I might call mathematics an informal science of formal systems. Actual belief is “organic” and enacted.
For any \frac{p}{q}, I can check whether its square is 2. I have a total function. No problem.
Precisely because I cannot check an “infinity of fractions” with this total function, I need an informal proof to settle my belief.
Checking a finite set with a total predicate function would be ideal, and we do this to make conjectures plausible. The informal proof aims at the infinite horizon.
Is that true. Fine. You are bringing a load of baggage into the discussion… the “zoom out”. I’ve other fish to fry, so I might leave you to it at least for now.
—“Round square” doesn’t really mean anything to me.
—Is that “round square” ?
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—To me it looks like belief is primary. “True” is really nothing more than a way people indicate belief that is always someone’s. It’s like an organism’s tendency to meet its environment one way rather than another. We sometime use marks and noises to indicate the direction of our comportment.
—Is that true, what you say ?
—It’s my belief about the word “true.” So I might call it “true” to indicate that it is my belief. I’d rather avoid the sign “true” in theoretical discourse to avoid what looks like pointless confusion to me.
—So you mean it’s just “true for you” ?
—I wouldn’t say that. I don’t define horses in terms of unicorns or belief in terms of truth. Doesn’t make sense to me.
—But surely your words fit with reality or not ?
—To me words are marks and noises. We can use sounds that somehow reliably — in the mundane empirical context —direct the eyes of others to something we want them to see. But they may look where we don’t want them to look. They may not react to what they see in the way we expect.
—OK, but does what you say there fit with reality ?
—When we start chucking chains of signs like “fit with reality,” the old “mirror metaphor” starts to look pretty helpless, at least to me. Does the chain “fit with reality” fit with the reality of the way signs work ?
—Surely the world is one way or another ?
—The world that I experience is sometimes pretty definite. But sometimes I’m not sure what’s going on or what to expect.
—Right. But the world “actually” is definite.
—Well, I find that the world often becomes more definite as I put some effort in figuring out what is going on. So I guess I act in the hope that I can reduce ambiguity in most cases if I have the time and energy. But life around what I focus on is there as a sort of blur.
—OK, but it’s not blurry for others.
—Sure. Some people have a better grip on this or that than I do.
—OK. But of course the world itself is definite.
—What is this phrase “the world itself” ? Is that like the environment of an organism from no point of view ? How I am supposed to make sense of that from my life as an organism with a point of view ?
—It’s innate. It’s logical.
—It’s language. It’s a common vague kind of talk. I think people call it “realism.” But I’m an “empiricist” or whatever you want to call it. Things are “more real” the more they are experienced, not the further away they are from experience.
—Is that true ?
This is philosophy of math.
But it’s cool if you aren’t feeling it at the moment.
Same here. COBOL was a fun language to learn. It is steam engine, but still getting used by many financial companies around the world.
Which OS did you use? MS DOS, WIN NT, UNIX, LINUX, Amiga, Atari, all dead but UNIX and LINUX still seems going strong.
Going back to Turning machine HALT problem, it is a clear demonstration, that machines and humans think and function different in solving the real world problems.
Humans have advantage of having mind which is the result of 6 billion years of evolution, and ability to use metaphors, abstractions for understanding the world, and communicating with other beings.
Machines are the invention of humans to deal with a single problem, although now they are getting complex. They cannot match human intelligence.
Yes. Basically I see computers as doing one part of human thinking very well.
The more exciting part is analogical, metaphorical, or poetic. And this touches on our fundamental creativity. The human being is a “poet” whose “poems” tranform the world, though largely with the help of the disciplined “computation” part of human thinking that we can have the computers do for us.
The “essence” of computation is arguably that it drives ambiguity down toward an absolute minimum. It crushes “poetry” into steel. Something is lost, but something is gained.
But the death of all “poetry” is the death of what is most human in us.
Something like that.
I bet those two things are connected !
“Analogy is the core of cognition.”
X is like Y even though X is not Y.
I learn from one blurry situation what I can use in another.
Apple IIe, MS DOS, various versions of Windows, ChromeOS, MacOS, and some Linux emulators. I frequently work in the terminal, and I’m really into these new Apple chips. Right now I am using the M4 chip.
I briefly fantasized about designing my own chip for the Markov language. You can get programmable chips. But it would be a huge difficult project, and too many other projects beckon. But basically it would be computation designed on the hardware level around strings of constantly changing length. Maybe the OS would just be a Markov machine emulator.
I wrote a C program that does this at the bit level in computer memory. That got me excited. But I use pointers to unsigned 64 bit integers. So it’s not like the hardware is designed for Markov. The hardware likes 64 bit chunks. In practice it’s more than fast enough, so it’s an aesthetic itch to have my own hardware for this.
This is it. Well said.
This is the guy I got that from talking about it:
He also wrote one of the most famous books on computation.
He has a great easy-going funny way to sharing his ideas. You can that his topic brings him joy. I can see why.
The more real “infinity” is the unpredictable “poetry” that pours out of us constantly.
Looks interesting video on the topic. Will watch it in my quiet hours. Thanks for sharing.
We can explore a different face of the same issue.
The vague initial question is: How many subsets of the natural numbers are there ?
How me even understand this question depends on our background and our philosophy of mathematics. I have to present the following mostly in the usual language, whether I have issues with some terms or not.
I think most positions will grant that there “at least” a “countably infinite” number of subsets. We can simply consider the bijection f(n) = \{n\}. This is just us considering each positive integer as a singleton subset. Of course we have the even numbers, the odd numbers, the prime numbers also. So on top of these singleton sets we have many infinite sets.
Do we have uncountably many ? The standard answer is yes. Every (infinite) sequence that outputs bits encodes a subset.
If we understand the natural numbers to start at 1, then the even numbers are encoded as 0,1,0,1,0,1,…
This sets up the proof. Any list of such sequences can be diagonalized in basically the most aesthetically pleasing and prototypical way. Rudin offers this particular set of all sequences of bits as the first example of an uncountable set in his famous PMA.
OK then, so what’s the problem ?
The set of all computable characteristic functions — computable sequences of bits — is countable. This set includes all “Turing machines” and not only the “good” ones that pick out a subset of the natural numbers.
The “good Turing machines” are a subset of the countable set of all Turing machines. So there are at most a “countable infinity” of computable subsets of the natural numbers. There are also at least that many, because it’s easy to design the machines that characterize the singleton sets mentioned above.
Since the computable subsets of the natural numbers are countably infinite, “most” of the subsets of the natural numbers are not computable. In the standard approach, this “most” is very strong. Think of an ocean of darkness dotted by the tiniest little stars.
My philosophical/aesthetic issue is that an “uncomputable” subset is an undefined subset. If you can specify a subset objectively —without ambiguity — then it is computable. So all of these “uncomputable sets” are a vague blur or fume that comes out of linguistic reasoning.
One beautiful attempt to save the continuum without leaning on this blur was Brouwer’s “choice sequences.” Basically we can think of a sequence of bits as an always-in-progress sequences-in-progress of “free choices.” If we don’t have a rule, we just decide on the next bit whenever we feel like it, if ever. The sequence is becoming but never finally fully arrives. Functions on these in-progress-sequences are of course themselves such in-progress sequences. We can put an alphabetical order on them. In this context, the subset of the natural numbers is being created or determined “in time.”
Instead of shining a light on what is already there, we create by fiat, but without the full power of a god. We never “complete” our “infinite” sequence.
So I have used “good” program more than once. But we don’t have a “computably objective” predicate or “good goodness-tester.” For that would be a “halts or not” predicate.
Some programs are so obviously “good” that we don’t feel the lack of a computable predicate. We have no doubt to settle.
Is this the essence ? If so, then “total function” must remain an informal notion. This just hit me as an implication, so maybe someone will show me where I am wrong.
A “Turing machine that always halts” is an informal notion. We have vivid important examples in the informal sense. I even see such informal examples as maybe the core of math. So the core of math is intuitive or informal ?
Why not just say plainly that ‘there is no transcendental signified’?
Mathematical theorems are not always important in themselves. They often serve as necessary steps in the proofs of other theorems. As far as I know, the Riemann hypothesis concerning the nontrivial zeros of the zeta function can be regarded as one such theorem. There is currently no proof of this theorem, but most mathematicians assume that it is true, and proving it would make the proofs of many other theorems rigorous. Therefore, some theorems may well be true even if we do not have a direct proof of them.
Not at all. I’ve long supported a more creative approach. I hope what I’ve had to say had the advantage of being at least cogent. What you have here is more of a just-so story. Find someone else to be your antagonist.
Most people would not call that plain language. Because the signs “show up differently” for different people, the “monological” approach does not make sense. I avoid forgetting my own point when I choose signs for this or that always particular dialogue.
We do not want to “reduce” Derrida to the naive version of “logical” positivism. To be honestly empirical about language is to feel the burden and risk in trading signs, to feel the “impossibility” of saying it plainly.
The signs you speak of are on the grammatical level, right along with Turing and Gödel.
The Transcendental sign is prior and ontologically universal to the signs that merely represent or obscure it.
Identity
Distinction
Exclusion
It’s possible that we are on the same-ish page.
I interpret Plato’s unwritten doctrine as a “primitive ontology.”
Basically it’s a theory of the thing, the thing in general.
A thing is a unity of differences. The distinct “face” of the thing in this moment excludes the specificity of the other “faces” of the thing.
The object is only here in a specific way by not being here in other specific ways.
The “identity” of the object is the temporal synthesis of all of these distinct “faces” that exclude one another.
One of the difficulties is that this phenomenology can sound like something mystical when it’s only an attempt to point at the mundane structure of shared objects that we constantly co-enact. I was trying to point with this above with the image of numerals.
To be a specific numeral is to not be the others in their differing specificity. But the numerals are “unified” through our enacting the unimportance of these differences. We “glue the numerals together in time” to enact the number.
Here’s Husserl with a great spatial example of this “temporal-gluing.”
“External perception is a constant pretension to accomplish something that, by its very nature, it is not in a position to accomplish. Thus, it harbors an essential contradiction, as it were. My meaning will soon become clear to you once you intuitively grasp how the objective sense exhibits itself as a unity in the unending manifolds of possible appearances; and seen upon closer inspection, how the continual synthesis, as a unity of coinciding, allows the same sense to appear, and how a consciousness of ever new possibilities of appearance constantly persists over against the factual, limited courses of appearance, transcending them.”
“Let us begin by noting that the aspect, the perspectival adumbration through which every spatial object invariably appears, only manifests the spatial object from one side. No matter how completely we may perceive a thing, it is never given in perception with the characteristics that qualify it and make it up as a sensible thing from all sides at once. We cannot avoid speaking of such and such sides of the object that are actually perceived. Every aspect, every continuity of single adumbrations, regardless how far this continuity may extend, offers us only sides. And to our mind this is not just a statement of fact: it is inconceivable that external perception would exhaust the sensible-material content of its perceived object; it is inconceivable that a perceptual object could be given in the entirety of its sensibly intuitive features, literally, from all sides at once in a self-contained perception” (Husserl, Analyses Concerning Passive and Active Synthesis, pp. 39-40).
The spatial object cannot be squeezed into a single “now.”
I like to use “moments” for a generalization of “adumbrations” or “aspects” that applies in non-visual contexts.
To me the issue is unpacking this.
The two basic strategies seem to be either away from or toward empirical-life.
The word “true” is one of most nebulous signs in philosophy. In ordinary life, it looks, in visceral terms, like an endorsement of comportment. One primate supports or inhibits the directionality of another.
We can say that a conjecture was “really a theorem all along” when a proof is accepted.
To me the goal here isn’t the “truth” about the situation as an “internal stuff” in our signs that “mirrors” an “external stuff.”