# Think differently part 2. :0

Not at all, it’s my pleasure to talk about philosophy of mathematics (although it might be considered off-topic for this thread?)

I might go a little overboard with my next response, but don’t blame me, eh?

So yes, this is the major dividing line in philosophy of mathematics. There is the realist view, of which probably the Platonist viewpoints is the most common, that our mathematical ideas are abstractions of existing concepts and rules (compare with the shadows on the wall of Plato’s cave). And then there is the anti-realist view, of which formalism and intuitionism could be considered common, in which mathematics has no objective truth and is merely the result of rules made up by us.

I have do make the disclaimer that consider myself to be in the formalist camp (as you might have noticed). I find it more beautiful if the thing we study is just how some rules interact with each other, than as a quest to find the truth. For one, I’m assured that it is impossible to find the truth (which I’ll explain below in a digression), and for another, one of my main interests is studying how different assumptions can give rise to different interesting mathematical models. If I were searching for the truth, most of that would become meaningless.

Perhaps it is a good point to stop and think for a moment what infinity actually is. How is infinity different from “finity”? Things like “unboundedly many”, “until forever” and “approaching zero” give some intuition to what it might mean, but they never get concrete. It does give us the idea that infinity has to do with the size of things.

So let’s start with the easy part: how do we know for a certain collection of objects, what its size is? Well, we do it by counting the objects. If I have n apples, then I could stick a unique number from 1 to n on each apple (that is, count them). How do we compare sizes? If I have a collection of m apples and a collection of n pears, I put them in pairs: for each apple, I choose a pear, and for each pear I choose an apple. If I run out of one of the two fruits, then I know that that collection had less than or equal objects than the other. In particular, if I can do it in such a way that I run out of the apples and the pears both at the same time, then I have two collections of equal size.

So now let’s lift this idea to the infinite. To begin with, we have a problem: using just finite resources (time, space, objects), I can never reach infinitely many things. This is actually something you could prove using the above description of size. The idea is that at each step x in my trying to reach infinity can only use a finite number f(x) of resources, so in total I have collected f(0)+f(1)+…+f(x) resources. But since everything is finite, this is still just a finite collection of resources: I didn’t get anywhere. Surely, if I do this for infinitely many steps, I can get infinitely many things, but who says there is enough time to do this for infinitely many steps? Could I ever claim to be at a point in my construction where I have done an infinite number of steps? It is not at all something obvious we should be allowed to do so.

So, we need to assume that there exists something that is infinite to get something infinite. This already makes it incredibly difficult to ever point to some actual thing being infinite. But if we disregard the Platonic difficulties with accepting infinity (which would need us to be convinced it actually exists), we could argue in the formal way that if such a thing as infinity existed, then we could study its properties. Usually we take the set of all finite counting numbers (0,1,2,3,…), also called ordinals, as our prototype for an infinitely large collection. We could “create” such a collection by a neverending process of taking the collection of finite ordinals up to a certain size 1,2,…,n, and then making a new collection 1,2,…,n,n+1 with just one more object. If we assume what would be the result of this neverending process, we get a collection with infinitely many objects (finite ordinals in this case).

Now that we have our infinitely many ordinals, I could describe any collection of things to have an infinite size, when I could label each object in my collection with a unique (finite) ordinal. If I run out of (finite) ordinals, then I know my collection is infinite.

In fact, I wish to digress a little yet again In fact, I wish to digress a little yet again, to discuss that there are actually many sizes of infinity. This is what gave rise to Axiomatic Set Theory, after Georg Cantor discovered this fact. There are collections that are properly larger than some infinite collection (in fact, for any infinite collection there is a strictly larger infinite collection). This might seem obvious at first sight, since we could just add more objects to an infinite collection, but this is not the case: simply adding some objects might not change the size.

For example, if I take all the even counting numbers (0,2,4,6,…), then there are just as many of them as there are general counting numbers (0,1,2,3,…): I could pair to each general counting number a unique even counting number, by pairing the number n to the number 2×n. By the above description of when two collections would have an equal size, this means that these two collections have the same size, even though “half” of the counting numbers are missing in my collection of even numbers.

However, there are properly larger collections, such as the collection of real numbers. The proof that this is true is one of the most beautiful mathematical proofs out there, and not too difficult to understand. It’s called Cantor’s diagonal argument, and I highly recommend reading it.

Here's the promised digression about that I believe it impossible to find the truth

Mathematical proofs are in the end based on some assumptions. Simply speaking, we can’t start without something. In the early 19th century, a school of mathematics arose that was specifically interested in describing a sufficient set of these starting assumptions, called axioms that were strong enough to prove all of mathematics in a very formal, logically sound way. The philosophy of this school was called logicism, made particularly famous by the attempts of Bertrand Russel and Alfred North Whitehead, resulting in the Principia Mathematica.

Now along came the 25 year old logician Kurt Gödel, who showed using a very clever trick, that it is impossible for any collection of axioms that is strong enough to do arithmetic in (that is, counting, addition and multiplication), can never be able to prove for all mathematically expressible statements whether it is true or false. Furthermore, he showed that it is impossible for such a collection of axioms to show that it would not lead to a contradiction: this means that if the axiomatic system was reasonable in that it would not have contradictory statements, then we would never be able to use this axiomatic system to prove so.

And sure enough, there were mathematical statements that have been proved to be unprovable using “standard” mathematical approaches (I’ll leave “standard” undefined, but read it as the kind of mathematics that 95% of mathematicians are concerned with). Two famous ones are the Axiom of Choice, which states that we could pick representatives from an infinite number of nonempty collections, or the Continuum Hypothesis, which states that any infinite collection of real numbers can only have two possible sizes. It has been shown for both of these statements that they could be true, and that they could be false, without making any difference to “standard” mathematics.

The question them becomes, are these statements Platonically speaking true or are they false? It will be extremely difficult to make a good case for either of these claims, since we cannot use “standard” mathematics to determine it. In my opinion this makes a strong case to refuse to assign any truth value to such statements, and instead consider them as formally following from certain (non-“standard”) assumptions and being negated by other (non-“standard”) assumptions.

I agree with this. In the abstract, the concept of infinity is something tangible, that we could work with, and create new concepts with. In the concrete, the concept of infinity does exist, since we can think of it. However, in the concrete, I am not convinced of the existence of infinity, i.e. the thing the concept talks about. To be clear: I’m ambivalent about it, I think it is very well possible that it exists (and would love it if it did), but it could equally well be that our universe is finite in nature.

Here I feel there is a flaw in the distinction between the concept and the thing itself. The space of “all the possible products of the human imagination” might be infinite (although I am not convinced of this either), but I think only the concept of this state space is part of the actual state space of our universe. Meaning that it might be that not every possible idea is in itself an existing things.

In my view, the thing that is “an idea that a human has” is an existing part of our universe, while “the idea” that a human has, is not an existing part of our universe.

Here you describe a process of creating something larger. As with the problems above, I find it not naturally obvious that such a process would have a result. There is a difference in the things a process can produce if we let it run forever, and the things a process could actually produce.

I also object to the state space of the universe expanding when ideas are born. In my view, those the “thinking of those ideas” are still part of the same state space that already existed. The object that is the idea itself, does not make part of the state space (unless it already was).

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I am reminded of a zen story where a monk is asked to be judicial in a disagreement between two people. He listens to the first argument and says “Yes.Yes. You’re right. You’re right.” And then listens to the second and says “Yes. Yes. You’re right. You’re right.”
And somebody says “wait a minute, you can’t agree with two totally different arguments.” And the monk says “Yes. Yes. You’re right. You’re right.”

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So, Naps, here’s a true story. I had a cool green copper wind chime with five or six tubes of different lengths when I was a teenager. One afternoon I was lying in my bed and spacing out listening to the wind chime ring its different and random clusters of notes here and there. For about five minutes I got into a zone where could anticipate what the next cluster of sounds would be. I heard them in my Inner ear before they happened. They were each different. I got so freaked out, I jumped up and took the wind chime down from outside my window and buried it in the back yard. In hindsight, sad, one should not give in to fear.

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I believe it is possible to have contact with the numinous, infinite, & divine. You know it when it’s happening. Just a drop of the infinite in a system goes a long way; can change everything.

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Hm. Assuming the visible universe (~10^27 meters in diameter) consisted of planck-length (~10^-34 meters) voxels and assuming they could be in one of two states (on/off, if you will), also assuming the universe was cubic with side length ~10^27 meters… (because otherwise it’s spherical and calculating 4/3 x pi x r^3 is probably not worth the hassle)

10^81 cubic meters, with each cubic meter subdivided into 10^102 voxels, that’s a total of 10^183 voxels and a grand total of 2^(10^183) possible states of the universe, or about 10^(10^182) A decimal number with 10^182 digits. Quite big, but still finite. xD

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Sometimes I feel that considering my mortality is a way to get a taste of the infinite. It is quite likely that there’s nothing after death, and that I will never exist again forever; not existing, no time or experience, but still never coming back again forever. Very strange

The universe isn’t cubic though, it’s flat

Unless there are infinite big bangs and infinite earths

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right, not steady state, but possibly cycling forever; with no beginning

It’s turtles all the way down!

Round like a circle in a spiral, like a wheel within a wheel
Never ending or beginning, on an ever-spinning reel
Jack ruby -Apathy

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turn like a wheel inside a wheel…

Thank you for such a thorough, thoughtful post! Really enjoying this discussion.

Rather than mathematics, my background is that of a writer / story-teller. I am also very interested in philosophy, psychology, and evolutionary biology, so a lot of my opinions and frameworks come from those backgrounds. However, it’s interesting that the same problems and questions come up - i.e. “are we finding truth or inventing it?” or “are we seeing reality as it really is or are we fundamentally limited to seeing what we are expecting to see?”

If one studies comparative philosophy long enough, something quite unsettling becomes clear: while any philosophical viewpoint allows one to gain insight into the universe in some particular way, it will necessarily blind the viewer in another way. As in your example above - things that make perfect explanatory sense from the Platonist point of view become nonsensical from the formalist view.

There is also the problem that - regardless of our philosophical background - our conceptual frameworks are innately, inescapably human. They are based on human experience and try as we might, we cannot avoid painting our reality with the colors and metaphors gained from that experience. And so our religious mythologies become filled with stern fathers, loving mothers - we make arguments for or against abstract concepts based on what our experience has trained us to expect (i.e “it can’t be turtles all the way down” or “God doesn’t play dice with the universe” etc.)

The philosophical framework of Zen gives us some interesting clues and tools in this regard, in that it asks us to explicitly focus on how our mental structures draw limits on what we are ready to accept, and urges us to actively break through those limits. However, just like any philosophical framework, Zen also has boundaries - in that - if you’re not using those tools to approach a particular goal (i.e. detachment / going beyond ego) then you’re “doing it wrong”

So, given that we are hobbled by our human perspective in all these different ways, is the whole process of making sense of the universe doomed? I don’t think it is - and again - I use a framework from evolutionary biology. At any one point there is a dialectic between our approaches and the reality outside of us. If our approach doesn’t fit - if it fails to explain the reality outside of us, we have to discard it, adapt, and evolve something else.

The point isn’t to get to some final answer at the back of the book, the point is to continuously evolve our frameworks, approaches, and answers to the reality before us because Reality is That Which Does Not Go Away When We Stop Believing In It - which is pretty darn close to the Scientific Method. I like this approach because - at any given point - we acknowledge our limitations. We aren’t seeing Reality as it really is - we acknowledge that it might be beyond our current ability to measure with the instruments we have at hand, or might even be more complex and strange than we can wrap our human minds around - but at least we are constantly building and discarding frameworks rather than being stuck in any one particular framework.

I think about these kinds of things a lot. How do you take something so huge and abstract and make it concrete? We look out at the darkness between stars and we can tell ourselves “That goes on forever”* - but again, we are limited by our human experience. Since nothing in our living experience is infinite, we are reduced to abstractions.

(*and yes, I know that our universe may not in fact be “infinite”: https://phys.org/news/2015-03-universe-finite-infinite.html)

For me, it helps to think about much smaller infinities like repeating decimal fractions. I know that 1/3 = 0.333333333… is in some ways just an artifact of base10 math, but there is something there about those 3’s going off into infinity that helps me make the abstract more concrete. The same thing goes for Pi - on one hand, I know that this irrational number goes on forever, on the other hand I know that they all fit in to that little tiny chunk of red between 3 and Pi.

It’s the same with fractals. Using a good computer program and diving deep into the Julia set, one gets that awe-inspiring feeling that it is bottomless. Now, some part of me knows that this is an illusion - that the software I’m using is only rendering so many iterations of the equation, and that my monitor is only able to display only so much granularity of that image, because of the limits of the pixel sizes. However, the thing that works for me isn’t the actuality of infinite depth but the POTENTIAL for infinite depth - that given these particular conditions - one could perform the calculations and they would theoretically go on forever.

So maybe that is the most useful way for me to think about infinity - something that has the potential for being an infinite process because there is no natural limit to keep it finite.

So, so back up and bring this around full circle - the idea I started with was that once the capacity for embodied awareness evolved, it began adding additional layers of complexity to the universe in ways that

• accelerated the universe’s natural process of adding those layers
• made novel resources and products available for complexity to use and
• created a situation where the process of adding additional layers of complexity had the potential to become an infinite process.

So from this point of view - if you think about the state space of the universe as not only the positions of the atoms in the universe, but also the motion of those atoms, and the procedural/interactive information that is made possible by all the processes and forces acting on those atoms - then you approach a framework where new ideas really do expand the state space of the universe.

What do I mean? Well for one - the presence of living beings has reshaped the chemistry of our planet many times, the most famous being the Oxygen Holocaust:

Once that great chemical change occurred, and a more energetic element was available as an energy source (i.e. oxygen metabolism) that then led to all these other evolutionary and metabolic pathways evolving, increasing the overall complexity of the system.

Or, let’s look at something like the hormone Oxytocin:

At one point in biological evolution, you had sexual reproduction, but no pair bonding, no parents taking care of offspring, etc. Along comes this new hormone (note: from an evolutionary perspective, this first evolved in sea birds, see this book for more: https://www.goodreads.com/book/show/262728.The_Naked_Neuron)
and suddenly you have these internal reward pathways that lead animals to develop new behaviors to nurture their young, form pair bonds, form larger societies of animals that are capable of more complex group behaviors than any individual. This increase in behavioral complexity then circles back and becomes the evolutionary pressure which leads to development in language complexity, group dynamics, and eventually the beginnings of theory-of-mind, etc.

Fast forward to modern times and we have scientists filling in the Table of Elements by creating new elements that were once only theoretical. We can even misuse those things, and create huge nuclear disasters (Chernobyl / Fukushima) that create long-term physical changes in our world. We are sending our machines and information about ourselves out into space. And now, we are working towards technology that is taking our embodied/physical awareness and transferring those processes into machines. Will we ever get a strong/general AI? I don’t know! But I do see it as something likely, and if we ever do succeed, it will create new configurations of matter, energy, and information - it will open up entire new situations where complexity can take root and grow - it will create its own new by-products and secondary/tertiary effects that we cannot predict.

So - yes - I do see our ideas as increasing the available state spaces which are possible in the universe in a very real / physical sense. I also see this entire process as something that has the potential to be infinite - even though I cannot hold the actuality of that infinity in my hand. Of course there are many factors that might limit it - our potential to destroy ourselves - the finite lifespan of our local star - even the potential heat death of the universe. But again, I can stipulate that this might be an infinite process because it has the property of constantly jumping beyond its limitations - creating new configurations of matter, energy, and information - making new forces, new materials, and new ideas available.

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Hi - so I was a little confused by your over-use of the pronoun THEY/THEM. Are you doing this because you’re trying to use gender-neutral pronouns? Or are you so beyond self/other that there is no distinction between your thoughts and your co-worker’s thoughts? Either way, it made it a bit difficult to understand who thought what and who said what…

I was also confused by the SWIM thing - I just looked it up and I’m guessing you meant Someone Who Isn’t Me… right?

Anyway, I guess I just wanted to point out that humans have this amazing capacity for non-verbal communication and theory of mind. We have entire centers of our brain cooperating to feed us theories of what might be going on inside other people’s brains.

Some people are naturally better at this than others, and I’ve certainly read a lot of anecdotal evidence, and had my own share of psychedelic experiences which let me know that our capacity to do this particular trick can increase in those situations.

So yeah, while it might be possible that some sort of telepathy was taking place, it’s probably a lot more possible that this was just one person’s mind being really good at making sense of the world. In other words

• the person not feeling well was giving off all these non-verbal signals that this is what was going on in their head
• another person absorbed those signals and - without even trying - created a complex mental model that predicted what was going to happen next, and then
• the thing they predicted actually happened

I’m not saying that this is somehow 100% proof that telepathy doesn’t exist, but it is the much simpler explanation…

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I like to fall back on that ‘ol rock & roller William Blake who tells us “If the doors of perception were cleansed, every thing would appear to man as it is: infinite.”

and

To see a World in a Grain of Sand
And a Heaven in a Wild Flower
Hold Infinity in the palm of your hand
And Eternity in an hour

The rubaiyat of Omar Khayyam -
“the flower once blown, forever dies”

The way that can be told is not the eternal way.

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Swim is clearly me lol and yeah he didnt see me, he was walking past talking to someone else and my half cubical blocked me from his vision. It was the timing too it wasn’t like 5 mins past it was like i thought it he said it level creepy.

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Stage 1 cleared. Now repeat “We would like to offer you a position at our company” and wait for the phone call.

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my random ass neo call and then text #matrix

So it’s funny you say wait for the phone call lolz
Guess I didn’t follow the white rabbit to clear stage 2.

This is more a taste of what empty means than what infinite means. Quite the opposite.

perhaps. You’re right