I'm hard pressed to think of a situation where this would come up in a conversation with non-scientists.
To be honest, I've been extremely disappointed by the scientific process in high-energy particle physics. People are so tethered to their theory convictions, that they accept derivations as physical realities and dismiss the importance of experimental validation. The old farts in the field are also completely averse to computational/numerical methods and dismiss the importance of production quality code development.
So while I embrace the sentiment that 'math is the essential language of physics', if you can't embed the mathematics into a simulation which describes reality, then what are you doing as a scientist?
http://www.nytimes.com/interactive/2015/07/03/upshot/a-quick...
You can always over fit for existing data. Useful theory's should suggest a novel experiments to test them.
I also know that much smarter people than me don't consider it a problem anymore, but it's exactly the kind of gap in the door that self-taught "experts" would use to say that the whole edifice of science needs to be torn down as incorrect and replaced with their personal theory.
I'm being very pedantic here but it's a bit better to think of it as subtracting infinity from another infinity.
You can easily describe renormalization without math because it's a physical concept. Regularization is more difficult. If the OP decided divergent integral regularization was an interesting conversation topic then I am on your side.
If you do this, you find that you need to choose a scale that "absorbs" the infinities of your calculations (yes, an "infinite" scale). But, the stuff that is not absorbed is a function of energy, and this actually agrees very well with experiment beyond the scale you normalized to.
For those with some time to spare, you might enjoy:
Clearing Up Mysteries -- The Original Goal
http://bayes.wustl.edu/etj/articles/cmystery.pdf
"Put most briefly, Einstein held that the QM formalism is incomplete and that it is the job of theoretical physics to supply the missing parts; Bohr claimed that there are no missing parts. To most, their positions seemed diametrically opposed; however, if we can understand better what Bohr was trying to say, it is possible to reconcile their positions and believe them both. Each had an important truth to tell us. But Bohr and Einstein could never understand each other because they were thinking on different levels. When Einstein says QM is incomplete, he means it in the ontological sense; when Bohr says QM is complete, he means it in the epistemological sense. Recognizing this, their statements are no longer contradictory. Indeed, Bohr's vague, puzzling sentences -- always groping for the right word, never finding it -- emerge from the fog and we see their underlying sense, if we keep in mind that Bohr's thinking is never on the ontological level traditional in physics. Always he is discussing not Nature, but our information about Nature. But physics did not have the vocabulary for expressing ideas on that level, hence the groping."
>we understand "infinity" just fine.
I enjoyed "Meta Math" by Gregory Chaitin:
http://arxiv.org/abs/math/0404335
"So the set of real numbers, while natural—indeed, immediately given— geometrically, nevertheless remains quite elusive: Why should I believe in a real number if I can’t calculate it, if I can’t prove what its bits are, and if I can’t even refer to it? And each of these things happens with probability one! The real line from 0 to 1 looks more and more like a Swiss cheese, more and more like a stunningly black high-mountain sky studded with pin-pricks of light."
Bah! Chaitin makes so much of his discovery that there is randomness in mathematics, but doesn't ever seem to have made the shift to thinking of structures as parametric over randomness.
To wit, generating random real numbers by simply flipping a countable number of coins and writing down the bits is trivial, and from that point of view:
1) An open set is just a set about which we can have finite information.
2) A closed set is just a set about which we can have complete information.
3) A compact set is just one we can quantify over and prove theorems about given only finite information.
4) A continuous function is one that has finite character: producing finite information about an element of the image only ever requires finite information about the argument.
These statements are a bit rough, but hey, if I ever have a gajillion dollars I can spend my newfound spare time merging the computational view on synthetic topology, Chaitin's algorithmic information theory, and the old "Age of Stochasticity" talk to get a view of mathematics that properly accounts for its finities and infinities.
http://www.stat.uchicago.edu/~lekheng/courses/191f09/mumford...
That guy is just a constructivist. Nothing new here.
It is no surprise that the set remains elusive, it doesn't exist.
It is at the same time a commutative additive group and a commutative multiplicative group such that the two group neutrals aren't the same. Furthermore the multiplication distributes over addition.
There exists a total order relation such that the addition distributes over the order relation and multiplication agrees with the order relation (if (0 leq x) and (0 leq y) then (0 leq x*y)).
Another axiom is required, which comes in many forms (all of which are equivallent) so take your pick, for example the supremum axiom https://en.wikipedia.org/wiki/Least-upper-bound_property
You can read about any of this in any texbook of real analysis (for example Spivak or baby Rudin).
If the axiomatic formulation of real numbers doesn't appeal to you, there is another way to define real numbers, namely to contruct them from natural numbers (and you seem convinced that natural numbers indeed exist). You can read about that in the appendix to chapter 1 of baby Rudin (Principles of Mathematical Analysis by Walter Rudin).
I don't think it's fair to say the reals 'most certainly exist' without being misleading to a layman. They exist given some axioms that are used overwhelmingly often in mathematics, but you can still do some interesting stuff without those axioms, or with their negation.
Along with it, certain properties of this object are assumed (among them the ability to choose an element of a non empty set - this is typically called the axiom of choice). For a full list you can take a look at http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html
But you can also assume ZF plus the negation of the axiom of choice, and get a system that is consistent if and only if ZF is consistent. It's not clear (to me, at least) that this other system will let you build the real numbers.
https://en.wikipedia.org/wiki/Ultrafinitism
...and "Set Theory, Should You Believe?"
https://web.archive.org/web/20110616020815/http://web.maths....
You can even write a set of them: {1, 2, 3, 4}
You cannot write the set of real numbers.
To the question: I'm only saying that the "set of real numbers" and the "set of natural numbers" don't seem to exist.
Despite numerous downvotes, no one has yet produced them here (or even a link to them).
Both real and natural numbers can be reasoned about despite their infinite size (and we even know that e.g. there must be "more" real numbers than natural numbers, even though both sets are infinite)
There are no bounds in the universe that can contain all the real or natural numbers.
and here is one of the natural numbers: ℕ
Axiom of choice says I should be able to select an element from the set of real numbers (assuming it exists; but, I believe the assumption to be counterfactual).
I think it makes sense though; if I (claim to) have a thing, I should be able to choose/pick/select/point-to it (seems to be almost[?] tautological).
But a) what it really indicates is that it would be unmeasurable whether we have free will, and b) it also doesn't really prove that, because while electron positions are unpredictable, molecular reactions, the sort that happen in the human brain are pretty predictable.
Basically, my response to this is "That would be relevant if we had black holes or large hadron colliders in our heads, but we don't, so...". But people really really want to believe in free will and will use any sliver of half-truth to believe they've proved it.
EDIT: To be clear, I do not believe free will has been proven OR disproven.
EDIT 2: More sophisticated wrong people argue from Bell's Inequalities rather than the Heisenberg Uncertainty Principle, but that also results from a misunderstanding of the theory and also neither proves nor disproves determinism.
What about those quantum effects that have recently been shown to potentially be part of brain activity:
https://www.sciencedaily.com/releases/2014/01/140116085105.h...
Perhaps someone can reframe that in terms of types of error, and people prone to more serious types of error (road rage) face different consequences. But that gets real complicated real quick.
I dunno. I suspect the majority of the time i'm on autopilot. Alarm goes off, get up, brush teeth, shower. I pretend i'm capable of not doing that. i just don't choose the other way.
The distinction still matters without free will, because it's still a predictor of future behavior. A driver who crashes because of a preventable cause shouldn't be allowed to drive until the problem is resolved. If the problem is falling asleep, the solution may be simple (get more sleep?) or hard (narcolepsy), while anger issues are probably pretty difficult to provably solve.
Belief in free will has some seriously harmful effects on the way we deal with these situations. Most notably, it's the basis for a focus on punishment rather than rehabilitation. Deterrents don't work, largely because they don't fix the causes for bad behavior. Rehabilitation is the only sensible approach if you don't believe in free will, whereas punishment makes a bit more sense if you believe that people have free will.
> Perhaps someone can reframe that in terms of types of error, and people prone to more serious types of error (road rage) face different consequences. But that gets real complicated real quick.
This is fundamentally a free-will-based way of seeing the problem. "Consequences" are completely irrelevant in a deterministic worldview unless you can prove that they act as a deterrent (that is, that they are part of the cause). Since consequences are provably not significant deterrents for many crimes, I don't think that works.
Your argument basically that "this idea of punishment/consequences (which only makes sense in a free-will-based ideology) doesn't work in a deterministic ideology, and punishment/consequences are good in my free-will-based ideology, therefore deterministic ideologies are wrong". But you have to hold a free-will-based ideology for the preconditions of your argument to make sense.
Very insightful. Thank you.
Myself, I do not even know what the will is, or what it could possibly be free from. The influence of the universe?
Fear sounds like an implicit assertion of choice, but i think you mean something more subtle.
Let my try to put it another way. People are complex systems that can be influenced. Water tends to flow downhill, we can set up troughs to control the flow of water. The flow is complex, and some water will splash out, but generally the water does follows the system. Likewise, we can set up legal troughs to guide people to the behavior we want. (I know argument by analogy is crappy, i'm just hoping we can agree on this point, and i think we do.) There's a whole other argument about why we would set up those legal rules, but i'd like to set that aside for a moment (if it's a big deal, or fundamental in some way, i'd like to hear your side. I'm not sure i can meaningfully engage that point)
It seems like, punishment would need to be interpreted purely as societal cost. So you tally up the wreckage from the car accident example above, say $25,000 and 2 lives.
I don't really see how we can have both manslaughter and murder, without free will. What extra thing did the angry guy do? He didn't make a choice, because (for the sake of argument) he has no free will. What is the characteristic that makes what he did worse? The cost to society is the same.
> Myself, I do not even know what the will is, or what it could possibly be free from. The influence of the universe?
My personal interpretation (which is probably wrong, but something i'm willing to go back and forth on to come to consensus) Free will the mind having a choice in taking action, rather than the body just doing whatever, and the mind rationalizing it after the fact. Perhaps every action we take is just like a heartbeat. I have no control over that. Perhaps when i hold my breath, i'm not actually choosing to hold my breath, but the complex system that is my body holds it's breath and my mind is claims that it was responsible for that holding of breath. Or typing this comment, or whatever. I don't think there's any way i can know. It feels like my mind is in control, but my mind gets all of its information from (and the software runs on!) my body. my mind seems to control the hardware, but maybe i'm just rationalizing after the fact.
That doesn't require the concept of free will. Because for manslaughter and murder the reasons are different for the death. One is deliberate, one isn't.
> It seems like, punishment would need to be interpreted purely as societal cost. So you tally up the wreckage from the car accident example above, say $25,000 and 2 lives.
You would also want to calculate predicted future costs from that person, mainly from the point of view of harm to other people. If someone like them is likely to kill on average one or more people in the rest of their life then they aren't going to be allowed to interact with general society again. This doesn't need the concept of free will to implement. It only needs a sufficiently sophisticated way of determining of what harm a person like them will likely cause. This is in addition to the harm they have already caused, so there is no reason for someone to become a one time killer. And yes, getting the predictions right may be really tricky, and the way the courts currently sentence people may be a good as or better than trying to predict what people do on average. But the way courts sentence at the moment doesn't require the concept of free will anyway, because people do what they do because of reasons, human reasons, and they can differentiate there.
I don't think so. I think determinism makes "punishment" as a concept, completely nonsensical--you can't punish someone for something they didn't choose to do. I think determinism pushes the focus to rehabilitation rather than punishment, which for many types of crimes (i.e. drug use, computer hacking, nonviolent robbery) is clearly more effective in preventing recidivism (and I'd argue that for crimes where it's not effective, that we haven't really explored rehabilititation because of the widespread belief in free will).
> I don't really see how we can have both manslaughter and murder, without free will. What extra thing did the angry guy do? He didn't make a choice, because (for the sake of argument) he has no free will. What is the characteristic that makes what he did worse? The cost to society is the same.
Agreed. But does this highlight a problem with a deterministic world view, or does it highlight a problem with the distinction between manslaughter and murder?
That is a really interesting point. I'm not convinced, but you've given me a ton to think about.
> There are a lot of people
> who will not commit crime
> only because they fear the
> consequences
Punishments from crime have decreased substantially - nobody gets transported or hung for sheep theft anymore. The world has never been a safer place. States with the death penalty are no safer than those without. Fewer people genuinely believe in a Christian (or whatever) hell than in years past. At the same time, religion has faltered as a source in many people's lives.
I would argue this means that circumstances that lead people in to crime have decreased rather than a suggestion that people are making choices to commit crime but don't based on deterrence. I appreciate there exist counter arguments to this too, but also feel that if we treated crime as a function of situation rather than morality, we'd find ways to decrease it more cheaply.
With no notion of free will, how do we distinguish between manslaughter and murder? When someone dies, the cost to society is the same either way. A consequence of no free will seems to be it's a distinction without difference. I disagree, but it might just be deeply embedded beliefs that are wrong.
I'd argue that the distinction is not indicative of a phenomenon that exists.
I think a much better model would be predictive and preventive rather than punitive--the level assignment to rehabilitation should be based on the probability and severity of recidivism, rather than the level of punishment being based on the severity of the crime.
I guess you could argue that we generally assume a deterministic model exists and try to find it, but I hadn't considered that before.
Not without input data, they couldn't. You can produce a model, but unless you collect data to validate your model against, you're just speculating in a void.
> There's something fundamental that we don't understand and the scientific community is very opposed to talking about it, dismissing any and all speculation as quackery.
We're talking about it right now, and I haven't heard anyone opposed to talking about it.
However, I'll fully dismiss opinions as quackery if they clearly misrepresent the facts, i.e. claims that Heisenberg's Uncertainty Principle or Bell's Inequalities prove free will are quackery. This isn't because the ideas are taboo or because I'm opposed to talking about them, it's because they're misrepresentations or misunderstandings of facts which are clearly established.
In my experience, when people claim that scientists are opposed to talking about their hypothesis, it's because their hypothesis has been thoroughly and concretely disproven.
Yes. What's not to see? I think appealing to something other than Turing machines shows a lack of imagination; it takes a lot of thought to even scratch the surface of how profound Turing machines are.
> I'm not suggesting anything supernatural. I'm suggesting that there's possibly physical phenomenon that do not fit our current materialist model.
Absolutely. Sure, there are of course physical phenomenon that do not fit our current materialist model.
Let's say we discover those additional phenomenon. Ultimately, we'd want to build a machine to model that phenomenon. Perhaps encode a model of that phenomenon in software. Once that happens, scribbling numbers on a piece of paper indeed could give rise to a conscious being.
If it's some system that can't be covered by scientific understanding, isn't that what supernatural means? It's either some internally consistent (but possibly very complicated) set of rules, or it's supernatural.
Rather than 'possessing' an infinite tape and infinite time, I prefer to think of turing machines as always being allowed to perform one more step, and always being allowed to use one more tape cell. I find this characterisation of time quite intuitive as-is, whilst the tape can be made intuitive by imagining tape factories on each end, which produce more cells whenever the read/write head gets close.
In fact, such "tape factories" have been used to build universal turing machines in the game of life http://rendell-attic.org/gol/fullutm/index.htm
Yes. I find that no less plausible than a warm sack of meat giving rise to a conscious being.
> I don't see how
> That means that a Turing Machine would be able to
> produce them
> the scientific community is very opposed to talking
> about it
Further, my big problem is with people trying to ignore the problem completely by claiming that there is no Hard Problem or even suggesting that consciousness is not real. It's a total head-in-the-sand position to take but it's the dominant stance.
How do you know there is a qualitative difference?
Free will is akin to an emotion. Just because physicists haven't explained it in excruciating detail doesn't mean it doesn't exist, or we'd all be running around debating the existence of fear and love and curiosity.
Someone who studies these things (a psychologist or neurologist) could make even more sophisticated predictions.
We may not have a complete low-level electrochemical model of these phenomenon, but we have a good enough model to make some predictions. But if anything, one of the most salient criticisms of free will is that it doesn't allow us to model and make predictions--in fact, it eschews predictions altogether.
No. Free will is the ability to make a choice not completely dependent on history.
Isn't that like saying I want to fly, so I don't have free will if I can't?
If you respond, "No, because the particles that make up the coin and the muscles that drive my thumbnail have a history that determines the outcome," then how about a slight variation? "I can't decide what to do. I have no idea what time it is, so if the minute digit on my clock is odd, I'll do X, otherwise I'll do Y."
Here I've explicitly refrained from using either a source of randomness or anything else with its own history to determine my fate. The decision to consult the clock might have been an act of free will on my part, but the outcome certainly wasn't.
Nobody would argue that they had free will if they didn't feel like they had free will. Nobody is positing baseless metaphysical nonsense without some reverence in their actual experiences. If you want to usefully discuss free will, you have to account for the sensation of free will, not just some abstract model that you've merely assembled grammatical statements from.
[0] To be more pedantic than enlightening, add before wavefunction collapse.
"Conspiracy" indicates free will on a large number of actors.
And a superdeterministic system could exist which differs from standard ways of approaching quantum mechanics. One obvious way in which it could exist is if our means of gathering data remain as limited as they are currently--i.e. we don't ever find a way around Heisenberg's Uncertainty Principle and therefore can't ever model quanta deterministically. Superdeterminism doesn't mean that we'd always be able to measure the deterministic phenomena.
Personally, I'm not even sure introducing non-determinism matters much. What would it even mean? That beyond determinsitic actions defined by our precise history, we get a bit of randomness permuting the whole thing? Does that really make it any freer?
Could you explain how that works? I'm unfamiliar with that, but I suspect that they may be using a different definition of "free will" than I am familiar with.
> Personally, I'm not even sure introducing non-determinism matters much. What would it even mean? That beyond determinsitic actions defined by our precise history, we get a bit of randomness permuting the whole thing? Does that really make it any freer?
I think free will and randomness are both forms of non-determinism. Non-determinism doesn't prove free will, but it allows for the possibility that it might exist (and post people intuit that randomness could be indistinguishable from non-determinism).
How are you defining it such that it must be non-deterministic and distinct from randomness? I don't really understand what that means. Assuming there's a single shared reality, what's the source of the non-determinism? Even if you say that's the free will itself, you'd be asserting that our "wills" are somehow capable of randomness.
Without some place in the sequence of events that isn't deterministic, I don't see how you can separate your current state from external forces. That is, if your decisions are a deterministic product of your current state and your current state is a deterministic product of other states before it (including states that existed before you did, such as your parents) then I don't see that as "the lack of external forces controlling my decisions and actions".
> How are you defining it such that it must be non-deterministic and distinct from randomness? I don't really understand what that means. Assuming there's a single shared reality, what's the source of the non-determinism? Even if you say that's the free will itself, you'd be asserting that our "wills" are somehow capable of randomness.
Part of the difficulty with talking about free will is that there isn't a clear definition of what it is. However, I do think I can say what it isn't--I think most people don't consider randomness or determinism to be free will. But I also think that it would be nigh impossible to distinguish randomness from free will without a clearer definition of free will.
The fallout of this belief is that I think we could disprove free will if we could prove determinism (but, I don't think we can prove determinism currently). But I don't think we can prove free will, even if we could disprove determinism (I don't think we can do that either). In order to prove free will we would have to disprove both determinism and randomness.
Is it because on a macro scale amd normal energy levels effects are negligible? Because there are well followed probability distributions?
Think of it this way: you set two clocks to the same time. The next day, they both have the same time on them. This isn't because the clocks are communicating via nonlocal hidden variables, it's because their precondition was the same and the sequence of events which followed was deterministic.
I am not a mathematician, but my understanding is that a considerable amount of research into how to properly address /use infinity in math was ongoing as recently as the early 20th century. Not that I disagree with your overall point though.
Cantor's work in the early 20th century was on infinite objects in the context of set theory. That work is also fairly well-understood but starts to get a bit esoteric. More recent research goes further in the same direction, and then you start getting the point where you can't prove whether a certain type of infinity exists or not, because the existence of an infinity that large would prove that set theory is consistent.
So yes, research is still ongoing into the nature of infinity, but it's the nature of a different infinity.
The "-1/12" "mystery" is well settled math (but mathematicians are always looking for more elegant or general perspectives on things), and (standard-issue) sloppy notation in physics.
The only "mystery" is why people who should know better intentionally use incorrect terminology to dazzle lay-people, creating a mystery where there is none.
I'm no mathematician either, so I'll stop there before I embarrass myself. The most accessible resource I found was this VSauce video [1].
^1 Basically large cardinal axioms give you additional types of infinity.
https://en.wikipedia.org/wiki/Continuum_hypothesis
I remember learning about them in the philosophy of maths, but have no idea if they have any consequences in Physics. And like /u/lasfter I'll stop there before I embarrass myself.
The best we can nurture is 'good' analogy, that still somewhat valid even if the analogy is expanded further
p/s: If only we have a list of Feynman-like experts in most of the fields :D
Take comfort, I believe you exist.
It's interesting the top-comment displays such a supercilious attitude while the takeaway from article, at least for me, was the exact opposite.
Maybe you could have helped at least some of the people by using it as an opportunity to teach rather than deride?
Of course gravity exists, but to suggest we understand it 100% through what models we have seems silly. But, I'm not a physicist, and I don't play one at dinner parties.
There have been attempts at avoiding the regularization magic altogether but from what I gather, they are developments from mathematicians, and physicists don't really care about it.
EDIT: I knew it was a magic related term, not black magic.
Mathematically, it's just as unsound as saying the
sum of all integers is -1/12.
Actually, that is something I noticed: Many people are scared off for good if they don't understand something at once. Most of my physicist peers don't understand things at once, it's normal. The question is how to deal with that - do you run away or do you try again, and what are your tools for that?
I'm not sure what triggers the need to argue in people. It's just so far removed from the ideal most of us strive towards in discussions, which is trying to understand things by all means possible, and not defend a preconceived notion. This sounds obvious and easy, but for many people, I found it's not. They identify so much with their state of knowledge and arguments (not to say opinions, which I can kind of get) that one has to tread very carefully. One wrong expression and they turn into fighting mode, and the discussion turns from a search for understanding and rightness into a match for which derailing is a valid technique.
Lack of focus is also an issue, and can be a derailing method. (Ask questions faster than they can be answered and don't listen to the person trying to explain. In the end console yourself that we can't really know anything and thus all questions are unanswerable. End monologue, walk away. God, these people.)
As far as physics topics go, it is the worst in GRT (general relativity) lectures followed by quantum things. Biologists told me they get strange argumentative older people in evolution lectures too.
</rant>
Not a specialist, but "infinity" is indeed a tricky thing to make sense of.
First, if you decide to use and define it only in a "theoretical" sense (constructed through the set of axioms you mentioned) things are fine, there are lots of things we have constructed in an abstract way in the past which have proved to be intellectually solid (think of the idea of God constructed by the scholastics).
The problem appears when you want to apply your idea constructed on a "set of axioms" back to the real life. The scholastics' idea of God had a very short life when that happened (more or less during the early Renaissance), and I wouldn't say that the mathematical idea of "infinity" would have a better life were people to view it in a more critical way.
Because were you to apply the idea of infinity to the real life (the life made up of stars, planets and Hacker News) that would mean that there are an infinity of planets and stars out there (some of us have started to be ok with that) but that would else mean that there are an infinity of Hacker News instances out there and there are an infinity of "me", the user paganel, making this comment on HN. This is hard for a lot of people to grasp, including me.
As an example, the natural numbers are infinite because they contain the even numbers and the even numbers are the same "size" (or cardinality) as the natural numbers themselves.
As a second example, the real numbers are infinite because they contain the interval (-1, 1), which has the same "size" as all of the real numbers.
More exactly, I say that translating the concept/idea of infinity from mathematics to physics is not idempotent (not sure if this it's the correct term, but you get the idea), saying that infinity "exists" in the physical world the same as it does in mathematics is kind of tricky when you think of the consequences through and through. To take your second example, applied to Physics that would mean that no matter what two small particles we choose as being "fundamental" there will always be an infinity of (presumably even more small) particles separating the two, and so on and so forth (it's "turtles all the way down", so to speak).
Like I said, I'm not a specialist, just a programmer, but I think we should be aware of the presumptions we implicitly make when discussing about different topics but using the same language (in this case the of concept of "infinity" in maths vs physics). Hume explained it better than me (maybe in an "An Essay Concerning Human Understanding" ?!) when he said that the mathematical idea of a straight line (I'm talking about the geometrical line) is just that, an human-constructed idea, there's no absolute straight line in the real world (not sure what he thought about the idea of infinity, probably not much, judging by his being critical against mathematical induction).
Stepping back in topic, from my point of view of a completely amateur physicist, I really appreciate her work. Whoever contributes to spreading knowledge is making the world a bit better than it was before (B.P.) I wish there were more people in the world keen to explain things rather than ridiculing others just because they don't understand something that for us is very obvious.
Just my two pence.