pattern recognition. We are so good at this, we even recognise patterns that don’t exist (insert link to mystical/theistic/alien conspiracy nonsense). Abstraction is one form of this – as you say, grouping objects requires that we can conceptualise objects and identify common features. Out of this pops mathematics (reference your article above, which describes the mechanics in detail.) By the way, you’re almost certainly correct about the universality of mathematics. For me, math is empirically-based. Hence, horse.
So far, so good. We have spotted some patterns, and have developed techniques of thinking abstractly. To me, all thus abstract stuff is clearly invented. None of it is real. It exist in our minds, and in our textbooks. It doesn’t even refer to anything real (although it was originally inspired by our thoughts about real things.) Hence, unicorn.
But hey, look what we did. By building an abstract language we have achieved universality. Not only can we build mental models of our universe, but of any universe. We can create things that never will exist. Double unicorn!
But hey, look what we can do when we apply math to our own universe. Not surprisingly, since the whole edifice was built from counting rocks, we can use it to say things about rocks. We can even surprise ourselves by calculating things about rocks that we didn’t know before. Double horse!
So unicorn-horse is my bet. Like the unicorn, we started with something real, then dived into fantasy. Unlike the unicorn, we can actually ride math! Hence math is so much better than unicorns.
“but there is, in fact, a single equation that is said to embody everything science understands about reality”
I have no idea what that guy is going on about. I look forward to your blog post to clarify this 🙂
October 18th, 2015 at 1:01 pm
“Unlike the unicorn, we can actually ride math!”
Doesn’t that make it a horse?
October 18th, 2015 at 1:27 pm
Some thoughts about patterns, pattern recognition, and abstraction.
“The universe is full of patterns and regularities.”
Yes. I think a basic question here is how we characterize statements about those.
“Abstraction is one form of this…”
I would say abstraction was one result of this; pattern recognition is distinct from, and comes before, abstraction. (I suspect in this context “pattern” and “regularity” are basically the same thing?)
“To me, all thus abstract stuff is clearly invented.”
So why are observations about true things not themselves true things?
If we observe that all objects dropped in Earth’s gravity well fall at the same rate no matter who does the dropping or where its done (assuming consistent experimental circumstances) have we made up a fact or determined a truth about falling objects?
(I continue to think this depends on the “reality value” we assign to abstraction. You feel it is less real than physical reality; I feel it can be more real.)
“I have no idea what that guy is going on about.”
It’s an effort by the Perimeter Institute to present the laws of physics in a simple infographic. Deep down in physics, there are only a few kinds of laws, and the underlying principles are fairly simple concepts. Those laws are all represented in the graphic (note Einstein’s contribution of GR, for example).
I’d planned to use it in today’s post, but went with the videos instead. Now I’m not quite sure how I’ll work it in — or if I’ll work it in — to a future post. Sean Carroll has something similar, IIRC, and I’ve been meaning to go research his and this one some more.
October 17th, 2015 at 12:31 pm
(FWIW: This is a completely different discussion, but FTR some form of dualism is an option on my menu, so to speak, so I do believe it’s possible art moves us for reasons that transcend anything math can describe. That we can be so moved is one of the things that makes me wonder about some form of dualism. When atheists say there’s no evidence God exists, I wonder if this is a data point for such. Is our ability to be so moved by love, beauty, or justice, evidence of something more? [shrug] As I said, a completely different discussion! 😀 )
October 17th, 2015 at 11:50 am
p.s. And, FWIW, I think we’ve been discussing the ontology of math more than the universality or inevitability of math. I happen to be totally into the ontology of math, so I’m enjoying the conversation, but I realize it’s more of a philosophical point and — technically — not really the intent of the post.
October 17th, 2015 at 11:59 am
Doesn’t universality arise from ontology though? Oh no, don’t answer, as this sounds like another huge question. 🙂
October 17th, 2015 at 12:24 pm
😀 I can’t answer, as I’m not sure I understand the question! If you mean that something’s actual existence means it exists for everyone, then yes, in that sense they are the same thing.
When I said we were discussing the ontology rather than the universality, I was referring to the nature of math’s existence. Even unicorns have an ontology, but the nature of their reality is different than that of horses. As such, both unicorns and horses are universal for us, albeit in different ways.
(In that sense of universal, I think we completely agree. Whatever math is, if you operate in the physical world — do engineering — you’ll invent ways of working with universal physical law. Those ways almost can’t help but converge on what we’d recognize as math.)