About math limitations
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Materialistic monism is a popular but not particularly robust view in my opinion, because those models can never be complete and always evolve in directions guided by the applications and predictions our minds seek. (Other views are available.)
I would encourage more people to apply the same principle at the higher level, changing “our brains are limited by physics, hence our mathematics is limited by physics” to “our model of reality, including physics, is limited due to our minds operating within that reality”. In other words, the model that makes you think your consciousness comes from neurones and is thus limited by physics is itself a product of your consciousness, and conclusions you make based on that model (such as “mathematics is limited by physics”) should be taken with a grain of salt (not just as possibly wrong but, worse, not-even-wrong) because you can’t construct a complete model of a system from within said system.
But the kicker is that this logic itself is rooted in mathematical reasoning (see Gödel’s incompleteness theorem), so take it for what you will.
If you start under the assumption that materialism is true you have to also eat the idea that free will doesn't exist, and that you can't trust any of your thoughts since they are: 1. Out of your control, since it's all physics. 2. Created by a brain that evolved to survive, not to create logic.
Thus the idea of materialism itself would be just an illusion created by your brain's need to survive.
This implies it is a self-defeating worldview from the get-go - because you've reached a contradiction: the brain generates the thoughts in a deterministic fashion and you cannot trust your brain, so you cannot trust the thoughts you have about the material reality including materialism.
And the armchair philosophers misunderstand Godel literally all the time, including this time. All it says is that there are statements that are true that cannot be proven within the constraints of a finite set of axioms. It's not a limitation of math, it's a limitation of what can be proven true within the confines of a set of axioms.
Meaning there could be another set of axioms under which the limitation to prove that such a statement is true disappears. But it doesn't make math any less true.
This article was written by someone with very poor philosophical training.
Isn't your argument an example of Appeal to Consequences?[1] It appears that you don't accept materialism not because you can show that it's incorrect, but because you don't like the implications?
In addition to it, I personally reject materialism because it fails to describe the universe properly, in particular the Big Bang.
The Big Bang is the beginning of all material universe - creatio ex nihilo. However, if there was a mathematical structure that sets the framework for the physical laws, there's a reality that transcends the material universe and exists prior to it. It's the only logically consistent worldview in my opinion.
This has even further implications but, you know, you can figure it out yourself.
Why we assume that an elegant model has a higher chance of being correct, or is otherwise somehow better, in the first place—it seems to be pretty much implied that beauty and elegance are desirable traits of a model—is a good and on topic question, but perhaps too much for this particular thread.
“At first there was nothing, then out of nothing the Big Bang happened and the universe began” is a popular misconception of the Big Bang theory. It’s completely wrong.
The Big Bang theory does not attempt to explain what things were like “before” the Big Bang. It’s not a theory of creation at all! It only offers a model starting with the inflationary epoch, when the universe was already expanding rapidly.
Prior to that, our existing theories break down and we need a different physical theory (commonly called the Grand Unified Theory or GUT). We don’t have it so we can go no further back.
The point is: back in the inflationary epoch the universe was all there, it was just much smaller. The point of the theory is to try to explain the extreme uniformity (isotropy) in the distribution of galaxies, gas, dust, and light we observe in the universe today (among other things, such as dark matter). It’s a much better model in terms of its simplicity and predictive power than any steady-state model of the universe that has been proposed.
But it’s got nothing to do with creation!
All I'm claiming is that before whatever happened, there must have been a mathematical structure that allowed it to happen following certain physical laws. That's just a fact. So these mathematical laws exist before anything. This implies a realm that's not physical. Anything that is contingent such as all material things (why is there anything at all?) must have a prior cause - and that's an assumption of science.
All theories presuppose these mathematical structures.
That is, even if the current big bang is just one in a series of several, or however else you try to explain existence, you have to presuppose the mathematical structure that forces it to follow certain physical laws.
I also don't think you fully understand inflation and why it's been proposed but I'll ignore it since it's not useful for the conversation. Also read up on the Borde-Guth-Vilenkin theorem as it basically proves that a beginning is necessary.
In addition, note that right now it seems almost certain that the universe had a beginning and any other explanation is not evidence based.
No,
> That's, like, just your opinion man ;)
We don’t know that they do. We assume they do, but our laws are not their laws.
You’re like the person who sees a birdwatcher recording their observations and asking “why do birds exist?” You’re asking the wrong question to the wrong person and demanding satisfaction.
You might want to study a bit of philosophy of mathematics, particularly the bit on the realism vs anti-realism debate. A great deal of people fall into the anti-realism camp, and they would disagree rather vociferously with your statement about “mathematical laws.”
Realism/Anti-Realism has no bearing to the matter. I personally think mathematics is a priori and non-causal, and not necessarily correspondent to the real world. This would imply I'm technically an anti-realist. Certain mathematics correlates to the real world, others do not and are simply (interesting) mathematical structures.
Neither viewpoints can explain why certain mathematical statements actually correlate to reality. Mathematical entities are not causal - they do not have any causal connection to the real world since they are (obviously) abstract.
Maybe a good place to start thinking about this problem would be to read Wigner's work "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".
What does random mean? Are you asking why the universe appears to be consistent and intelligible at all? Where would we be if it weren’t?
The weak anthropic principle covers this nicely. Perhaps it is an unsatisfying answer, but scientists and mathematicians aren’t usually in the business of answering ontological questions.
Maybe a good place to start thinking about this problem would be to read Wigner's work "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".
I have studied all of this. Wigner’s account is very important for the debates it inspired but it’s far from the last word on the topic. There have been numerous replies addressing all of Wigner’s claims. Try reading Hamming’s replies to start.
"From all of this I am forced to conclude both that mathematics is unreasonably effective and that all of the explanations I have given when added together simply are not enough to explain what I set out to account for." - Hamming
In many cases, physicists are not so successful. Look at the current debates in particle physics and the grand unified theory. Progress has largely stalled.
In other cases, models are revised continually as new information comes to light. For example, the age of the universe as we know it may double [1] from 13.79 billion years to 26.7 billion years.
You might call this “unreasonable effectiveness,” I call it a process of messy refinement and rethinking over thousands of years, much of it later shown to be invalid, despite mathematical correctness.
[1] https://phys.org/news/2023-07-age-universe-billion-years-pre...
Minor correction. Goedel's theorem applies to decidable (a.k.a. computable) sets of axioms. e.g. Peano's axiom, when written in first order logic, contain an infinite set of induction axioms that follow a specific and recognizable (i.e. decidable) pattern.
In fact, Goedel's theorem applies even beyond that including any set of axioms whose syntax can be specified by some (first order) mathematical predicate, even if that predicate is not computable in any way.
> Meaning there could be another set of axioms under which the limitation to prove that such a statement is true disappears. But it doesn't make math any less true.
Indeed, when studying logic it is common to consider the set of all true first-order (arithmetic) statements, which necessarily forms a complete and consistent set of axioms. Of course, given Godel's incompleteness theorem it must be the case that such a set of axioms isn't computable. Not only that, but it cannot be even defined in a first-order (arithmetic) way, which leads the the corollary of Tarski's undefinability theorem[0].
[0] https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theo...
P.S. There is no free will and one's thoughts are indeed out of one's control. I'm not certain that implies one should never trust one's thoughts. Though it does make sense to take them with a grain of salt. See for example, "The past is not true" (https://news.ycombinator.com/item?id=36798854).
P.P.S I'm not saying your argument is invalid. It can both be the case that materialism is contradictory and there is no free will.
That my thoughts are not under my control is the counter argument to the claim "But then how is it that you feel like you do have free will?" The answer to this is that, upon more careful introspection, it seems I don't even really feel like I have free will either.
We don't understand enough about thinking/consciousness to reduce them entirely to physics (the literal brain).
> The answer to this is that, upon more careful introspection, it seems I don't even really feel like I have free will either.
Why did you write the above reply then? I posit that I have free will because I perceive my typing this as occurring due to my free will. Maybe everything is (effectively) deterministic, but at the very least I exist in a reality where I can think and act in ways that demonstrate free will. That's more fundamental than any scientific finding, which is necessarily an approximate view of reality. Think of a computer that can somehow emulate computers such that apps in the emulated computers have no way to discover a distinction (timing and side effects are emulated too, I guess). What difference does it make to those apps that they're running in an emulated computer or the base?
We don't understand enough to know whether (or how) they reduce entirely to physics. But if the materialist position is correct, then matter (including energy particles) and the laws of physics all there is. So thinking and consciousness have to reduce entirely to physics, because there's nothing else for them to reduce to. We may not know how, but we know they do - if the materialist position is correct.
> > The answer to this is that, upon more careful introspection, it seems I don't even really feel like I have free will either.
> Why did you write the above reply then?
Because he had to, of course! (/s, or not, depending on your philosophical position...)
I find materialism weird because it seems like a given that all we do is motivated by our minds, including physical models—but then we are tempted to make a leap and claim that metaphors from those models (particles flying, strings vibrating, fields permeating, wave functions collapsing) are objective facts about environment rather than metaphors that try to predict how our environment would behave (in order to satisfy our minds’ curiosity and increasingly achieve some goal relevant to our minds), and that our environment and our mind itself completely reduces to those metaphors. I blame it on lack of philosophical sophistication.
That is not to claim it’s one way or another—just that we should at least be aware that it is a leap without positive proof.
True. And sometimes I even say "he/she" or "they". But 1) the vast majority of people on HN are male (I'd guess 90%, but I have no hard data), 2) the default "she" for doesn't fit on a male-dominated site, especially when referring to a real individual, 3) "he/she" sounds more inclusive, but it still misses nonbinary people, and 4) "they" just sounds off to me when referring to an individual.
I could have, instead of saying "he", gone up-thread and used the name of the poster, but... sometimes I'm lazy.
After reading his comment, I had the thought and an urge to type up a response to attempt to clarify that I indeed understand the idea that thoughts not under my control and my lack of free will are somewhat independent aspects (it is certainly possible to imagine both having free will at some subconscious level and also seeming to have no direct control over one's thoughts that appear in consciousness), and add that my primary reason for doubting free will is physics based.
No serious counter-thoughts or other considerations appeared, i.e. it didn't seem dangerous or harmful to reply. So I ended up succumbing to my urge type up a reply and in fact did so.
TL;DR: I couldn't help myself.
You can make a hypothesis that perhaps you have no free will, but you should acknowledge that it is a philosophical thesis unfalsifiable within the scope of physics.
But my act of calling out a word isn't just mental, it is physical. It causes physical vibrations in the air, and choosing different words causes different physical vibrations. And according to physics we can trace back the events leading up to the vibration in the air, which are caused by vibrations in my larynx and the shape of my mouth, each pulled in turn by contractions of muscle fibres that are happen when they are bathed in acetylcholine released by neurons attached to those fibers. The neurons are in turn linked together by electrical and chemical connections, all in turn following the laws of electrical and chemical potentials though their interactions with each other, each in turn following the laws of physics.
If I had a free choice, then somewhere that free choice interacts with the physical environment. If the same situation were somehow to truly arise again, as in the state of my brain is the same, the wind touching my skin is identical, the radiation falling from the sun on my face is identical, and I have a free choice, then the evolution of the state of me and the entire worlds diverges at some point, the point at which I make a free choice.
Quantum mechanics does allow divergence similar to this. In the observation of measurements in quantum mechanics, quantum theory says that identically prepared experiments can yield different results in which the probability of each result following the Born rule. However these results are random, and not based on a free choice. If we try to use measurement as a way to cause the state of the world to diverge based on my free choice, then I could be using my free choice to violate the Born rule, which could then in turn let me freely violate the conservation of energy and conservation of momentum, etc.
I acknowledge that our understanding of quantum physics isn't complete. The so-called vacuum catastrophe remains unexplained as just one example. And what the heck is qualia anyways? So it may be possible that a later theory does, somehow, give an opening for free will.
But just as the theory of the earth being flat was refined to be a sphere and then later refined into an oblate spheroid, each refinement of physics theories cannot stray too far from the previous one, as all the previous experimental results have to be maintained. Thus any updated version of quantum physics will need to maintain something close to unitary evolution of quantum states and it will need something close to the Born rule to make quantum events that appear to follow a random distribution, and will want to at least nearly respect conservation of energy and momentum.
Thus I remain skeptical of any refinement of physics ever enabling free will. But yeah, maybe it isn't entirely impossible.
Where determinism truly fails is at the transcendental argument level. The act of affirming determinism is self-defeating like affirming materialism: to rationally accept something as true you must have the freedom to weigh the evidence, but if determinism is true there's no true deliberation as everything is the result of prior causes, including its acceptance.
So you cannot trust your own belief system and thus your worldview is inconsistent at its core.
It also fails to account for our mental lives, given the experience we all have.
Like, maybe the world is inhabited by leprechaun's holding up cardboard facades of the city when I'm walking through it always hiding themselves. Or maybe my brain is in vat, hooked up to a computer that is making me believe that I'm experiencing typing out this comment. Or maybe I have no brain at all and I'm just an improbably quantum fluctuation in unfathomably large dead universe that just for a fraction of a second creates an experience of typing on a keyboard and the sensation of having a history that never was, a Boltzmann brain.
I accept that all the above are possible, but I don't "believe in them". I do believe in determinism (plus randomness) with about the same conviction that I believe the keyboard that I am tapping on exists.
> It also fails to account for our mental lives, given the experience we all have.
So this is what got me quite recently.
I remember a while back Henk Barendregt remarked something to the effect that everything that happens either follows deterministically from past event or happens randomly, and in neither case is there any free will.
I remember accepting that from what I know of physics this does seem to be the case. I'm not aware of any violations of the Borne rule that would allow for any event to be neither random nor deterministic. But, that the time, I remember thinking that it did seem strange that I feel like I have free will.
But recently, I've been listening to Sam Harris's Waking Up app, and, in addition to meditation practices, he has a section where he argues, convincingly in my opinion, that if you pay attention, there isn't actually any feeling of free will either.
For instance, if I am sitting and meditating and I am trying to pay attention to my breathing, or trying to pay attention the the bird chirping outside my window, then often, usually only after a few seconds, I suddenly find myself thinking about what I'm going to make for dinner, or remembering that time I was talking with my aunt about her birdwatching hobby, or any number of a million possible intrusive thoughts.
I certainly do not freely will these thoughts into existence. My will is supposedly to be paying attention to what I am hearing, or the experience of breathing. Yet inevitability I eventually end up distracted by whatever pops into my head at any given moment.
And after spending some time paying attention to my experience, I cannot help but see that Harris is right. He suggests taking the example of picking a movie, any movie. Just pick one and pay attention to the experience of freely choosing a movie title. For me, just now, I picked "Ghostbusters". But it was simply the first title that popped into my head.
Is this free will? The title popped into my head the same way that thoughts intrude into my meditation, just appearing in consciousness with no source in sight. No particular explanation as to why one title appears and not another.
Try again and think up three songs and choose one of them. Take as much time as you want. Make this choice as freely as you like. But also watch as the titles just appear in your consciousness from nowhere to be seen. Watch as you /deliberate/ about which one to pick. Reasons, like all thoughts, just start popping into one's head from nowhere to be seen. "I really enjoy X song" or "X song is quite popular" or "I haven't listened to X in a while, maybe I'll pick it then go listen to it". All these reasons for freely picking a title are just popping into consciousness in just the same way that the titles appeared in the first place. Maybe the thought pops into your head that you should just roll a die to choose one of the three.
Upon examination, my experience of deliberation is just the same as my experiences of other kinds of thoughts. I simply experience reasons popping into my head relating to the choice at hand, but I have no real control over which reasons will appear. They all just start to show up. Maybe eventually a thought appears, "Ah, that is a good reason to do X". Did I just freely come to a choice? Am I in control of that thought? That last thought just popped into my head in the same way that every other thought did, and I don't recall choosing to come to a final conclusion.
Anyhow upon this recent introspection on my own mental life, it now feels completely compatible with the lack of free will. While it doesn't conclude that I for sure have no free will, it doesn't actually look like I have it. The thoughts that enter my consciousness don't directly appear to have any origin. They just appear in consciousness in a way not so different from the appearance of the sound of a bird chirping outside my window does. I don't have any direct control over what thoughts will appear in my consciousness. And what type of thoughts that do appear in my head seem to be at least somewhat influenced from past experiences. i.e. they do kinda look like something that is offered up based on some combination of determinism and randomness.
It's clear circumstance and your memory will impact your present - that doesn't mean you did not choose to reply to my comment with your thoughts.
The whole point of a belief system is for it to correlate with reality and truth, and if there's an obvious contradiction in it, it clearly doesn't.
In this case, thinking all of reality is material implies that you cannot believe your thoughts, including the thought that all of reality is material. But that was your original belief, meaning through your own worldview you've reached a contradiction.
Free will or not, I don't see how I can change my belief system in such a way to exclude the possibility that I'm a brain in a vat, so in that sense I don't see how I can ever 100% believe my experiences match reality short of attempting to tautologically define my experiences as what reality is.
Since you claim have free will, I'd be interested in how you experience the sensation of freely choosing a movie or song title and where exactly your free will comes into play within those sensations, and of how your sensations of the task differ from mine.
> This article was written by someone with very poor philosophical training.
Strange to throw shade on the author when your position is that an idea that's been taken seriously by philosophers for 2000+ years is obviously contradictory. Maybe some humility is in order?
The whole purpose of a worldview is to be able to trust it as one trust one's own eyes - because a good mental model of reality affects how one acts and moves around the world.
If the worldview implies a contradiction it is not a worthwhile worldview to hold, because you know it to be logically false like any other set of statements that lead to a contradiction.
On the other hand, if materialism is false then your thoughts are (1) definitely out of your control, since they aren't even constrained by physics, and (2) exist due to some mechanism that's not just not understood so far, but not understandable in principle (otherwise it will be inside the realm of science).
How is that better?
I invite you to try!
Non-materialist philosophy is very compatible with physics. The problem is people who start talking physics as philosophy/religion and believe that metaphors in current physical models (electrons flying, strings vibrating, fields permeating) are not metaphors but actual final objective reality, thinking that natural sciences make existential statements.
You are, of course, free to make such statements—but you shouldn’t try to pretend that physics backs it up, because it doesn’t. With these theories you are engaging in philosophy, and as long as you refuse to acknowledge that it will be difficult to discuss them with you constructively.
All natural sciences rely on the fact that "This follows from That" everywhere in the universe and that thoughts are manifestations of This following from That in the brain. In order to have a thought that doesn't follow from That, This must not follow from That in some cases because the brain magically willed that Something Else follow from That. With your theories, you are engaging in denying physics, whether you realize it or not.
Yes, some scientists believe in materialistic monism. Some scientists also believe in whatever version of God (perhaps this would be news to you, but it happens all the time and there’s no paradox at all between being a scientist and religious simultaneously). However, both of the above are philosophical or theological beliefs; they can inform what natural sciences does, but not the reverse.
Natural sciences create models and make predictions that if this, then that would likely follow. However, there is never a conclusion that if “this -> that” is demonstrated then a model is “true” in some sort of absolute objective sense—the model is merely a transient metaphor, another model will replace it, and no model can ever be “true” because the map is never the territory.
But Language Came First, in Order to have a Permutation of the Languages an Individual Could Speak.
Would it not Follow that On the Hunt one Could Described a Series of Logical Steps that Much up to Logic, without Being Called Logic. To Predictably Bring down the Bison?
Or Know Logically that they Could Frighten the Bison to Run off a Cliff and save themselves the Trouble of the Hunt in the First Place.
That it would be Counter Intuitive to Describe the Hunt Mathematically, but the Hunt also had to Be Something that could be Repeated. So What other form of Language Prose exists regardless of Written Language that Allows for Something to Be Repeated?
A Story, but how is the Story Composed to Pass down the Lesson to Frighten the Bison to Save the Trouble of the Hunt? But So too what Mechanism in those People would Recognize that Running the Entire Heard off the Cliff would Lead to Waste and Depreciate the Yearly Yield of this Strange Tradition? Logic without a Name.
So Mathematics is a Late Understanding of Language that Provides a Condensed Symbol System to Perform Operations based on Some Mathematical Logic. Proven by Some Test we Call Theorems. To Inform that these Symbols are Not just Random Symbols, but Correspond to Reality via Some linking from the Individual doing the Maths to the Process where the Maths is used to Inform what Was Once the Hunt, but is now used in the Creation of a Product instead. Just a Slice on a Document Filled with Other Language to Inform the Creation of the Product where Math is Just an Additional Quality that informs that Processes Outcome.
Interesting to Note Math was a Late Addition to Computer Science.
Noticing that no Comments went beyond the Format which is Designed to be Difficult to Read on Purpose. And no Comment associated with the Contents of the Post.
That and it's a Format Designed for a Parser. Disappointing Really
But this is not a limitation of mathematics at least not any more so than there being no largest prime number is a limitation of mathematics. It is a limitation on abstract systems, such as Turing machines.
The halting problem is not that you can not deduce whether any given Turing machine halts, but that there is no Turing machine that deducing this.
Gödel's (ö and o are different letters, by the way) theorems are far more wide reaching, as they appy to large classes of axiomatic systems, but they are only making claims about the systems in relation to themselves.
Behold: 2.
(I'll let someone better educated expand on prime ideals)
It's not merely that computers aren't guaranteed to be able to calculate these. BB(n) for some n <= 745 is independent of the ZFC axioms of set theory.
Furthermore, Gödel's theorem and Turing's problem are related. It is undecidable whether a Turing machine halts, it's not just that you can't write a Turing machine that decides whether another Turing machine halts.
Without Gödel's theorem, you could try to enumerate all possible proofs in your complete and consistent axiomatic system until you find one that proves whether an arbitrary Turing machine of your choosing halts, and you'd have conceptually solved the halting problem. (Then, you could encode that whole algorithm in a Turing machine (TM), and have that new TM be your decider that solves the halting problem, if you wanted).
Because of Gödel you can't do this, whether on paper, with a TM, or even in principle. Your axiomatic system is not complete, so there are things it simply cannot prove. Like the halting problem, or in the case of ZF the value of BB(745).
This is what motivated me to come up with a new definition of real numbers, namely, they are objects (I call them oracles) that answer Yes or No when asked if the number ought to be between two given rational numbers. Abstracting out what properties such an object should have, one can come up with a space of these oracles, define an arithmetic, and prove that they satisfy the axioms of real numbers.
For details: https://github.com/jostylr/Reals-as-Oracles/
In many ways, this is giving a definitional support to the use of interval analysis which is, of course, a very practical concern. It also brings our some cool stuff about mediants and continued fractions (nothing new about that, but nicely motivated).
It also fits in with the adjacent post about busy beaver numbers and its conclusion about knowing a number is in an interval.
No, it is not. Or rather, which differences matter depends on your application. A circle of radius 1 is a very natural thing to me. I don't think 3/4 is more natural.
Dedekind cuts can similarly by recast as a function that gives -1,0, or 1 if a number is below, the same, or above the real number. This would be a great improvement over the usual presentation.
But beyond that difference, consider the solution to f(x) = 0 in terms of the intermediate value theorem for an increasing function. The Dedekind cut solution is A = {x | f(x) < 0}. That is the answer. From the perspective of Dedekind cuts, there is nothing further to compute and it has provided nothing to compute with.
The oracle answer would be the rule that says yes to a<b if f(a)*f(b) <=0. This gives a direct mechanism for testing intervals and the idea of narrowing intervals by picking a point inside a Yes interval is very natural from this perspective. I do not see that suggestion popping out of the Dedekind cut approach.
The rational interval approach specifically is targeting narrowing solutions to the answer with error bounds built into it. If I tell you that 3 is in the set A, you have no intrinsic sense of how good of an approximation that is. But if I tell you that the target is in the interval 3 to 3.1, then you have a sense of the precision. Oracles promote having accuracy as foundational.
So a real number is a function from pairs of rationals to a two-element set (plus some sanity conditions)? Why is that better than the other constructions?
The lazy evaluation is that defining the real number is about having a way of answering the question when presented with the rational interval, but one does not actually need to have the answers until asked.
From this perspective, a program that computes Yes or No when given a rational interval via the rule "a<b Yes iff a^2< 2 < b^2" can be said to be the square root of 2. For the usual presentations of the other definitions, they cannot be embodied in a computer. One cannot literally have all the infinite elements of a Cauchy sequence, or the infinite sequence of digits in a decimal representation, or the uncountably many elements of a Dedekind cut, represented in a computer memory. One can have a function in memory.
The other definitions can also be presented, in various ways, as functions as well, but I think, fundamentally, what we want in practice from a real number is an interval of small enough size to be of use in whatever we are doing with the real number. That is what the oracles facilitate.
But what if I want to represent an uncomputable number?
Or regardless of that, under any reasonable encoding of programs that can be held in memory by a computer, there are only countably many programs.
There are a number of real numbers that one can define which can depend on whether we can prove something or not. It may turn out that we can never prove it and the number is never resolved.
My attitude, which may not be satisfactory, is that we do what we can and we should have a language/framework for facilitating that. I think the oracle approach highlights what we know and marks what we can't compute clearly. I call it the resolution of the oracle. I don't want known precision to be lacking just because of a poor definition of what a real number is.
As for the example being algebraic, it is a particularly nice example and it is the same example in every example of a Dedekind cut. Another example of such a rule, one which is not algebraic, would be whether pi is in an interval or not. Given a<b, one rule could be that it is No if b < 3 or a > 4 or sin(a)sin(b) > 0. Alternatively, it would say Yes if a >= 3 and b<= 4 and sin(a)sin(b) <= 0. To compute this, one needs to compute sine of a and b sufficiently precisely to determine their sign.
The flavor I am trying to convey is having a definition convey a useful goal. The interval approach says that we are trying to generate precision about our inaccuracy. I think this is something which would greatly benefit those learning about real numbers. Most of the time, the error is presented as secondary and an annoyance, the concerns of the error propagation in further computations is pushed to the side, and it is all relegated to experts or computers. The expansionary nature of arithmetic in error propagation is pushed to a usually unsatisfactory discussion about significant digits.
My goal here is to change the mental framework so that these concerns come to the forefront. Ideally, they also come with useful tools to handle the uncertainties such as ways to compute how narrow the input intervals need to be when doing a computation. And maybe, just maybe, students could become more comfortable with fractions.
Do you mean physically? Basic shapes like circles, squares and triangles allow us to hold irrational numbers in our hands as distances. Children playing with blocks can sense that root 2 does not conform nicely with other (rational) distances.
If I want to compute pi + e, an infinitely accurate version is pi + e. That's about it. So what we are actually looking for in this computation is an estimation algorithm, one which can be made as accurate as we wish, but finitely so. The natural way to express this is with rational intervals as rationals are precise and intervals give a containment of the numbers.
For arithmetic, we can have a mechanism for figuring out how precise the input approximations need to be in order to get a given precision for the final computation. The perspective presented here naturally leads to that as a matter of defining and establishing the arithmetic of oracles.
As for playing around with physical representations of irrational numbers, keep in mind that there is no way to prove that, say, something that looks like a unit square to us is really a perfect square down to infinite precision. And without that, we can easily have that the unit square is only very close to such a figure, but is actually a rational rectangle with a rational diagonal that very closely approximates the square root of 2.
Suppose you want to solve the Halting problem for Turing problems that can execute one step per second. If I have a machine that takes 1 second for the zeroeth step step, and 0.5 seconds for the first step, and 0.25 seconds for the second step, and so on, I can just run the "normal" machines' entire infinite history in 2 seconds. If I want to check if the twin prime conjecture is true, I'll just check every integer. If I want to check if an arbitrary normal machine halts, I just run it on my unusual machine.
As far as I can tell all of the arguments against the reasonableness of such an unusual machine rely on some physical argument. For example, computational speed requires energy, an increasing speed would require an unbounded amount of energy, which is both impractical and would also presumably collapse into a black hole.
So, I guess I should be careful a bit. The Halting problem is a perfectly well defined mathematical question and it has a definite answer. But why is it an interesting question at all? That's a question about the nature of the world.
Yes. Absolutely, Church-Turing is a claim not just about math and computation but about physics too, because any and all math "runs on" hardware or wetware or some other system that obeys the laws of physics.
But I disagree with your assessment of the halting problem. It is not possible to have, even in theory, a machine that solves it. This is from the classic proof of undecidability of the problem, which involves, just as you have, supposing the existence of a machine to oracularly answer the question. You can then construct a logical paradox using a machine that enters an infinite loop if its input is a halting machine, and halts if its input is a non-halting machine, and simply run this machine on itself. It cannot both halt and not halt, so you have to back up and invalidate the assumption that the halting-problem machine could exist. No appeal to physics is required for this proof.
Ok, so now the fun part: Let this mathematical impossibility speak into the domain physics by way of Church-Turing. Your accelerating machine is not only mathematically impossible, it _must_ also be physically impossible, else we have to throw out Church-Turing. We deduce that the laws of physics are somehow constrained so as to make it impossible.
What makes it _not_ a Turing machine? It violates one of the assumptions! Essentially, it can reach the transfinite ωth step in a finite amount of time. The assumption that Turing machines can't do that is well-hidden, but essentially it enters every time someone says "and then". "It reads the tape _and then_ updates its state _and then_ moves the head _and then_ etc." You never actually reach the ωth step if each step takes a finite amount of time [edit: an amount of time bounded from below by any fixed positive amount of time].
However, I agree that whatever class this kind of machine belongs to has its own Halting Problem. It's just not the same Halting Problem that our beloved Turing Machines face.
Why is the seemingly-relevant question the Normal™ one and not this worse one? That is a question about physics, and not mathematics.
Anyway, I hope I've convinced you that the appeal to physics is inside "Turing Machines are a reasonable model of how computations work." It's so weird to think about other more powerful machines exactly because they go against deeply-ingrained intuition about what the world is really like.
Moreover, physics really seems to prevent you from doing any "sick" by monkeying around either on the quantum or gravitational side. Did the laws of physics have to do that? I have no idea.
Perhaps someday we'll wind up elevating the (extended, quantum) Church-Turing thesis to a physical principle and rule out classes of physical law because they'd lead to violations of computational difficulty, in much the same way we rule out classes of physical law because they lead to violations of conservation of energy.
This makes me wonder, will we be able to develop new mathematical frameworks that bypass these issues? And if so, what will they look like?
Not being able to prove the consistency of a system within the system does not entail that the system is inconsistent (or not fully consistent).
We can't say it's inconsistent just because we can't prove otherwise. We can say it may be incomplete (or inconsistent, and we haven't noticed yet)
Consistency is there as a technical detail: an inconsistent system can prove every true theorem, by virtue of being able to prove every theorem.
I think you mean substrate.
So it isn't clear that maths is limited by physics. It might or might not be.
Oh, You discovered that Earth is not the center of the universe and Sun doesn't go around Earth? Please go a step further to realize that there is no absolute frame of reference that shows what goes around what. Earth going around the Sun is just as equally true as Sun going around Earth. And yes, there is nothing wrong in saying Earth is the center of universe as you can claim any place in the universe as "the center". And you spent countless amount human energy and money to establish the half-truths like this, and call it Science and progress?
The worries that we have today are, to me, preferable and much less violent, bloody, or tragic than the worries of yesterday.
I think diversity of opinion can contribute useful perspective, but I'd encourage not calling most people brainwashed who disagree. Even if that were true, it's hardly conductive to changing anyone's mind.
Sometimes people have different values and cannot agree. Assuming good faith can help cross that bridge.
Here are some triggers: People are going to die anyway, people want to make an infinite amount of babies at all cost, we really want others to have less if as a result of it we can have more for ourselves. We love animals, then we eat them. We love nature, then we burn and cut it all down for personal gain. etc
I really believe eugenics is the dirtiest word in the dictionary. If you try talk about it Ill probably get angry much sooner than reasonable.
If anything is applied it becomes a defining ingredient of our society. Products we buy and services we use partially define who we are both personally and collectively.
To me that means that before you create a product or service, before you unleash the curiosity, you might as well ask yourself what kind of people you want to create.
Forget tinkering with whatever random thing is thrown in front of you. Put them all in a bag, carry it around and carefully examine some when you have the time.
There has to be something in the bag that modifies the humans in your preferred way.
What is wrong with people? What is their problem? You decide!
We have a law here that roughly says for all government construction projects there must be 1% spend on art. The most convenient way to deal with this "problem" it to take the 1% and have some artist create one of those unsightly giant blobs of metal. The size justifies the price. No one needs this, few can appreciate it and with a lot of effort you could create something or many things that more people would enjoy.
We also have science budgets. The most convenient way to deal with that problem is to have as large as possible experiments where size also justifies the cost. If one was to make a lot of effort one could create many experiments each with a weight in practicality. The very idea might be offensive for pure science. But say, improve wind, wave, solar, improve batteries, etc or even a big project like an attempt to fix scientific publishing, the study replication problem, the archiving of research data, etc Or you can spend everything on one giant particle collider. It would be interesting but when we sort the set by usefulness it wont be in the top 100k.
One might look at great scientists and see what impact their work had on the real world. Faraday pops to mind. You could ignore everything Tesla did after his AC motor and it would still be stunning.
And then Relativity Theory. A pile of free floating nonsense with no connection to anything. I could be accused of not understanding the elegance of it but I don't have to as nothing practical followed. There was no real world revolution of the kind that makes great scientists. Just saying it doesn't make it true.
Pure science is to slow to meaningfully fill the void of possible questions. We have by comparison only a hand full of real problems to solve. We might actually make some progress there if we care for it. Those answers will trigger an infinite chain of new questions - practical questions.