Indulge me in a little thought experiment. Imagine a rocky planet a long way from here, orbiting a star rather similar to our own sun, at just the right distance for water to remain liquid. Let this planet have an atmosphere containing all the elements needed for interesting biochemistry to take place – carbon, hydrogen, oxygen, nitrogen, phosphorus, sulphur and so on. Now imagine that on this planet lives a single lifeform – one composed of trillions of tiny plant-like organisms amassed in one huge layer that covers the entire planet. This lifeform quietly converts the sun’s energy into food, and while it does so, it thinks. What kinds of thoughts might it ponder?

Let me tell you more about this lifeform. It has lived on this planet for a very, very long time. It is blind, as it has no need for sight. It cannot hear, or touch, or taste, as those senses serve no purpose for its survival. It cannot move, as it has nowhere to go.

But it does grow, and it senses the sun’s light on its surface, creating the food that sustains it. This feeling gives it pleasure. And it can think too – not quick thoughts, like our own, but slow, deep thoughts of an alien plant-like nature. It has no brain, but the enormous distributed interconnectedness of its body makes it rather like a brain in its structure.

What kind of mathematics does this being think about? It knows nothing of numbers, because numbers are a tool for counting, and this creature has nothing to count. It does not think about objects, because it knows of none. It has never imagined addition or multiplication or set theory or things like that. It has never conceived of things, and would be quite unable to grasp the concept of things.

It does know about the passing of time, however. It knows about change too, as it senses the sun’s rays slowly moving across its exposed body as day turns to night and back. It feels the cycles of the day, of the seasons, of the years. It knows about surfaces – not the simple, flat triangles and circles of Euclid, but the smooth, undulating, complicated surfaces of its own form. It knows about volumes as well, but not cubes or pyramids, or anything so childishly simplistic and unreal. It knows nothing of lines or points, as these do not exist in its world.

So it understands geometry and form; magnitude and change. It can do mathematics! And so this is what it does. After all, it does not know words or images or communication – things that humans spend most of their time thinking about.

Yet how does it measure the world without numbers? It must conceive of magnitude in some kind of continuous way. Not fractions or decimals, which are a crude way of approximating an analogue world with digital measures. But somehow it must invent an entirely new way of thinking about metrics.

What would this mathematics look like? That is my question for today.

Ha! This is weirdly similar to a post I’ve been making notes for (so don’t feel like I’m copying you when I post it 🙂 ). I, too, have tried to imagine an intelligence that thinks in ways as far as possible from ours. (The underlying question being the potential — or inability — to communicate.)

Wouldn’t the passage of days give it something to count? The concept of a “day” could also lead to the concept of shorter and longer time periods. All of which lead to the natural numbers. The idea of units of time can also lead to the idea of

partsof units of time — fractions, the rational numbers.Thoughts themselves can be recognized as countable objects.

Any perceptions of magnitude (e.g. sun’s heat) lead to scalars and the real numbers. Perceptions of volume or extent lead to the idea of space. Recognition of the difference between a surface and a volume lead to the idea of dimensionality in space.

There is, given any concept of extent and magnitude the potential to invent basic geometry. Euclidean space is based on extremely simple, almost

a prioriconcepts (given the experience of spacial extent). Circles and spheres are extremely basic concepts that seem almost inherent in physical space.The natural numbers depend on set theory which depends on the concept of classes of things. Such concepts can very likely arise

a priorifrom pure thought given minimal experience with the physical world (perhaps even without it — thoughts can be countable, classifiable objects).The natural numbers lead to the basic operations: addition, subtraction, followed by multiplication and division. The rest tend to follow.

Magnitudes lead to real numbers and so does contemplating the ratio between a circle’s diameter and circumference.

Mathematics is so much a product of thought that it’s quite possible beings with limited physical experience, but with high mental ability, might actually be advanced theoretical mathematicians. As you know, there is an ancient philosophical debate on whether math is an obvious invention or a discovery. Given cognition and the most minimal experience of reality, math seems almost inevitable.

(Sorry for the length, but this is a topic I’ve been pondering recently, so there’s a lot “on tap” as is were.)

“I, too, have tried to imagine an intelligence that thinks in ways as far as possible from ours.”Let me introduce you to the mother-in-law Wyrd.

Hariod: I would have hit the like button on your comment, but there is no such option. So here it is: “Like” [click].

Heh! I’m afraid I know this alien creature only from the TV machine. The one I had, but briefly, I got along with fine (although, come to think of it, she was a bit of a dour old bird, but to everyone, including her daughter, so I didn’t take it personally). And both my parents got a long fine with both

theirmothers-in-law, so no soap opera drama there, either.Boring. I know.

I had the same thought, only mine was a lot less articulate…more in the plant-like region of mathematics.

The “plant-like region of mathematics”? I’m not sure what you mean there. ❓

In any event, given we both lean Platonist, it’s not surprising we’d be thinking along the same lines! ❗

Ha! I don’t believe there is a plant-like region…just a plant-like mind—mine. 😉

LOL!

Wyrd, I agree that once you hit on the idea of “things”, our version of mathematics falls into place, seemingly inevitably. My thought experiment was trying to remove the idea of discrete objects as a tool for thought.

You can still understand magnitude without integers and real numbers. Real numbers are, after all, a digital approximation of an analogue quantity. What if we could think in analogue terms?

The problem is that there are objects of thought, so even granting a thinking mind with no qualia, a thinking mind could still recognize objects of thought.

The thought I just had, and the thought I had before that — those are distinct and enumerable. They’re members of a class; thoughts I’ve had in the past. That we think serially leads to the experience of time, and there are also objects of time.

So thoughts alone should lead to mathematics, which isn’t surprising since math is the “pure science” — a vast amount of which (e.g. all of string theory!) — is entirely abstract.

I would say magnitudes

arereal numbers. A magnitude is a quantity. A given object can have variant magnitude, or multiple objects of a given class can each have their own magnitude. That leads directly to comparison operators (‘A’ < ‘B’).So are ratios between things — two sides of a shape, or a circle’s diameter and circumference, for example. Even if you don’t invent a way of “spelling” them, magnitudes — almost by definition — are real numbers.

(FWIW, when I talk about the real numbers, I mean the real real numbers, not their written approximations. The term “real numbers” refers to the quantities (magnitudes) themselves, not their representations. We recognize there are quantities we cannot write precisely in a given base. Consider the (rational) real number represented by 1/3. In base ten it cannot be written precisely as a decimal number (0.333…). But in base three, it can (0.1). This is part of the problem with floating point in computers.)

It is those “real” real numbers I’m trying to grasp hold of here.

From this and what you say to Paula, you are trying to imagine a (single individual) species that is aware of magnitudes, but does not use “our clunky decimal approximations.”

How does it compare the sun’s heat today with yesterday? (If it cannot, then I’m not sure I’d agree it’s a thinking being.)

Perhaps it remembers magnitudes as “the energy at that moment in time” (i.e. it recalls specific time-space events — which raises the issue of labeling those events). Maybe it can compare event ABC with event XYZ.

Maybe its daily experience forms a curve in its mind — a single mental gestalt representing a passage of time versus heat. (How does it label yesterday’s time-heat curve? Or the one from the days before that?)

Regardless, if it’s going to think about these things, it’s going to invent some kind of symbol or labeling system as a handle on magnitudes. It’s also going to invent operators on those magnitudes (e.g. “greater-than” or “difference”).

Pretty much what humans have done, too. Decimal representation has well-known problems, but there are other ways we’ve invented for dealing with magnitudes. Ratios is an obvious one (and for rational numbers a much cleaner one).

Once, in my younger years, after taking some LSD, I stumbled upon a whole new branch of mathematics. It was embedded in simple operations like (2 x 4 x 6). I saw that beneath each of the simplest steps in this operation, a hidden process enabled the valid execution of the step. It is not a form of geometry or algebra or calculus. It is a whole new form with a unique set of rules, a code that makes possible the very concept of numerical relationships. My hidden branch of mathematics was like an infinite spiral staircase full of trap doors opening and closing beneath each step in the surface operation. But, as you might expect, the vision was not recoverable once I came down off the acid. But perhaps if I could get to your planet …

I once wrote something absolutely amazing on the back of a beer mat. In the morning it had turned into illegible scribbles.

I think I remember that. Was it in the Upper Haight back in ’67?

This post suggests that numbers aren’t needed for mathematics? If so, elementary children everywhere will make you their hero. Magnitude doesn’t need a number, does it? What it needs is something to compare it to. A world without numbers would be a world of comparisons, ratios, not of numbers but of the things that our numbers represent. We’ve come to think of numbers as nouns rather than adjectives. Keeping this in mind, they already don’t exist, not even in our world. “5” is no more real than “very.” In the world that you are suggesting you only need two “quantities:” nothing and everything, and your creature could think of its world as a world of comparisons and there would be no long division in its world. I think. 🙂

I’m thinking that my creature would have a more subtle and intuitive way of thinking about magnitude than we do. Instead of our clunky decimal approximations, it would understand real numbers directly. I don’t know how though. Perhaps it could use a direct representation, like a graph.

Thinking about this increases my confidence that the foundations of mathematics are empirical. (I was already feeling pretty confident about it after I had finally understood what tensors were.)

But this also raises the question of what exactly is intelligence? If something can only receive information from the environment, not really manipulate it, is it what we would call intelligent? In fact, can intelligence as we understand it evolve without the ability to manipulate the environment? It doesn’t seem a coincidence that a primate species, among the most dexterous on Earth, is also the most intelligent. (At least according to that species’s definition of intelligence.)

Dolphins? They interact with their environment, but don’t manipulate it in the ways that we do. They probably can’t do mathematics, however, other than perhaps some counting up to a few numbers. Perhaps not even that.

I no longer believe in any kind of Platonism. I’m increasingly confident that mathematics is empirical, and so is logic. What then, is the boundary of mathematics? If mathematics is a collection of abstractions that we derive from our observations of the world, what are its limits? Is everything that we “know” mathematics? Where do we draw a line, and why?

What do you guys mean when you say “empirical”? If you’re denying Platonism, then I’m pretty sure you don’t mean that math is a real thing out there to be discovered by testing (since that would be Platonism).

I take it you mean something along the lines of

ad hocbased on experience? That is to say, “made up.”What I mean by empirical is the second thing you said – ad hoc rules based on experience of the world. In other words, made up.

Example: suppose I am counting sheep. Here comes a sheep. Let me call it one. Here comes another sheep. Now there are two. One plus one equals two.

But suppose the universe didn’t work that way. Try this alternative version of reality. Here’s a sheep. Baa. Here comes another sheep. Put them together and they coalesce into a single sheep identical to all others. Now one plus one equals one. The rules are different.

Things that appear obvious and self-evident to us are examples of conservation laws at work. If these rules change, our thinking changes.

Imagine a third universe in which I place a sheep into a field, and next time I look there are two sheep, then no sheep, then a billion sheep. The rules are different again.

Our arithmetic is clearly inspired by the universe we live in and the way we experience it. Of course, we can imagine and invent different sets of rules, just like I did here. These are not empirical, but they are still “made up”. Mathematics is a set of rules that we invented because it helps us to do stuff.

It’s odd how the sense of empirical and theoretical seem inverted here! Normally it’s theoretical things that are “made up” rather than empirical things.

“Things that appear obvious and self-evident to us are examples of conservation laws at work. If these rules change, our thinking changes.”Certainly! Is science equally made up? We study the physical world, see a pattern, develop a theory for that pattern, test the theory. This applies to conservation laws

andto mathematics. Both are based on observed patterns in the physical world. Both have theories explaining those patterns.“Our arithmetic is clearly inspired by the universe we live in and the way we experience it.”Of course! As is our science (and just about everything else). Why is math different?

You describe two alternate realities. In both cases the same mathematical sensibilities described them aptly. The rules of how objects

behavemight differ, but the underlying math remains the same. It’s math that describes those rules in the first place.I see a difference between the language we invent to

usemath and its underlying concepts. Those concepts are as fundamental as gravity and conservation laws.Morefundamental, since math describes those things. (As you know, the laws of conservation are based on Emmy Noether’s work in mathematical symmetry.)Math is so fundamental that people like Max Tegmark think reality might

bemath. Fanciful idea, but not an incoherent one given quantum physics. If math is made up, one has to explain its “eerie effectiveness.” Time and again supposedlya priorimathematical ideas turn out to apply to some physics experiment. Physicists often discover some weird math theory fits and helps explain observational data. That’s… kinda weird!I do absolutely agree math depends on the nature of the universe! To me that makes it a physically real thing we discover. It seems at least as real as gravity or light.

Wyrd, to reply to your comment above, I am trying to conceive of a being that does not have a notion of objects, or things, or events. After all, these are things that we experience directly, but our alien plant does not. I will try to answer your quastions here.

“How does it compare the sun’s heat today with yesterday?”

What is today and yesterday? Those are arbitrary labels that humans assigned to intervals of continuous time. My plant does not arrange its thinking in this way.

“it recalls specific time-space events”

Those “events” are not real. Life is continuous. We created the idea of those events for our own convenience, primarily so that we can talk about them with other people.

“Maybe its daily experience forms a curve in its mind — a single mental gestalt representing a passage of time versus heat”

That’s what I have in mind.

“Regardless, if it’s going to think about these things, it’s going to invent some kind of symbol or labeling system”

But what if it doesn’t? Can it do mathematics without such symbols?

“there are other ways we’ve invented for dealing with magnitudes. Ratios is an obvious one”

And yet simple magnitudes like pi can’t be represented in our decimal system. Irrational numbers won’t fit in the box we created for them!

By the way, I don’t have any answers to the question I’m asking. I’m just asking if we can think without the idea of objects. Thanks for taking part in the thought experiment! I think that what you say about a “single mental gestalt” is close to what I have in mind. Humans are hopeless at this type of thinking. We have to break everything into bite-sized chunks, but maybe our planet-sized plant can handle such big thoughts.

“I am trying to conceive of a being that does not have a notion of objects, or things, or events.”I understand. I just don’t see how it’s possible. For example, consider the question of comparing the sun’s current heat with the heat felt at some previous time. In general consider the problem of comparing different experiences of any type of magnitude.

“What is today and yesterday? Those are arbitrary labels that humans assigned to intervals of continuous time. My plant does not arrange its thinking in this way.”I’m referring to “today” and “yesterday” conceptually. Your being feels its star’s heat, is aware of the passage of time, and understands change. Therefore, it must have some sense of the passage of daily heat cycles. It may also have a sense of seasonal cycles.

Without saying how it would

referto those things, it must have someexperienceof them. Key questions: Can it compare previous sensations with current ones? How does it refer to the things it’s comparing?“Those “events” are not real. Life is continuous. We created the idea of those events for our own convenience, primarily so that we can talk about them with other people.”And so we can

thinkabout them in order to talk about them. Language enables communication, but it also enables thought.Events are as real as the numbers on the real number line. Life is continuous (at least at the macro level), but any curve has instantaneous values at any given point along the curve.

“But what if it doesn’t [invent a symbol or labeling system]? Can it do mathematics without such symbols?”I don’t see how.

Thoughtisn’t possible without symbols. Our entire experience of reality is symbolic. Just ideas created in our mind due to sensory inputs.What kind of math is possible without relationships between magnitudes? And then how do you refer the quantities in question?

“And yet simple magnitudes like pi can’t be represented in our decimal system.”Right. I quite agree your being would

notinvent positional notation (and certainly not in base ten!). I think we agree it deals in mental gestalts that could be quite complex.It might, for example, think of magnitudes as vectors or volumes (or areas or angles or…). The ancient Greeks figured out π was weird; your being might also. It might realize the necessary gestalt involves a circle and its diameter.

“I’m just asking if we can think without the idea of objects.”It depends on what you mean by objects. Without distinct external physical objects that can be manipulated? Then sure. I think math comes from the bare minimums of thought and experience (along with the enabling intelligence).

But I don’t think thought without

anyobjects is a coherent concept. You can’t think without thinking ofsomething. And thatsomethingis necessarily some kind of object (even if subjective and abstract).I don’t know how to square the circle 🙂 I’m just asking questions, not necessarily hoping to find answers.

Why would you ask questions and not want answers?

Wyrd – yes, an empirical observation (how the world works) leads to the construction of a theory (one plus one equals two.) That’s how I see it, even if it sounds a bit backward! And of course we invent science too!

I’m familiar with Tegmark’s idea (I am hesitant to call it a theory.) I used to subscribe to the idea that the universe is mathematical, given the success of mathematical laws in describing it. But mathematical laws can be used to describe all sorts of things, such as economics, gambling, or board games.

Yet Newton’s (beautiful, simple) mathematical theory of gravity turned out to be wrong and had to be replaced by Einstein’s (complicated, difficult) theory. Even that is incomplete. The

realtheory is going to be messy and hard to comprehend.The thing is, even if the physical world was just noise and chaos at its most fundamental level (which it might be), it would still scale up to an orderly world at some level (the law of large numbers at work.) Since we don’t know what’s happening deep down, I think it’s far too premature to declare the universe to be mathematical in nature. All we know right now is that mathematics is an extremely powerful tool that enables us to make predictions.

“But mathematical laws can be used to describe all sorts of things, such as economics, gambling, or board games.”Agreed! We seem to draw different conclusions from that, though.

“Yet Newton’s (beautiful, simple) mathematical theory of gravity turned out to be wrong and had to be replaced by Einstein’s (complicated, difficult) theory.”Point of order! Einstein didn’t prove Newton

wrong. Newton’s laws work extremely well in their proper domain. EinsteinextendedNewton because those laws are (like the laws of Special Relativity) restricted to a specific domain.“All we know right now is that mathematics is an extremely powerful tool that enables us to make predictions.”It does more than that! As you said yourself, it

describesthe world. And it describes it extremely well. As you also pointed out, Newton’s laws are beautiful and elegant and describe a vast array of diverse physical phenomena.General Relativity is more abstract (highly mathematical), but it’s also beautiful and elegant. (My personal belief is that GR is right and complete and that we’ve gone down an “epicycles” road with QM. The long-time failure to work out quantum gravity suggests to me it might be a non-starter.)

In both cases, elegant, beautiful mathematics precisely describes aspects of reality.

That’s… weird! 😄

On Newton vs Einstein, my point was that Newtonian dynamics is a good rule of thumb that lets us do lots of useful calculations. But it’s not how the world really works, and it lets us down at high speeds and high gravitational forces. Why should we expect Einstein’s equations to fare any better?

As for QM, there’s plenty of evidence that rules out more mundane explanations. Like it we may not, but it won’t go away. As for quantum gravity, the world really must work that way, but since we lack any kind of empirical data, it’s a tricky problem to crack.

Most mathematics is an approximation of some dirty problem that can’t be solved exactly. I do wonder if physics is ultimately the same. We are simply building mathematical models.

But Newton’s laws

arehow the world really works at that level. Newton’s laws very precisely describe the motions of the planets (save for Mercury) and lots of other things from cannon balls to water waves.Nothing in relativity says Newton was

wrong— just that it doesn’t apply to all situations. Exactly like SR isn’t wrong in an accelerated frame, it just doesn’t apply to that context.WRT QM, that’s a tangent I’m not sure is appropriate here. Obviously matter and energy are quantized, but there’s no reason so far that space or gravity have to be also. All we really know is that GR and QM disagree, so at least one of them is — at best — incomplete. The search for QG is based on the assumption that

everythingis quantized. It may not be. Assuming it is may turn out to be a fatal flaw in our physics.In fact, most mathematics is an

idealizationof a physical problem (the old “imagine a spherical cow” thing). It’s the physical world that seems to be the approximation. There is no such thing as a mathematical circle in the real world, for example.Wyrd, “it’s the physical world that seems to be the approximation” LOL. I must use that some time. Seriously, to answer these questions, I feel another blog post coming on. Watch this space!

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