Tag: Math

CH and GCH

I’m not even close to being an expert in mathematics, but I have some ideas regarding the continuum hypothesis and the generalized continuum hypothesis.

CH

Regarding the continuum hypothesis, I think one possible way to have a cardinality between and might be to define something like the real numbers but where certain elements of the sequence composing the number or patterns of elements are equivalent to certain other elements or patterns of elements.

For example, you specify your numbers in base 10 and define that all 8’s are equivalent to 9’s, so 0.7652861066492217 would be equal to 0.7652961066492217, 0.7652861066482217, and 0.7652961066482217. (Not sure if it would become an issue that 1=0.999…=0.888…, 2=1.999…=1.888…, etc., so you might not be able to have any whole numbers, but that’s just an artifact of this particular example.:P Also, making one numeral equivalent to another is probably the same as working in base 9 instead of base 10, which shouldn’t impinge on its cardinality. So maybe that’s a bad example, or maybe this disproves my whole tack.)

Another example would be to use binary and define that all sequences of {0, 1, 0} are equivalent to {0, 1, 1}. So 0.3125, or {{}, {0, 1, 0, 1}} would be equivalent to 0.4375, or {{}, {0, 1, 1, 1}}. (In this notation I’m defining real numbers as sequences of two sequences, where the first sequence is the integer part and the second sequence is the fractional part. Also, a better example wouldn’t allow for numbers/sequences where you have to choose between multiple paths of substitution. Not sure that’s even possible in base 2.)

This would create a set that’s smaller than the real numbers but larger than the integers. I have a hunch that you’d need to define your equivalencies in a way where it they happen an infinite number of times in the set of all real numbers.

GCH

Regarding the generalized continuum hypothesis, my idea is that an integer is a sequence of digits, while a real number is two sequences of digits, and in between any two consecutive integers is an unlimited number of fractional parts of the numbers, where the fractional part is the second sequence. So, naturally, to get to the next cardinality, you’d just have to add a third sequence. Between any two consecutive real numbers would be an unlimited number of possible values for the third sequence. So basically the set of this type of number is an infinite set of infinite sets of infinite sets, and it’s all ordered. For ℵ₃ you’d need numbers comprising four sequences, and so on and so on.

P≟NP

I don’t have a solution for P≟NP, but I may have some insight as to what direction we can take to arrive at a solution.

My theory is that there are certain concepts and shortcuts in our mathematical/computational language that hide what we’re really doing on an absolute level.

An example would be working with numbers. Essentially a number is a sequence of elements (or if it’s a real number, two sequences—one before the decimal point and one after; if it’s a complex number, 4 sequences). The simplest way to qualify the range of the sequences would be to use base 2 and consider the range to be {0, 1}, but you can use any base you want.

A number is not just a sequence, though; certain sequences being “numbers” implies certain relationships between them and the possible operations on them. If, instead of “*”, we listed the algorithm applied to the sequences involved that multiplication reduces to, we would be using a language that more directly reflects what we’re actually doing.

I think that finding a language that wholly reflects what we’re actually doing when we compute—and not because it’s possible to directly reflect what you’re actually doing in the language but because it’s necessary to—would be the first stepping stone in solving P≟NP.

I think of particular importance would be how we define selecting a limited amount of members from a larger set. An analogy would be the difference between procedural programming and constraint programming; or maybe between constraint programming and something like functional programming except where the executor doesn’t do any sophisticated things to figure out how to solve problems; or maybe between regular functional programming, where the executor is sophisticated, and a kind of limited functional programming, where you have to spell out how to do anything sophisticated/combinatorial/quadratic/whatever within the code.

So, for example, when defining prime numbers in this new mathematical language, the very act of defining them would necessitate implementing some kind of algorithm for selecting them.

But then, you’d think that if it’s necessary to define an algorithm for selecting the primes just by the nature of the definitional language, then it wouldn’t be arbitrary which algorithm you select, yet there is definitely more than one possible algorithm to implement. Maybe the reconciliation of those two facts is that this new language would make it obvious how all solutions that select primes (at least from the same range, be it finite or infinite) are equal or isomorphic to each other. And if it can do that, then it wouldn’t be too far a leap to expect it to make it obvious, for our purpose of proving P≟NP, how the solution to any NP-complete problem leads to the solution to any other NP-complete problem.

Not that this gets you all the way to solving P≟NP, but the point is that once you have this language, you’re way closer to the essence of all problems and solutions, which should naturally lead to insights into solving P≟NP. It might still be a lot of work, but at least we’d have a starting point, whereas it seems that currently we’re clueless as to where to even start.

Is the Universe Infinite?

As with a lot of simple yet deep philosophical questions and statements, the question is basically nonsensical but appears to make sense because of our tendency to be duped by language. To a certain extent, we tend to think that grammatically correct sentences must makes sense. And that’s to say nothing of the semantic problems raised by the verb “to be” (see Alfred Korzybski).

The universe is neither finite nor infinite.

What does the term “infinite” mean exactly? Basically, it’s a mathematical term that means that a value is so large that any finite value is smaller than it. Of course, the problem with this definition is that to have an actual value it must be finite—otherwise you have a formula for creating values.

For example, any actual whole number must be finite, but the number of whole numbers that “exist” is said to be infinite. Of course, you can’t possibly ever represent, count or observe every possible real number. Not even if you had all the time in the universe. Not even if you had an “infinite” amount of time, whatever that might mean. Because no matter how many numbers you’ve counted, you can always count more, by definition.

So  the set of all whole numbers is “infinitely” large simply because you can execute an algorithm (however you want—by hand, on a computer, in your mind, whatever) to generate more successive (or non-successive, if you prefer) whole numbers for as long as you want. The algorithm itself does not contain all the whole numbers and is not infinite in content, so how can you execute it for as long as you want? The answer is that the algorithm essentially runs in a loop.

In the case of generating successive whole numbers, the algorithm could look something like this:

  1. Start with some number. If you think about it, this number is actually nothing other than a sequence of digits
  2. Copy the contents of the current number to the next number
  3. Start working on the last digit of this number
  4. If the current digit we’re working on is 0, change it to 1. If it’s 1, change it to 2, etc.
  5. If it’s 9, change it to 0, change the working digit to the one before the current one, and go to step 4. If you can’t do this because we’re on the first digit, then prepend a 1 to the entire sequence and change every subsequent element to 0
  6. Go to step 2

..Or something like that. Whatever. The point is that all infinite sets or infinite values (such as the size of an infinite set) actually boil down to algorithms for generating those things that run in loops. If you’re wondering about infinite sets other than the number line, in general any infinite set is ultimately a prescription for finding new elements that belong in the set indefinitely, or at least until you stop.

So when we ask, “Is the universe infinite?”, we’re basically asking if the universe can be generated by a mathematical algorithm in a loop. And even if it could be—which it obviously can’t, because that would create a universe so regular and ordered that it would be uninteresting, not this one—that would only make the universe as big as the time God or whoever spent executing that algorithm. And that’s to say nothing of the fact that mathematical algorithms deal with numbers only, and numbers are purely quantitative and abstract and can’t possibly generate quality or substance. (That’s why the universe can’t fundamentally be made up of math, but I digress.)

I said earlier that the universe neither finite nor infinite. So why is it not finite? Because it’s unlimited. Just like the infinite contents of an infinite set don’t actually exist anywhere, because you can’t define infinite existence except as a looping algorithm or some kind of paradox, the universe doesn’t exist in an “infinite” sense. But neither does it have any boundaries to its existence. The more you look, the more you find, forever.

How can this be true and the universe not be infinite? The answer is that existence itself is relative. If you think about it, in order for something to be said to “exist,” it must be able to affect you in some way. If it can’t affect you, then you have no way of knowing it’s “there” and therefore you can’t rightly posit that it exists.

The concept of “existence” is a tricky one. Emmanuel Kant said, in response to the ontological proof of God’s existence, said that “existence is not a predicate.” While his reasoning surrounding this statement was valid, the statement alone isn’t exactly true. Existence is a predicate, it’s just not a normal one. If existence weren’t a predicate, why would we say that a unicorn—or anything else—is either “existent” or “non-existent”? That’s exactly how predicates work.

You could say that the unicorn that’s non-existent can’t have any predicates because it doesn’t even exist, but if you think about it, all objects we can possibly think or talk about are mental objects; they exist primarily in the mind. They may or may not “point” to objects outside of us.

How do we know if a mental object points to something outside of us? Presumably, we can’t directly know of anything that exists outside of our minds. We only infer as a result of sensation. So how do we know the chair exists even while we’re not sensing it? If we expect that, when we will our muscles to contract in certain ways we call “walking into the dining room,” we will see a chair with specific properties there, then we say that that chair “exists” and that our concept of the chair therefore points to something outside of us. But insofar as we can think of or talk about the chair, it exists in our minds.

We don’t even know if reality outside of our minds (if there is such a thing) is made of objects, or if it’s just some continuous field that wouldn’t even look like objects if we could have a “view from nowhere” (or, to be more epistemologically coherent, at least a “more objective” viewpoint). Indeed, “the chair” is just an arbitrary collection of atoms that we separate as “a chair.”

Let’s say the chair is made of wood and, due to attrition, some wood particles on the bottom of the chair’s legs get scraped onto the floor. Exactly which particles belong to the chair, and which belong to the floor? Where does the chair end and the floor begin? What if a child marked the chair 3 years ago with a magic marker? Are those ink particles now part of the chair, or not? If you break apart the chair with a hammer piece by piece, or burn it to the ground, at what point during the process does it cease to be a chair? Etc.

Since any two things we can possibly compare and contrast to each other (presumably using thought) must necessarily be ideas, the schism between the ideational (that in our minds) non-ideational (that outside of us) must necessarily be the biggest possible schism we can imagine—or, arguably, bigger than any schism we can possibly imagine.

So, back to the existence of the chair. To say that it exists is necessarily merely to say that we expect to perceive particular sensations in response to willing (what we think are) our muscles to do certain things. (We don’t know for certain that we have muscles, but we know for certain what we’re willing since that’s a part of our mind and therefore is directly known.)

If you posit something extant that can’t possibly affect us, any possible description of that thing is equally valid, since none of it is provable/demonstrable or falsifiable.

So, to validly posit that something “exists” must imply positing that it can potentially affect us in some way. If we will X, we expect to sense Y, hence Z exists. E.g., if we will walking to the dining room, we expect to have the visual sensation of a brown geometric form whose shape is determined by our perspective, hence the wooden chair exists. Of course, there are a million other ways we could less directly test its existence, and we can guess they’d all work because reality seems to be self-consistent, but that’s beside the point.

The reason existence is relative is that not every object that exists in the multiverse, according to some kind of fully objective view from nowhere, is potentially available to us at any given time. Most of it isn’t most of the time. Most of it will never be. But anything is experientially available to some entity at any given time (probably some entity you don’t have access to on a certain level), and on the most ultimate level, all entities are one, so the fact that it’s available to them and not to you is a relative fact.

From the perspective of this view from nowhere, every possible experience exists. I said/implied earlier that there’s no such coherent thing as a view from nowhere, which is exactly why we can’t say, based on this view that every possible experience exists “somewhere,” that the universe is infinite. The best we can say is that it’s unlimited or unbounded because your viewpoint constantly changes and therefore the breadth of objects that become extant to you constantly changes. (TBH, in actuality I contradict myself by saying we can’t have a view from nowhere and then saying that in view from nowhere every possible experience exists. But oh well—”I am large, I contain multitudes.” :P)

I don’t know whether the separation between what’s existent to us and what’s not is discretized/bounded according to finite universes within an unlimited multiverse, in which everything in our particular universe is existent to us at once but in the big picture we have access to more than just this universe, or if it’s more of a continuum. Maybe what’s existent to us is everything in our past light cone.

By the way, in case it seems odd that I would appeal to epistemology or the subjective in defining the meaning of “exists,” I will justify it here. First, trying to define “exists” purely ontologically/objectively is problematic and fraught with paradox. Explaining it epistemologically is much more cogent and tidy. Second, our epistemology comes before our knowledge of external reality both in chronology and in logical primacy, so it’s the natural place to ground such a deep concept as “existence.”

Third, to state that something exists is necessarily to talk about a concept, because what we’re talking about isn’t a thing in reality in order for us to be talking about reality when we talk about it. And it can’t be that things that don’t exist are fundamentally concepts while things that do exist aren’t fundamentally concepts, because then you could never decide whether a thing exists or not. Its existence status couldn’t be subject to change. Therefore it has to be that both things we talk about that exist and things we talk about that don’t exist are fundamentally concepts. So it makes sense that “existence” is a conceptual quality/category we ascribe to those concepts.

QED.

Why Mathematical Platonism Is Silly

Basically, mathematical Platonists feel that, because there is so much complexity to math, it must be something “discovered,” as if from some Platonic mathematical realm. The problem with that is that the derivations in math cannot be any other way; they are what they are out of necessity (given math’s fundamental axioms, at least). That means that, even if there is some Platonic mathematical realm, the “discoveries” of math can’t come from it because that would imply that, were the realm somehow any different, the discoveries of math could be somehow different—that just working out the logic in our minds or by hand or on computer we’d come up with different results. But that would be incogent. The reason we get the results we get is that they’re the only results that are cogent. So, even if there were such thing as some Platonic realm containing all mathematical objects and relations, it would be completely superfluous because the discoveries of math can’t “come from” it, and therefore, given Occam’s razor, it makes little sense to assume its existence.

Another way to tackle the issue starts with an analysis of the meaning of the term “exists”. In order to coherently claim something exists, you must imply that it’s in some way, at least in principle, detectable or otherwise noticeable. If something is not noticeable under any potential circumstances, then what does it mean to say that it exists? To claim that something exists includes defining what the basic form is of the thing that exists; otherwise you’re not saying what it is that exists, and it might as well be the most contentless thing imaginable, with the limit being nothingness. And how can you imagine the form of something without imagining interacting with it in some way to see the form? (See my argument for “form is function” in my previous essay and here.) And if the thing you posit exists can’t be interacted with (or, more specifically, can’t affect you) even in principle then imagining this observation of it is self-contradictory when you take the whole context into account, i.e., the whole world, from you to the claimed extant. Not to mention that the idea that something that exists that doesn’t affect us is a) unfalsifiable, and b) in violation of Occam’s razor.

So, if to say that something exists is to imply that it can affect us, then it makes no sense to say that mathematical “objects” (or whatever they are) are “exist” in some Platonic mathematical realm, because if they actually affected us then it would be hypothetically possible for them to affect us in some other way, thus implying some other hypothetical nature in which they exist. Instead, mathematics is all tautology, as it all necessarily follows from its fundamental axioms. Interaction/affecting is a process of action in time, and the objects of mathematics are timeless and unchangeable, so they can’t affect us in order for us to observe them.

In his book The Emperor’s New Mind, Roger Penrose argues for mathematical Platonism on the grounds that a given point is or is not in the Mandelbrot set independently of what mathematician or computer is examining it. By “examining it”, of course, he means executing the algorithm that determines whether a point is in the Mandelbrot set. I would say that, since there’s no way for an independent truth of which points are in the Mandelbrot set to “make its way into” the results of a completely deterministic algorithm, that truth must be an aspect of, or an indirect reflection of, the algorithm itself (including the rules for multiplication of complex numbers). It is simply illustrated in a way by which it appears very complicated, while its abundant self-similarity across place and scale is one sign of its actual underlying simplicity. Basically, humans are not smart enough to see “through” the imagery to its underlying simplicity so our minds are tricked.

Let’s now tackle this problem from the opposite direction, starting with the fractal image and then deriving the algorithm. Let’s consider two reasonable suppositions: 1) The greatest measure of compressibility of a set of data is the smallest algorithm that can recreate that data, and 2) A set of data only actually contains as much information as its most compressed state; the rest is redundancy. If you made a program that could read a set of data and return the smallest algorithm that creates that data (though it might take a quintillion years to do that) and you fed it a Mandelbrot image, it would certainly (eventually) spit out the algorithm that created the image in the first place. Therefore, a Mandelbrot image actually, on a fundamental level, contains no more information than the algorithm that created it.

This thought experiment brings us to another interesting point: Penrose could have used for his argument any algorithm that produces an apparently complex set of data. For example, a pseudo-random number generator would generate an image with much more apparent complexity than a Mandelbrot image (in that it appears to be way less compressible, hence it appears to contain more information), yet Penrose doesn’t use a pseudo-RNG for his argument because it’s more obvious in that case that the only meaning in the data is in the algorithm that produces it. Yet the obvious structure of a Mandelbrot image is not any more evidence that the information exists in some Platonic realm than a pseudo-RNG-generated image is, because it’s no surprise that a simple algorithm could produce a structured image, since the image, being wholly a reflection of the originating algorithm, must therefore be a manifestation of complexity in simplicity. So, it’s apparent that Penrose was duped in this case by the mere interestingness (or whatever) of the patterns composing the Mandelbrot image.

Another argument for mathematical Platonism I’ve come across goes something like this: Math must exist prior to matter logically, if not chronologically, in order for matter to even exist because matter’s existence as such is wholly dependent upon mathematical laws. To this I have to say that mathematical laws aren’t something matter requires, as if they’re a separate thing from matter—the mathematical “laws” characterizing matter’s behavior are only ways to formally describe the behavior, and they’re merely abstractions. Reifying abstractions as something objectively existing is silly. In what form could they possibly exist?

A mathematical model of matter is basically a reduced simulation of matter. The math is merely a way of representing the matter’s behavior, and the matter is not separate from its behavior. Again, form is function. As I said in my previous essay, how can you know the form of something other than through how it interacts with the observer? And how it interacts with the observes is its function. And the functionality of matter and energy is the physics of it.

The degree to which matter behaves according to mathematical principles is the degree to which matter behaves both consistently and cogently (i.e. self-consistently). Of course matter behaves consistently, because it’s still the same stuff from one moment to the next, and the nature of its composition determines how it behaves. And to imagine that matter behaves in any way but cogently would be an incogent imagining, and thinking incogently is useless and irrelevant to reality, so of course matter behaves cogently.

Mathematical laws aren’t detectable even in principle except indirectly via the behavior of matter, so it’s unwarranted to assume that they have an existence independent of matter. And they’re not really even detectable via the behavior of matter because they could hardly have been anything different; they’re merely cogent or self-consistent thinking, codified.

Another argument (or perhaps merely a description) of mathematical Platonism I’ve seen briefly describes Platonism in general and then adds math to that realm in terms of some kind of basic or archetypal mathematical forms. The exact nature of these forms is irrelevant, because the premise of Platonism itself is silly.

Some Platonic forms, such as beauty, are merely abstractions derived from what many objects seem to have in common and then apparently reified as things-in-themselves by way of language. “Beauty” as being independent of anything beautiful exists only as a linguistic construct.

Other, more concrete Platonic forms, such as the ideal horse, are simply categories people hold in their minds as a result of seeing many similar objects which are given a common name, especially where there is not a smooth continuum of objects’ forms ranging from the ideal in question to completely different forms. There are many different reasons objects would take common forms in islands of similarity, and none of them is because there exists some Platonic form somehow supernaturally dictating their manifestations. For example, all horses are relatively similar to all other horses (and thus categorizable under one name) because of the evolutionary mechanics of speciation.

What’s more likely: That forms exist as templates in our minds used to categorize objects, created largely without our noticing over time through observation and teaching, especially in the early stages of learning; or that they exist in some unobservable, independent realm of abstractions without any conceivable sort of grounding, and that we psychically access a form in this realm every time we identify something? Especially considering how pragmatically useful it is to employ these categorizations, thus implying their likely arising from natural processes of cognition, and considering how naturalistically the islands of similarity in objects arise, thus making their definitions in an independent realm superfluous. And to say nothing of the areas of object differentiation where there are no islands of similarity, only continuums of object forms ranging between objects of completely different configurations, and also to say nothing of the ubiquitous continuums between areas of object forms where there are distinct islands of similarity and areas where there aren’t; for example, extruding from the island of horse forms are forms such as the zorse, a zebra/horse cross, a horse that just lost one of its limbs, horses with some sort of obvious genetic mutation, etc., horses still in the womb ranging through all the phases of ontogeny, etc.

Platonism is obviously a very naive and antiquated way of thinking characterized by a lack of self-reflection regarding language, abstraction and the process of identification, and mathematical Platonism is an even more problematic extension of that.

This essay was loosely based on a much more awkward and obtuse essay I wrote 21 years ago that can be found here: http://local.inhahe.com:8008/book/rough%20drafts%20%26%20notes/html/platomath.html.

I wrote a little bit more about mathematical Platonism, particularly about why it’s not true that “pi is infinite,” here: https://philosophy.inhahe.com/2020/10/29/a-better-solution-to-zenos-paradox-of-motion/