ChatGPT tells me that PA+"PA is consistent" is not quite enough. I believe that it has digested enough logic textbooks that I'll believe that claim.
> I think just PA+"PA is consistent" is enough?
It's not clear to me how. I believe PA+"PA is consistent" would allow a model where Goodstein's theorem is true for the standard natural numbers, but that also contains some nonstandard integer N for which Goodstein's theorem is false. I think that's exactly the case that's ruled out by the stronger statement of ω-consistency.
On the first question, how do you encode omega-consistency as a formula of PA? Just curious, it's not at all obvious to me.
I was also wondering about that, but going by the Wikipedia definition, it doesn't seem too complicated: you say, "For all encoded propositions P(x): If for every x there exists an encoded proof of P(x), then there does not exist an encoded proof of 'there exists an x such that ¬P(x)'." That is, if you can prove a proposition for each integer in the metatheory, then quantifiers within the target theory must respect that.
https://en.wikipedia.org/wiki/%CE%A9-consistent_theory#Relat...
The Wikipedia article says T is omega-consistent if "T + RFN_T + the set of all true sentences is consistent", which should mean the same thing as "T + RFN_T is true".
The work linked here doesn't show that PA is inconsistent, however: what it does is to define a new, weaker notion of what it means for PA to “prove its own consistency” and to show that PA can do that weaker thing.
Interesting work for sure, but it won't mean anything to you unless you already know a lot of logic.
That is: Real numbers describe real, concrete relations. For e.g., saying that Jones weighs 180.255 pounds means there's a real, physical relationship -- a ratio -- between Jones' weight and the standard pound. Because both weights exist physically, their ratio also exists physically. Thus, from this viewpoint, real numbers can be viewed as ratios.
In contrast, the common philosophical stance on numbers is that they are abstract concepts, detached from the actual physical process of measurement. Numbers are seen as external representations tied to real-world features through human conventions. This "representational" approach, influenced by the idea that numbers are abstract entities, became dominant in the 20th century.
But the 20th century viewpoint is really just one interpretation (you could call it "Platonic"), and, just as it's impossible to measure ratios to infinite precision in the real world, absolutely nothing requires an incomputable continuum of reals.
Physics least of all. In 20th and 21st century physics, things are discrete (quantized) and are very rarely measured to over 50 significant digits. Infinite precision is never allowed, and precision to 2000 significant digits is likewise impossible. The latter not only because quantum mechanics makes it impossible to attain great precision on very small scales. For e.g., imagine measuring the orbits of the planets and moons in the solar system: By the time you get to 50 significant digits, you will need to take into account the gravitational effects of the stars nearest to the sun; before you get to 100 significant digits, you'll need to model the entire Milky Way galaxy; the further you go in search of precision, the exponentially larger your mathematical canvas will need to grow, and at arbitrarily high sub-infinite precision you’d be required to model the whole of the observable universe -- which might itself be futile, as objects and phenomena outside observable space could affect your measurements, etc. So though everything is in principle simulatable, and precision has a set limit in a granular universe that can be described mathematically, measuring anything to arbitrarily high precision is beyond finite human efforts.
Rationals > irrationals on computing them. You can always approximate irrationals with rationals, even Scheme (Lisp, do'h) has a function to convert a rational to decimal and the reverse. decimal to rational.
This entire discussion is about mathematical concepts, not physical ones!
Sure, yes, in physics you never "need" to go past a certain number of digits, but that has nothing to do with mathematical abstractions such as the types numbers. They're very specifically and strictly defined, starting from certain axioms. Quantum mechanics and the measurability of particles has nothing to do with it!
It's also an open question how much precision the Universe actually has, such as whether things occur at a higher precision than can be practically measured, or whether the ultimate limit of measurement capability is the precision that the Universe "keeps" in its microscopic states.
For example, let's assume that physics occurs with some finite precision -- so not the infinite precision reals -- and that this precision is exactly the maximum possible measurable precision for any conceivable experiment. That is: Information is matter. Okay... which number space is this? Booleans? Integers? Rationals? In what space? A 3D grid? Waves in some phase space? Subdivided... how?
Figure that out, and your Nobel prize awaits!
There are plenty of ratios that are ratios of other things than integers, so they are not rational numbers.
Ratios of collinear vectors are scalars a.k.a. "real" numbers, ratios of other kinds of vectors are matrices, ratios of 2D-vectors are "complex" numbers, ratios of 2 voltages are scalars a.k.a. "real" numbers, and so on.
In general, for both multiplication and division operations, the 3 sets corresponding to the 2 operands and to the result are not the same.
Only for a few kinds of multiplications and of divisions the 3 sets are the same. This strongly differs from addition operations, which are normally defined on a single set to which both the operands and the result belong.
In practice, multiplications and divisions where at least one operand or the result belong to another set than the remaining operands or result are extremely frequent. Any problem of physics contains such multiplications and divisions.
This requirement is necessary to allow the definition of a division operation, which has as operands a pair of values of that physical quantity, and as result a scalar a.k.a. "real" number. This division operation, as you have noticed, is called "measurement" of that physical quantity. A value of some physical quantity, i.e. the dividend in the measurement operation, is specified by writing the quotient and the divisor of the measurement, e.g. in "6 inches", "6" is the quotient and "inch" is the divisor.
In principle, this kind of division operation, like any division, could have its digit-generating steps executed infinitely, producing an approximation as close as desired for the value of the measured quantity, which is supposed to be an arbitrary scalar, a.k.a. "real" number. Halting the division after a finite number of steps will produce a rational number.
In practice, as you have described, the desire to execute the division in a finite time is not the only thing that limits the precision of the measured values, but there are many more constraints, caused by the noise that could need longer and longer times to be filtered, by external influences that become harder and harder to be suppressed or accounted for, by ever greater cost of the components of the measurement apparatus, by the growing energy required to perform the measurement, and so on.
Nevertheless, despite the fact that the results of all practical measurements are rational numbers of low precision, normally representable as FP32, with only measurements done in a few laboratories around the world, which use extremely expensive equipment, requiring an FP64 or an extended precision representation, it is still preferable to model the set of scalars using the traditional axioms of the continuous straight line, i.e. of the "real" numbers.
The reason is that this mathematical model of a continuous set is actually much simpler than attempting to model the sets of values of physical quantities as discrete sets. An obvious reason why the continuous model is simpler is that you cannot find discretization steps that are good both for the side and for the diagonal of a square, which has stopped the attempts of the Ancient Greeks to describe all quantities as discrete. Already Aristotle was making a clear distinction between discrete quantities and continuous quantities. Working around the Ancient Greek paradox requires lack of isotropy of the space, i.e. discretization also of the angles, which brings a lot of complications, e.g. things like rigid squares or circles cannot exist.
The base continuous dynamical quantities are the space and time, together with a third quantity, which today is really the electric voltage (because of the convenient existence of the Josephson voltage-frequency converters), even if the documents of the International System of Units are written in an obfuscated way that hides this, in an attempt to preserve the illusion that the mass might be a base quantity, like in the older systems of units.
In any theory where some physical quantities that are now modeled as continuous, were modeled as discrete instead, the space and time would also be discrete. There have been many attempts to model the space-time as a discrete lattice, but none of them has produced any useful result. Unless something revolutionary will be discovered, all such attempts appear to be just a big waste of time.
The Heisenberg uncertainty principle is just a trivial consequence of the properties of the Fourier transform. It just states that there are certain pairs of physical quantities which are not independent (because their probability densities are connected by a Fourier transform relationship), so measuring both simultaneously with an arbitrary precision is not possible.
The Heisenberg uncertainty principle says absolutely nothing about the measurement of a single physical quantity or about the simultaneous measurement of a pair of independent physical quantities.
There is no such thing as a quantity that cannot be measured, i.e. the set of its values does not have the required Archimedean group algebraic structure. If it cannot be measured, it is not a quantity (there are also qualities, which can only be compared, but not measured, so the sets of their values are only ordered sets, not Archimedean groups; an example of a physical quality, which is not a physical quantity, is the Mohs hardness, whose numeric values are just labels attached to certain values, a Mohs hardness of "3" could have been as well labeled as "Ktcwy" or with any other arbitrary string, the numeric labels have been chosen only to remember easy the order between them).
Various proposals exist in which the fundamental physical structure of space form a kind of "quantum foam". And so, somewhere near the Planck length, it doesn't even make sense to talk about distance.
What you have in mind about "ranges of values" has nothing to do with Heisenberg, but with the formalism of quantum mechanics as developed by Schroedinger, Dirac and others (not counting the matrix mechanics of Heisenberg, which was an inferior mathematical method, soon forgotten after superior alternatives were developed, and which is something completely else than the Heisenberg principle of uncertainty).
In the various variants of the formalism of quantum mechanics, the correspondent of a single value in classic dynamics is a function, the so-called wave function, but even those functions cannot be called "ranges of values". There are several interpretations of what the "wave" functions mean, but the most common is that they are probability densities for the values of the corresponding physical quantity.
Depending on the context, in quantum mechanics the value of a physical quantity may be unknown, when only the probability of it having various values is known, but the value may also be known with absolute certainty, usually in some stationary states.
In both cases the value of the physical quantity can be measured, but in the former only an average value can be measured, which nonetheless may have unlimited precision, while in the latter case the exact value can be measured, exactly like in classical dynamics.
Granularity is implied by some, but not all, post standard model physics. It's a very open question.
> You can measure any value to arbitrary precision
Quantum mechanics lets you refine one observable indefinitely if you are willing to sacrifice conjugate observables.
You can measure that one observable to an arbitrarily (not infinitely!) high precision -- if you have an correspondingly arbitrary duration of time and arbitrarily powerful computational resources. That measurement of yours, if exceedingly precise, might require a block of computronium which utilizes the entire energy resources of the universe. As a practical matter, that's not permitted. I'm certainly unaware of any measurement in physics to more than 100 significant digits, let alone "arbitrary precision".
In fact, as it turns out, no experiment has ever resolved structure down to the Planck length. A fundamental spatial resolution is likely to be quite a lot smaller than the Planck length; even the diameter of the electron has been estimated at anywhere from 10^-22m to 10^-81m.
The question I was responding to asked whether reals are "necessary" -- plainly, insofar as reality is concerned, they are not.
My concern with the reals goes the other direction. The reals are not closed under common operations. A lot of the work is done in complex numbers, which are. The rationals are not closed under radicals, and radicals seem pretty fundamental.
You can define a finitist model despite this. But it's ugly, and while ugliness is not physically meaningful, it tends to make progress difficult. A usable solution may yet arise; we shall (perhaps) see.
Now, are the reals necessary?
During the last decades, there have been published many research papers exploring the use of discreteness instead of the traditional continuity, but they cannot be considered as anything else but failures. The resulting mathematical models are much more complicated than the classic models, without offering any extra predictions that could be verified.
The reason for the complications is that in physics it would be pointless to try to model a single physical quantity as discrete instead of continuous. You have a system of many interrelated quantities and trying to model all of them as discrete reaches quickly contradictions for the simpler models, due to the irrational or transcendent functions that relate some quantities, functions that appear even when modeling something as simple as a rotation or oscillation. If, in order to avoid incommensurability, you replace a classic continuous uniform rotation with a sequence of unequal jumps in angular orientation, to match the possible directions of nodes in a discrete lattice, then the resulting model becomes extremely more complicated than the classic continuous model.
Everything that is known about numerical analysis is based on discreteness, instead of continuity. Every useful predictive model that we have about the world, for example for forecasting weather, depends on numerical analysis.
I consider this a success!
Forecasting weather, designing semiconductor devices, or any other such activities are all based on continuous mathematical models, which use e.g. systems of equations with partial derivatives or systems of integral equations.
Only in the last stage of solving such a problem, in order to use a digital computer the continuous mathematical model is approximated by a discrete mathematical model, which is constructed using a standard method, e.g. finite elements, boundary elements, finite differences, most of which are based on approximating an unknown function with an infinity of degrees of freedom with a function determined by a finite number of parameters, then by approximating the operations required to compute those parameters by operations with floating-point numbers.
Such an approximating method is something extremely different from formulating a discrete mathematical model of physics that is considered the exact model.
Even the graphics for a game are based on continuous models of space, not on discrete models, where it would be very difficult to implement something like the rotation of an object or perspective views.
The failures to which I have referred are the attempts to create such discrete models of physics, e.g. where the space and time are discrete not continuous.
These attempts have nothing in common with the approximating techniques, which are indeed the base for most successes in using digital computers.
Time-scale calculus is a pretty niche theoretical field that looks at blending the analysis of difference and differential equations, but I'm not aware of any algorithmic advances based on it.
In a sense this is wrong. Some say the indispensability of modern math, which includes the incomputable reals, shows that such abstract objects are required for science.
Of course not, but that would invalidate the entire project of some/many here to turn reality into something clockwork they can believe they understand. Reality is much more interesting than that.
It all gets mind-bendingly mind-bending really fast, especially with people like Harvey Friedman ( https://u.osu.edu/friedman.8/foundational-adventures/fom-ema... ) or the author of this post trying to break the math by constructing an untractably large values with the simplest means possible (thus showing that you can encounter problems that do not fit in the universe even when working in a "simple" theory)
(i saw the username and went to check audiomulch website, but it did not resolve :( )
The idea that uncountable means more comes from a bad metaphor. See https://news.ycombinator.com/item?id=44271589 for my explanation of that.
Accepting that uncountable means more forces us to debatable notions of existence. See https://news.ycombinator.com/item?id=44270383 for a debate over it.
But, finally, there is this. Every chain of reasoning that we can ever come up with, can be represented on a computer. So even if you wish to believe in some extension of ZFC with extremely large sets, PA is capable of proving every possible conclusion from your chosen set of axioms. So yes, PA is enough.
If you're not convinced, I recommend reading https://www.amazon.com/G%C3%B6del-Escher-Bach-Eternal-Golden....
I imagine their reputation problem (nobody wants to post things there because it just gets deleted) plus the 15 point requirement precludes many people from upvoting over there.
It was interesting because they were asking "is this all you need? What about the Peano axioms? Is there anything more primitive than the inductive data type?"
I bring it up because it's good not to take for granted that the Peano axioms are inherent, instead of just one design among many.
I believe that the natural numbers are more primitive than inductive data types, since all inductive data types may be constructed from the natural numbers alongside a small number of primitive type formers (e.g. Π, Σ, = and Ω).
I think there are two primitive sets for dependent type theory. The one with omega, and then the one with inductive types. None of them need axioms like the Peano axioms.
"That's because there are nonstandard models of PA which contain infinite natural numbers. So PA may be able to prove that it produces a proof, but can't prove that the proof is of finite length. And therefore it might not be a valid proof."
If PA proves "PA proves X" then PA can prove X. This is because the key observation is not that there are nonstandard models, but rather that the standard natural numbers model PA.
Therefore if PA proves "PA proves X", then there is in fact a standard, finite natural number that corresponds to the encoded proof of "PA proves X". That finite natural number can be used then to construct a proof of X in PA.
My logic is very rusty at this point, but if someone could give me an argument that you cannot move the 'Provable' outside the 'forall', I would really appreciate that, without making reference to Goodstein sequences. In other words, that in general for propositions 'P' it is not true that proving "forall n, Provable(P(n))" implies you can prove "Provable(forall n, P(n))".
Not true.
From PA we can construct a function that can search all possible proofs that can be constructed in PA. In fact I outlined one way to do this at the end of my answer.
With this function, we can construct a function will-return that analyzes whether a given function, with a given input, will return. This is kind of like an attempted solution to the Halting Problem. We know that it doesn't always work. But we also know that it works a lot of the time.
From will-return we can create a function opposite-return that tries to return if a given function with a given input would not, and doesn't return if that function would. This construction is identical to the one in the standard proof of the Halting Problem.
Now we consider (opposite-return opposite-return opposite-return). (Actually you need a step to expand the argument into this recursive form. I've left that out, but that is identical to the one in the standard proof of the Halting problem.)
PA can prove the following:
- PA proves that if PA can prove that opposite-return returns, then it doesn't. - PA proves that if PA can prove that opposite-return doesn't return, then it does. - PA proves that if it can prove everything that it proves that it can prove, then PA must have a proof of one of the two previous statements. - Therefore PA proves that if it can prove everything that it proves that it can prove, then PA is inconsistent.
This is a form of Gödel's second incompleteness theorem.
And, therefore, there must be a distinction to be made between "PA proves" and "PA proves that it proves".
If we assume PA to be sound, then indeed everything it proves is true.
> Not true.
Now you're saying PA is unsound.
But your article wasn't about PA proves "PA proves X", it was about "forall n : PA proves G(n)".
For PA not to prove "forall n: G(n)", there is no soundness issue, only a ω-consistency issue.
If PA proves that it proves a statement, PA cannot conclude from that fact that it proves that statement.
If PA proves that it proves a statement, and then fails to prove it, PA is unsound.
There exist collections of statements such that PA proves that it proves each statement, and PA does prove each statement, but PA does not prove the collection of statements.
Our understanding of the last is helped by understanding that "PA proves that PA proves S" is logically not the same statement as, "PA proves S". Even though they always have the same truth value.
I don't believe this is true. I don't know what result you're using here, but I think you're mixing up "provable" and "true".
In particular your line of reasoning violates Lob's Theorem, which is a corollary of the second incompleteness theorem.
Boyer and Moore wrote an automated theorem prover that goes with this theory. I have a working copy for GNU Common LISP at [2].
"It is perhaps easiest to think of our program much as one would think of a reasonably good mathematics student: given the axioms of Peano, he could hardly be expected to prove (much less discover) the prime factorization theorem. However, he could cope quite well if given the axioms of Peano and a list of theorems to prove (e.g., “prove that addition is commutative,” . . . “prove that multiplication distributes over addition,” . . . “prove that the result returned by the GCD function divides both of its arguments,” . . . “prove that if the products over two sequences of primes are identical, then the two sequences are permutations of one another”)." - Boyer and Moore
btilly•1d ago
It covers both the limits of what can be proven in the Peano Axioms, and how one would begin bootstrapping Lisp in the Peano Axioms. All of the bad jokes are in the second section.
Corrections and follow-up questions are welcome.
Cheyana•1d ago
burnt-resistor•12h ago
lmpdev•11h ago
Can we not ruin every technology we develop with ads?
ccppurcell•11h ago
blipvert•6h ago
brookst•10h ago
smoyer•7h ago
Incipient•6h ago
Also is this somewhat not just lining Google's/ad networks pockets? They won't care about even a large-ish number of people using this add on.
0_gravitas•5h ago
The larger adtech companies will have some form of bot-activity-detection, _but_ plenty will let it all just slip on by. (I've personally written a Babashka script at $job that checks performance of our publishers, and flags any fraudulent-seeming activity)
chii•6h ago
tshaddox•6h ago
louthy•6h ago
Fewer people went to pay. That’s the crux. Companies wanting mass market penetration, rather than just accept the natural size of the market.
im3w1l•12h ago
kjellsbells•8h ago
Very reminiscent of the New Math pedagogy of the 1960s. Built up arithmetic entirely from sets. Crashed and burned for various reasons but I always had a soft spot for it. It also served as my introduction to binary arithmetic and topology.
Art9681•9h ago
nemomarx•8h ago
gjm11•1h ago
(I am not suggesting that that very specific sequence of events happened in this case. It's just an example of how "I was looking this up for other reasons and then I saw something about it on HN" could genuinely happen more often than by chance, without any need for weird tracking-and-advertising shenanigans.)
RcouF1uZ4gsC•8h ago
billyjmc•4h ago
anthk•12h ago
https://justine.lol/sectorlisp2/
Also, lots of Lisp from https://t3x.org implement numerals (and the rest of stuff) from cons cells and apply/eval:
'John McCarthy discovered an elegant self-defining way to compute the above steps, more commonly known as the metacircular evaluator. Alan Kay once described this code as the "Maxwell's equations of software". Here are those equations as implemented by SectorLISP:
ASSOC EVAL EVCON APPLY EVLIS PAIRLIS '
Ditto with some Forths.
Back to T3X, he author has Zenlisp where the meta-circular evaluation it's basically how to define eval/apply and how to they ared called between themselves in a recursive way.
http://t3x.org/zsp/index.html
btilly•4h ago
When it comes to bootstrapping a programming language from nothing, the two best options are Lisp and Forth. Of the two, I find Lisp easier to understand.
ikrima•9h ago
Then every modal logic becomes some mapping of 2^(N) to some set of statements. The only thing that matters is how predictive they are with some sort of objective function/metric/measure but you can always induce an "ultra metric" around notions of cognitive complexity classes i.e. your brain is finite and can compute finite thoughts/second. Thus for all cognition models that compute some meta-logic around some objective F, we can motivate that less complex models are "better". There comes the ultra measure to tie disparate logic systems. So I can take your Peano Axioms and induce a ternary logic (True, False, Maybe) or an indefinite-definite logic (True or something else entirely). I can even induce bayesian logics by doing power sets of T/F. So a 2x2 bayesian inference logic: (True Positive, True Negative, False Positive, False Negative)
Fun stuff!
Edit: The technical tldr that I left out is unification all math imho: algebraic topology + differential geometry + tropical geometry + algebraic analysis. D-modules and Microlocal Calculus from Kashiwara and the Yoneda lemma encode all of epistemology as relational: either between objects or the interaction between objects defined as collision less Planck hyper volumes.
basically encodes the particle-wave duality as discrete-continuum and all of epistemology is Grothendieck topoi + derived categories + functorial spaces between isometry of those dual spaces whether algebras/coalgebra (discrete modality) or homologies/cohomologies (continuous actions)
Edit 2: The thing that ties everything together is Noether's symmetry/conserved quantities which (my own wild ass hunch) are best encoded as "modular forms", arithmetic's final mystery. The continuous symmetry I think makes it easy to think about diffeomorphisms from different topoi by extracting homeomorphisms from gauge invariant symmetries (in the discrete case it's a lattice, but in the continuous we'd have to formalize some notion of liquid or fluid bases? I think Kashiwara's crystal bases has some utility there but this is so beyond my understanding )
ricardobeat•9h ago
There’s probably ten+ years of math education encoded in this single sentence?
ikrima•9h ago
Wish me luck!
ikrima•7h ago
https://github.com/ikrima/topos.noether/blob/master/discrete...
bobsh•4h ago
gjm11•1h ago
[EDITED to add:] This is worth noting because today's LLMs really don't seem to understand mathematics very well. (This may be becoming less so with e.g. o3-pro and o4, but I'm pretty sure that document was not written by either of those.) They're not bad at pushing mathematical words around in plausible-looking ways; they can often solve fairly routine mathematical problems, even ones that aren't easy for humans who unlike the LLMs haven't read every bit of mathematical writing produced to date by the human race; but they don't really understand what they're doing, and the nature of the mistakes they make shows that.
(For the avoidance of doubt, I am not making the tired argument that of course LLMs don't understand anything, they're just pattern-matching, something something stochastic parrots something. So far as I can tell it's perfectly possible that better LLMs, or other near-future AI systems that have a lot in common with LLMs or are mostly built out of LLMs, will be as good at mathematics as the best humans are. I'm just pretty sure that they're still some way off.)
(In particular, if you want to say "humans also don't really understand mathematics, they just push words and symbols around, and some have got very good at it", I don't think that's 100% wrong. Cf. the quotation attributed to John von Neumann: "Young man, in mathematics you don't understand things, you just get used to them." I don't think it's 100% right either, and some of the ways in which some humans are good at mathematics -- e.g., geometric intuition, visualization -- match up with things LLMs aren't currently good at. Anyway, I know of no reason why AI systems couldn't be much better at mathematics than the likes of Terry Tao, never mind e.g. me, but they aren't close enough to that yet for "hey, ChatGPT, please evaluate my speculation that we should be unifying continuous and discrete mathematics via topoi in a way that links aleph, beth and Betti numbers and shows how our brains nucleate discrete samples of continuum reality" to produce output that has value for anything other than inspiration.)
gjm11•1h ago
Aleph and beth numbers are related things, in the field of set theory. (Two sequences[1] of infinite cardinal numbers. The alephs are all the infinite cardinals, if the axiom of choice holds. The beth numbers are the specific ones you get by repeatedly taking powersets. They're only all the cardinals if the "generalized continuum hypothesis" holds, a much stronger condition.)
[1] It's not clear that this is quite the right word, but no matter.
Betti numbers are something totally different. (If you have a topological space, you can compute a sequence[2] of numbers called Betti numbers that describe some of its features. (They are the ranks of its homology groups. The usual handwavy thing to say is that they describe how many d-dimensional "holes" the space has, for each d.)
[2] This time in exactly the usual sense.
It's not quite true that there is no connection between these things, because there are connections between any two things in pure mathematics and that's one of its delights. But so far as I can see the only connections are very indirect. (Aleph and beth numbers have to do with set theory. Betti numbers have to do with topology. There is a thing called topos theory that connects set theory and topology in interesting ways. But so far as I know this relationship doesn't produce any particular connection between infinite cardinals and the homology groups of topological spaces.)
I think ikrima's sentence is mathematically-flavoured word salad. (I think "Betti" comes after "beth" mostly because they sound similar.) You could probably take ten years to get familiar with all the individual ideas it alludes to, but having done so you wouldn't understand that sentence because there isn't anything there to understand.
BUT I am not myself a topos theorist, nor an expert in "our human brain's sensory instruments". Maybe there's more "there" there than it looks like to me and I'm just too stupid to understand. My guess would be not, but you doubtless already worked that out.
[EDITED to add:] On reflection, "word salad" is a bit much. E.g., it's reasonable to suggest that our senses are doing something like discrete sampling of a continuous world. (Or something like bandwidth-limited sampling, which is kinda only a Fourier transform away from being discrete.) But I continue to think the details look more like buzzword-slinging than like actual insight, and that "aleph/beth/Betti" thing really rings alarm bells.
ikrima•3m ago
ikrima•9h ago
https://github.com/ikrima/topos.noether/blob/aeb55d403213089...
doodlebugging•6h ago
In the "Why Lisp?" section there is a bit of basic logic defined. The first of those functions appears to have unbalanced parentheses.
>(defun not (x) > (if x > false > true)
I have a compulsion that I can't control when someone starts using parentheses. I have to scan to see who does what with who.
You later say in the same section
>But it is really easy to program a computer to look for balanced parentheses
Okay. This is pretty funny. Thanks for the laugh. I realize that you weren't doing that but it still is funny that you pointed out being able to do it.
This later comment in the "Basic Number Theory" section was also funny.
>; After a while, you stop noticing that stack of closing parens.
I really enjoyed reading this post. Great job. Though it has been a long time since I did anything with Lisp I was able to walk through and get the gist again.
kazinator•5h ago
Are you saying that parentheses introduce the problem of having to scan to see what goes with what?
As in, if we don't have parentheses, but still have recursive/nested structure, we don't have to scan?
Retr0id•5h ago
User23•4h ago
doodlebugging•2h ago
I automatically count and match even today, 40 years later.
btilly•4h ago
And I'm glad that someone liked some of my attempts at humor. As my wife sometimes says, "I know that you were trying to make a joke, because it wasn't funny."
whatagreatboy•4h ago
Brian_K_White•2h ago
doodlebugging•2h ago
Timwi•2h ago
There are two places where you accidentally wrote “omega” instead of “\omega”.
btilly•2h ago
I am away from my computer for the day, but I will fux it later.
gjm11•1h ago
I think the problem is that it's already fuxed.
edanm•1h ago
I also find it super weird that Peano axioms are enough to encode computation. Again, this might be trivial if you think about it, but that's one self-referential layer more than I've thought about before.
One question for you btilly - oddly enough, I just recently decided to learn more Set Theory, and actually worked on an Intro to Set Theory textbook up to Goodstein sequences just last week. I'm a bit past that.
Do you have a good recommendation for a second, advanced Set Theory textbook? Also, any recommendation for a textbook that digs into Peano arithmetic? (My mini goal since learning the proof for Goodstein sequences is to work up to understanding the proof that Peano isn't strong enough to prove Goodstein's theorem, though I'll happily take other paths instead if they're strongly recommended.)