A wild theist platonist appears, to ask about the path
I recognize the title could be more informative. At the same time I believe it says what is important.
I believe in a deity, I believe in mathematical entities in the same way.
The community of LessWrong (from whenceforth: LessWrong) is deeply interesting to me, appearing as a semi-organized atheist, reductionist community.
LessWrong seems very interested in promoting rationality, which I applaud. The effort does seem scattered, though, and this is the reason I post.
One has Eliezer’s website with some interesting posts. The same of this community. The community links to some posts when you are coming for the first time into it, and you also have a filter for top posts. One has the blog. And recently, the center for modern rationality (in the same page as harrypoter fanfiction about rationality).
The point being there is no defined roadmap to go from AIC (average irrational chump to make an analogy to Game—which also seems to come up around quite a bit) to RA (again, rationality artist).
I write this post as to maybe generate a discussion on how the efforts could be concentrated and a new direction taken.
Should the creation of the Center for Modern Rationality envision this same concentration, this post may and should be disregard.
If it does not, then I leave it to your consideration.
Hang.
I don’t understand what this means. Do you?
What does it mean to “believe in” the number 2, for example? And even among mathematical realists one does not usually find the belief that the number 2 is going to do anything; it won’t reach into your life and provide you with greater two-ness, as it were. So if you believe in a god in the same way that you believe in the number 2, whatever that may be, what is the purpose of this entity? The number 2 has its uses; you can add it to itself and get 4. What similar operation can you perform on your god, such that the belief is a useful one?
What a great question! Traditionally, the operations performed on a god are:
asking for goods and services
grounding morality on
ascribing mysterious feelings and special mind states to the influence of
giving thanks for blessings of existence
basing the feelings of security and purpose in life on
invoking as a semantic stopsign for metaphysical questions
None of these, except the last one, could be performed on a god-as-a-mathematical-like-entity, I think.
Actually I think grounding morality can be performed on a god-as-a-mathematical-like-entity if you wanted to. For certain settings of God you even get interesting and neat properties, which can be pretty useful (in a sense similar to this) if FAI is not near or possible and you question moral progress.
You can also use God to avoid certain kinds of blackmail and do other neat superrational tricks. Who knows it may even be the best implementation for this that we can currently build on some human brains.
Though in practice the reason we have Jesus is so we can ask “What would Jesus do?”, which is easier to answer than “What would the ideal rational agent with unlimited computational resources do?”.
’Course, Jesus says “Be ye therefore perfect, even as your Father which is in heaven is perfect.”; I think we still have a moral obligation to figure out the theoretical foundations of justification for perfect agents.
In Stoicism, we call this type of person a sage. It is actually a very practical concept to make use of. During before-sleep meditation, I’ll playback my entire day in fast-forward mentally, but alongside me I imagine a semi-transparent sage-me and I “watch” as our two paths diverge (with the sage-me living a perfectly virtuous life and me falling far short).
Interesting; I am annoyed and relieved that no Stoic seems to have nominated any particular historical person as a sage.
I don’t think I could pull off that kind of meditation, due to my having too much structural uncertainty about ethics and meta-ethics. What’s that Borges quote? “I have known that thing the Greeks knew not—uncertainty.”
BTW random LW people here is the SEP on Stoicism.
I notice that like LessWrong the Stoics are big on Logos and instrumental rationality and related ethics but their (meta-)physics and theology strike me as fuzzy and underdeveloped.
For this, there’d have to be a well-defined God, provably unique up to isomorphism.
Why so? How well-defined? I find it useful to base normative epistemic arguments off of the existence of Chaitin’s omega, even though there isn’t a unique omega and even though we barely know any bits of any of them. Similarly one could base moral arguments off of just the knowledge of the existence of a normative standard against which moral agents could be compared or by which moral agents could in theory be judged; postulating such a standard is itself a non-trivial meta-ethical position.
I’m not sure exactly what point you wish to illustrate with the Chaitin’s omega example. Yes, its value depends on the TM coding. But when a specific one is chosen, the value is unique.
Well, I can certainly ask the number two for goods and services, so if that’s a useful operation to perform, there ya go.
My chances of receiving those goods and services won’t increase if I do so, but that’s something else again.
Similarly, I can ascribe mysterious feelings and special mind states to the influence of a mathematical structure, thank it for the blessings of existence, base my feelings of security and purpose in life on it, and (as you note) invoke it to avert metaphysical questions.
I admit, I don’t quite understand how to ground morality on a mathematical structure, but then I don’t quite understand how to ground morality on a traditional god, either. (I recognize that many people claim to do this.)
I’ve never quite understood how grounding morality on a traditional god is supposed to work.
Well then, you could also multiply gods by constants and add them together, producing a vector space over a divine basis.
Grounding morality works straightforwardly, I think: God said thou must not kill, love thy neighbour, etc.
Well, yes, I understand that various commands, preferences, etc. are ascribed to gods, and that followers of those gods attempt to obey those commands and satisfy those preferences.
I’ve just never understood what morality has to do with any of that.
I mean, sure, presumably a suitably knowledgable (let alone omniscient) god is capable of giving moral commands, in that it would know what the moral things to do are, in the same sense that it is capable of telling me what stocks to purchase in order to maximize my earnings, or how most efficiently to breed cows. But to conclude that therefore wealth, morality, or cow-breeding is grounded on god (in a way that poverty, immorality, and cow-genocide, for example, are not) has always seemed odd to me.
(Divine command theory, where you obey God because He’s God as such (and not because He’s God and He commands things because they’re good), is not the most popular way to tie God into your meta-ethics, and it has various semantic problems. In better-justified meta-ethics God is useful as a necessary final cause of existence but it’s not immediately derivable what properties He has that make Him a justified final cause, nor how we as creatures should orient our actions towards Him—these are matters of ethics that are somewhat decoupled from “grounding” morality in God in a higher level sense. God is used in such meta-ethics in a way similar to how an oracle machine is used in theoretical computer science, that is, He’s an important part of a larger interconnected framework. One can’t evaluate theistic meta-ethics without knowing what the other parts are.)
Yeah, the better-justified version you describe strikes me as, if not necessarily better justified, at least more intelligible.
That said, now that I think about it a bit more, I’m enough of a consequentialist to have serious difficulty thinking straight about what it even means for a choice to be moral in the presence of a force capable of, in practical terms, divorcing my actions from their consequences. (Of course, not every theistic theory posits such a force, and it is possible to be in that position in a nontheistic context as well.)
I might quibble about your use of “popular” above, though, unless you really do mean it advisedly.
That is, it seems likely to me that Divine command theory is indeed the most popular approach, in the same sense that the most popular theory of ballistics predicts that when I drop a rock as I walk down the sidewalk, it will hit the ground a step or two behind me even though no halfway serious student of ballistics would predict any such thing. (Modulo extreme winds, anyway.)
But I’d love to be wrong about that.
I don’t know what meta-ethics are held by the Christian masses—does it actually come up very often?—but Catholic doctrine tends strongly towards Thomism, which isn’t divine command theorist, and Catholicism is the largest sect of Christianity. I suspect that most Catholics would be dimly aware that divine command theory isn’t quite right, upon considering the issue. I don’t think that my “average Catholic” friend has ever considered meta-ethics in a detailed enough way such that she could distinguish between divine command theory and some alternative meta-ethical theory. After all, in all theistic meta-ethics morality stems from God in some sense, it’s just the exact way in which it does so that is contentious. The sort of distinctions that are necessary to make are I believe quite beyond the philosophical competencies of your average Christian.
I think it goes something like this: (1) God created morality and cow-breeding, and (2) put the knowledge of it into humans [or, alternatively, the knowledge of morality was the result of eating the apple and knowing Good and Evil, I’m not sure], and (3) one of the important points of morality is that humans should have free will, and so (since God is moral) they do, and thus (4) they are free to practise immorality and cow-genocide.
If game-theoretic principles (like Nash-equilibrium) are mathematical structures and contractarianism (such as Gauthier’s ethical theory) is true, then mathematical structures “ground morality”.
Morality consists of courses of action to achieve a goal or goals, and the goal or goals themselves. Game theory, evolutionary biology, and other areas of study can help choose courses of action, and they can explain why we have the goals we have, but they can’t explain why we “ought” to have a given goal or goals. If you believe that a god created everything except itself, but including morality, then said god presumably can ground morality simply by virtue of having created it.
Yeah, that is the dominant view, but Gauthier actually attempts to answer the question “why be moral?” (not only the question of “what is moral?”) using game-theoretic concepts. In short, his answer is that being moral is rational. I don’t remember whether or not he tries to answer the question “why be rational?”; I haven’t read Morals by Agreement in years.
There are (at least) two meaning for “why ought we be moral”:
“Why should an entity without goals choose to follow goals”, or, more generally, “Why should an entity without goals choose [anything]”,
and, “Why should an entity with a top level goal of X discard this in favor of a top level goal of Y.”
I can imagine answers to the second question (it could be that explicitly replacing X with Y results in achieving X better than if you don’t; this is one driver of extremism in many areas), but it seems clear that the first question admits of no attack.
An entity without goals would not be reading Gauthier’s book.
Well, look at things like TDT/UDT for starters.
Though TDT and UDT weren’t designed to be moral as such; it just turns out that non-self-defeating behavior seems to necessitate some degree of something like morality, largely because self-ness is a slippery idea.
Dear Mr. RolfAndreassen.
Maybe I should have said that I believe in a deity in the same way I believe in mathematical entities. Natural language is tricky.
I question the assumption that something needs to do something else in order to exist. Take, for example, mathematical facts. They just “are” if you want. Some of them (but not all) are accessible trough our formal systems of mathematics. Some are not (certainly you are familiar with Godel’s proof).
You may assert that the number two has its uses and thus assert the existence of number two. But what uses can you assert for mathematical truths that are not accessible? Do they stop existing because they are not accessible, or do they “pop into” existence, if I may, once they are?
The mere fact that the mathematical truths are before they are accessible (Again, godel’s incompleteness theorem) says that mathematical truths exist, and therefore so do the parts that they are comprised of.
If you wish to be formal, it’s “Dr”. If you prefer informality that’s fine.
I can assert them as axioms and use them to generate new formal systems. Consider Euclid’s fifth, for example, which two millennia of geometers have failed to prove from smaller axiomatic systems; but which yields any number of theorems when taken as an axiom, or when either of its negations are so taken.
Again, I do not understand this usage of the word ‘exists’. You cannot prove Euclid’s fifth axiom, or at any rate nobody has succeeded in doing so. Is it true? But its negations yield equally fruitful formal systems. What then is the sense in which it exists? Do you just mean that you can write it down on paper? Then likewise the adventures of Frodo Baggins exist. Are we to take it that the competing facts “Exactly one parallel line through a point not on a line”, “Exactly zero lines”, and “An infinite number of lines” all exist at the same time? What does this mean?
And even that aside, I object even more strongly to saying that a god exists in this same undefined sense. From an axiom you can at least derive theorems; an axiom is part of a formal system. Of what formal system is your god a part?
He (she?) believes that a deity exists, where “exists” is meant in the same sense in which mathematical entities exist, rather than in the sense in which physical objects exist?
“At least one natural number greater than 152 exists.”
What does it mean, actually? In a formal sense, the sentence means that its formalised version is a theorem in PA, or whatever axiomatic system deemed relevant.
“God exists.”
What does this mean, actually?
Well, in the first place, I’m really unclear on what it means to say that “the number 2 exists”. I understand what it means to say “There are two sheep”; but to assert the existence of the number 2 seems to me rather vague. What would the world look like if it weren’t so?
But that aside, my further objection is that the god doesn’t seem to have any of the attributes of mathematical entities: It is not connected to any theorems, as it were. It is as though I had started the Peano Axioms by saying “0 is a number”, and stopped. Why make this assertion? I can conclude that 0 ‘exists’, but so what? The existence axiom is only useful in the presence of the other axioms.
Consider the question of whether there exists a largest pair of twin primes. Is this question meaningful to you?
Yes, but I don’t think this uses the word ‘exist’ in the same way. There’s a particular set of formal steps I can go through to convince myself that two numbers are twin primes, and a different set of steps which will convince me that some particular pair is the largest such pair, or that there isn’t a largest pair. But this seems different from what is meant by saying that the number two exists.
I’d say not. I tend to use two independent terms when discussing the nature of a thing’s existence; I will discuss first whether or not something is real; and then whether or not that real thing exists.
To be real; a thing must be an accurate description of some pattern of behavior that things which exist conform to. (I realize this is dense/inscrutable, more in a bit.) To exist; a thing must directly interact in some fashion with other things which exist; it must be ‘instantiable’.
So numbers, mathematical constructs, words; these things are real but they do not exist. We can recognize them in how the things which do exist behave. The concepts are not themselves instantiated—ever—but we can handle them symbolically. If I hold three pebbles in my hand, that means there is a precise arrangement of pebbles; it has a precise relationship with two the arrangement we’d call “two pebbles” and the arrangement we’d call “four pebbles” and so on. But you’ll never see/hear/touch/smell/taste the number 3. It’s physically impossible for that to occur; because the number, ‘three’, does not exist. Pebbles on the other hand do exist; you can take a pebble and throw it into a lake.
I find that this differentiation between different meanings of the term “to be” makes the discussion vastly simpler. It eliminates whole swaths of silliness (like TAG for example); I am perfectly free to say “I can prove using the Laws of Logic that the Laws of Logic do not exist.”
An excellent and useful distinction.
This is a philosophical mire. Do pebbles actually exist? But they are composed from quarks, electrons, etc, and these are in principle indistinguishable from one another, so a pebble is only defined by relations between them, doesn’t it make the pebble only ‘real’?
On the other hand, when I play a computer game, do the various objects in the virtual world exist? Presumably, yes, because they interact with me in some fashion, and I exist (I think...). What if I write a program to play for me and stop watching the monitor. Do they stop existing?
I refer to this as the Reductionist Problem of Scale. “Psychology isn’t real because it’s all just biology. Biology isn’t real because it’s all just chemistry. Chemistry isn’t real because it’s all just Physics.” I don’t see this as so much of a ‘minefield’ as a need to recognize that “scale matters”. In unaided-human-observable Newtonian space, there is no question that pebbles are “totally a thing”—they are. You can hold one in your hand. You can touch one to another one.
Of course; if you look solely at the scale of subquarks, then this distinction becomes unintelligible.
No. Interacting with the symbol of a thing is not interacting with the thing itself. They are, however, fully real—just like you yourself are fully real, but do not exist (you are not your body; you are not your brain; you are not the electrons and chemicals that flow through it. You are the pattern that is so-comprised. But that pattern itself is entirely non-physical in nature; it is non-instantiable and does not itself interact with anything—nor can it ever.)
I… am not rightly sure how you could come to the conclusion that this is a relevant question to the definition I provided. I did not say “to exist, things must be observed”—I said “to exist, things must interact with other things”. Pebbles interacting with lakes are interacting. Regardless of whether someone watches them.
If a tree falls in a forest, the tree exists. Regardless of whether it makes a sound.
Hmm. Under your definition, “to exist, a thing must directly interact in some fashion with other things which exist”. For this to be non-circular, you must specify at least one thing that is known to exist. I thought, this one certainly-known-to-exist thing is myself. If you say that under your definition I don’t exist, then what can be known to exist and how can it be known to do so?
There is nothing circular about the definition—merely recursive. “GNU” stands for “GNU is Not UNIX”.
As soon as you observe two things to directly interact with one another, you may safely asssert that both exist under my definition.
This is, frankly, not very complicated to figure out.
Recursive definitions must bottom out at some point. The ones that do not are called circular.
You didn’t say so before. Now, we two are interacting now (I hope), so we do exist, after all? And what about the characters in the virtual world of a computer game I mentioned before? I certainly saw them interacting.
So sorry for my stupidity.
See Corecursion, Non-well-founded set theory, Barwise&Moss Vicious Circles.
Cool, thanks!
I’m not convinced this distinction holds up all that well. For example, would you say that software “exists”? How about supply functions? Nations? Boeing 747s? People? Force fields?
Edit: yes, what gRR said.
No. But it is real. Software is a pattern by which electrons, magnetic fields, or photochemically-active media are constrained. The software itself is never a thing you can touch, hold, see, or smell, or taste; it never at any point is ever capable of directly interacting with anything. Just like you and me; we are not our bodies; nor our brains; nor the electrons or chemicals that flow through the brains. We are patterns those things are constrained by. I am the unique pattern that, in times past, created the password to the LW account, Logos01; and you the pattern that (I presume) created Eugine_Nier. But neither of us, physically, exist. This is important to notions of substrate-independence; where goes your pattern, is you. (Remember your Ship of Theseus problem.)
Right now I am downloading onto a VM on my workstation the 12.04 release of Lubuntu. This software is being pulled over ethernet to be delivered to a virtual harddrive image where it will be configured and installed. If I say I have LibreOffice installed too, it is clear I am talking about a specific release/instance/copy. We talk about identity in terms of software “Have you tried the latest Halo? It’s awesome! ^_^”—and two people can apparently own exactly the same game. But of course these are multiple copies of the pattern. It’s even possible to talk about backups and restores. This is because the only thing that matters about defining whether something is or is not the software is that pattern.
Real but do not exist. If every last person of the US packed their bags and got onto a rocket and shot themselves to Mars, it’d still be the United States of America. Even if every last person died while on that rocket and their kids were the ones who took over for them. Substrate independence once again demonstrates this.
Exist. It is possible to see/hear/touch/smell/taste a Boeing 747 (I hear they taste like burnt chocolate and chicken.) It is possible for two Boeing 747′s to be run into one another; or for a comet to strike one. It is not possible for a factory to churn out political constructs or minds. (Though it is possible for them to assemble all the pieces that would, when activated, allow for the presence of a mind.)
While you CAN take all the individual components of a Boeing 747 apart and put them back together again to make the same object; or over time transfer pieces into / from it (Theseus’s Grandfather’s Axe) -- what you can’t do is just “declare” a different physical object to BE that original Boeing 747. You can’t have five of the same Boeing 747. That is because it is a thing which directly interacts with other things.
See the above. If some temporal accident causes me to split into two, both of those people would still be ME. (Though their cohabitating the same space would cause divergence of identity over time.) Again, this is because what I physically am is irrelevant to determining my identity (and identity is the conformance to a specific pattern).
“In physics a force field is a vector field that describes a non-contact force acting on a particle at various positions in space.” You see that word, “acting”? To “act upon” something is quite literally definitional to being said to “interact with” a thing. By the definition I have provided of ‘exists’, and the definition of ‘force field’ as found on Wikipedia, force fields definitionally exist.
Granting this distinction for the moment, I still think that this points to how a platonist would answer your original question:
The platonist might reply: If the number 2 didn’t exist in the platonic sense, then one implication would be that you would be unable to construct proofs of that number’s existence within the formal systems that you’re thinking of.
In other words, the platonist might argue that formal systems such as PA are accurate (if incomplete) maps of the “territory”, which is the objectively real platonic realm within which abstract mathematical objects exist. If the statement that the number 2 exists were false of the territory, then, since the map is accurate, the corresponding formal statement would not hold within the map. Hence, an empirical implication of the number 2′s not existing would be that you would have the subjective experience of observing the failure of attempts to prove the formal statement that the number 2 exists.
This seems to me rather confused, because it is easy to construct a formal system with which we have precisely that experience. Consider this variant of the Peano Axioms:
0 is a number.
0 has a successor, which is also a number.
The successor of the successor of 0 is not a number. (Alternatively, “is 0”. I think this gives us arithmetic modulo 2.)
(Add the axoms of reflexivity, transitivity, and so on.)
Now clearly, in this formal system I cannot prove the existence of the number two, because its nonexistence is an axiom. Shall I conclude, then, that the number two doesn’t exist, on this account? By what standard are we to judge between formal systems in which 2 is provable, those in which it is disprovable, and those in which it cannot be proved either way? Do we take a vote? Is it a question of appeal to human intuition?
I included the qualification “the formal systems that you’re thinking of”. Were you thinking of that formal system when you wrote
?
This is a separate question from your original one. To answer your original question, the platonist need only point to a particular formal system (e.g., PA), and say that the nonexistence of the number 2 would mean [ETA: rather, would imply] that there would be no proof of 2′s existence in that particular system.
But the platonist would also find your new question interesting. For example, when you wrote “There’s a particular set of formal steps …”, how did you come to settle on that particular set of formal steps?
One position would maintain that some particular formal system is implicit in the statement of the twin prime conjecture (TPC). That is, when someone asks “Are there infinitely many twin primes?”, they are speaking in an abbreviated fashion, and they really mean “Does PA prove that there infinitely many twin primes?”, or “Does ZFC prove that …”, or something like that. This position would claim that number-talk has meaning only when the speaker has some specific formal system in mind.
The difficulty with this position is that people seemed to be making meaningful assertions about numbers for thousands of years before settling on any formal systems of arithmetic — indeed before they even had the concept of a formal system. (Euclid’s is presumably too unrigorous to count.) Ancient number-theory texts such as the Introduction to Arithmetic by Nicomachus often didn’t even include anything like what we would call a proof, not even in the sense in which Euclid’s arguments are called proofs. Nicomachus pretty much just flatly asserts number-theoretic propositions, perhaps bolstering his claims with a few examples. Yet somehow readers would reflect on these propositions and agree with them, even though Nicomachus didn’t communicate which formal system he was working within.
Another position would maintain that people absorb some formal system implicitly from their culture, and their number-talk is automatically in terms of that formal system. Even if that culture lacks the concept of a formal system, nonetheless all its number-talk is governed by some formal system.
But it is not even known whether the TPC is decidable in any of the standard formal systems. Suppose that the TPC were proved undecidable in, say, ZFC. Would the question really then lose all meaning? Consider a physical computer that brute-force factors one odd number after the next, and prints every consecutive pair of odd numbers that have no nontrivial factorizations. Would it really become meaningless to ask whether there is a bound on how many entries the computer would print, regardless of how many physical resources it were given? Granted, this is necessarily a counterfactual, but . . .
I am not a platonist, but I still can’t call myself a formalist, because I can’t bring myself to declare confidently that the TPC would become meaningless if it were proved undecidable in ZFC. Indeed, even if the TPC happens to be undecidable in every formal system whose axioms would seem to us to be “intuitively correct” of numbers, it still seems to me that our concept of number may suffice to determine a truth value for the above counterfactual. Either that computer would churn out pairs forever, or it wouldn’t.
I wasn’t, but I rather strongly opine that the word ‘exists’ should not be applied to a state that can change with the vagaries of what formal system I happen to be thinking of. At an absolute minimum, it should be qualified along the lines of “X exists within formal system Y” or “The existence of X is a theorem of formal system Y”. At which point I can return to my original question: What is the formal system of which “God exists” is a theorem or axiom? I also note that “Given axioms X, God exists” is somehow a rather less impressive claims than a floating “God exists”. Yet it’s so much more specific and satisfactory.
This is at least an answer to the question, “What is meant by ‘exist’?”; it gives us a definite procedure for deciding what does and doesn’t exist. But I opine that it’s not a very satisfactory one. Why that formal system and not some other one?
I don’t think so, but I’m not sure I understand the relevance, but what I’m objecting to is the word ‘exists’. Suppose you established that the computer was going to print these two numbers and then stop. That is an experimental prediction which we can test. (Updating our belief in the proposition upwards with every second that the computer prints nothing more.) I still don’t see the value in asserting on these grounds that something exists. Why not stick with what is observable, namely that the computer halts, or doesn’t halt?
If you want to say that ‘exists’ is a short form of “the computer halts”, fine; but it does not seem to me that this is what platonists usually intend to say. And, to return to the original problem, it is still completely unclear what “God exists” is shorthand for.
The platonist certainly agrees. The test I described would only work for “accurate” maps of the territory. The platonist would consider PA to be an accurate (but incomplete) map of the actual natural numbers, while the formal system you described is not.
Actually, I was a little sloppy, there. When speaking on behalf of the platonist, I shouldn’t have written “would mean”, but rather “would imply”. The point is that I wasn’t defining what “the number 2 exists” means, but rather describing what the world would be like if the number 2 didn’t exist.
At any rate, I don’t mean to be giving a “definite procedure”. For the platonist, PA is an accurate, but incomplete, map. Consider that there probably exist physical things of which we will never find any empirical trace. Similarly, the platonist expects that there exist mathematical objects that remain entirely unmapped by our formal systems or intuitions.
If you would be willing, on these grounds, to assert with some confidence that the computer will never print any more pairs, why would you demure from asserting, on these same grounds, and with this same confidence, “A largest pair of twin primes exists, and this most-recently printed pair is it.”?
(Just to clarify: The program I described never halts, even if the TPC is true. The program continues to run; it continues to brute-force factor progressively larger odd integers. It just never prints anything further to its output tape beyond a certain point.)
Ok, but what is the test which distinguishes between accurate and inaccurate maps? It seems to me that the reasoning here has become circular: It is asserted that the number 2 exists because otherwise, formal system X would be unable to prove it; and also that formal system X is a good test because the number 2 exists. I feel that at this point, you ought to abandon the formal systems and just go for straightforwardly asserting that the number 2 exists in a mystical, intuitive sense which is not open to rational disproof, but which you can use to test formal systems for accuracy.
Because this does not seem to me to match the meaning of the word ‘exists’. If I say that Planet X exists, I can point to it with a telescope and I expect in principle to be able to travel there. If I say that a species of animals exists, I am asserting the possibility of shooting one and eating it. What am I asserting when I say that a number exists? If it’s that a particular computer will print so many numbers and then stop, then I think this is not the same class of assertion as in the two previous examples, and it ought to have a different word.
Both maps are accurate maps of different parts of the territory.
Yet they both make assertions about the number 0, its successor, and its successor’s successor.
They use the word ‘successor’ to mean slightly different things.
I was a little sloppy on this point earlier, so I want to be more careful about it now:
Are you saying that “Species X exists” means exactly the same thing as “I could shoot and eat X in principle”? Or are you just saying that, if you show that you can shoot and eat X, you’ve shown that X exists? For example, is it meaningless to assert the existence of a species outside of your lightcone?
It becomes very tricky to talk without appearing to commit yourself to the existence of numbers.
For example, you say above that the computer is printing numbers. How is it doing that if numbers don’t exist?
That might seem like nitpicking, and you probably could find a perfectly adequate rewording of that sentence that avoids even the appearance of implying the existence of numbers. But it has proved very difficult to give a fully satisfying nominalist rewording of all talk that is ostensibly about abstract objects.
ETA: The question you raise about how we could know, on a platonic account, that a given formal system is an accurate map of the natural numbers is a good one. It’s a large part of why platonism is a nonstarter for me. But I don’t think that it’s incoherent in the same way that you seem to.
Well then, let it arrange groups of pebbles instead.
I gently suggest that you need to check your association maps, here. If you can give no account of this very fundamental thing, how we know that particular formal systems are the right ones to use, then isn’t it time to go from “platonism is a nonstarter for me” to “platonism is wrong”?
How would I learn about such a species? It may not be meaningless, but I don’t see how to connect it to any experience.
What do you mean by an “associate map” [ETA: oops, “association map”] in this context?
Unfortunately, there are many very fundamental things for which I am unable to give any account worth the time. Strangely, it seems that the more fundamental something is, the more difficult it is to account for it satisfactorily.
At any rate, you can take “platonism is a nonstarter for me” to imply “I think that platonism is wrong”. (What else would “nonstarter” mean?)
As in Belief in the Implied Invisible, a physical theory with strong empirical support could commit you to believing in such a species with high probability.
I mean that you should check that you’re not compartmentalising your beliefs. If we don’t regularly test the implications and associations between our beliefs, we can end up asserting contradictory things, like the theist scientist. That said, this:
seems to indicate that the exercise isn’t necessary for you here.
At any rate it apparently allows you to defend platonism to some extent. Perhaps ‘non-finisher’ would be a better term?
Yes, but then I could broaden my shooting definition slightly and say “At one time it was possible to shoot this species.”; in other words I would give the past empirie that convinced me of the species’ existence. Or alternatively, I could go for “in principle” and say that we just need some closed timelike curvies, or other means of extending the lightcone.
I didn’t see myself as defending platonism, so much as defending a certain kind of existence-talk that I, along with the platonists, think is legitimate. I agree with them that one should be able to consider the possibility that a largest pair of twin primes exists. Furthermore, it should be possible to do this without having a formal procedure for deciding the question in mind, even implicitly.
Couldn’t an empirically well-supported theory give high probability to the claim that there exists a species X whose future and past lightcones never intersect yours, and which is not accessible to you by any closed timelike curves or other means of extending the lightcone?
And besides, wouldn’t it be kind of weird if your ability to use the word “exists” were constrained by how sophisticated your physics is? It would seem to follow that someone who believes in a classical Newtonian universe allowing arbitrarily fast travel can legitimately ponder the possibility that dragons exist 10^100 light years away, while you cannot because you (let’s say) believe in a universe in which it is and was always impossible for you to interact causally with anything that far away.
Ok, but here is your original reply:
And all your replies since have taken the form “the platonist might say”. Can I suggest that you should defend the theory you actually believe in? At this point you seem to agree with me that theorems within some particular formal theory is not a good reason to say that numbers exist. What then do you mean by asserting the existence of the number 2? No more platonism, if you please, since you don’t believe in it.
No, actually, I don’t see how it could. If, by construction, I never, either in past or future, interact with any of the species or with anything that it has interacted with, then I don’t see how I can get an empirically supported theory of their existence.
Ok, just use the past interaction then.
I haven’t been able to come up with a full-fledged theory of what mathematics is about that I’m happy with. I can say some vague things, but there is no reason to burden you with their vagueness. I spoke from the perspective of platonism so that at least there would be a concrete theory of mathematical meaning on the table. I have no alternative concrete theory to offer in its place.
Nonetheless, all the difficulties I raised above about trying to avoid talk about the existence of the number 2 are difficulties that I really think are problems for your position. That is, they are issues that keep me from being convinced of your view.
I don’t see a principled way to avoid talking about the existence of a largest pair of twin primes in a sense that is independent of any particular formal system. You’ve said that you would allow talk about such a pair’s existence only if such talk amounted to statements about what certain sequences of computations would yield. However, this appears to commit you to the existence of sequences of computations. It doesn’t seem helpful to reduce this sense of existence to derivations within formal systems, because that commits you to the existence of formal systems — an existence, moreover, that appears to be in a sense that is independent of any particular formal system, lest an infinite regression drain all appearance of meaning from any kind of existence-talk.
Compared to all these abstruse abstract objects (sequences, computations, and formal systems), talk of the existence of numbers seems very innocent to me.
Furthermore, as I’m arguing in the “physics” thread of our conversation, existence-talk about physical objects seems to me to suffer from some of the same obscurities that existence talk about mathematical objects does: Namely, in neither case does a purely positivistic reduction of “exists” really work.
I admit that I’m confused about what “exists”, at bottom, really means. But I don’t know how to get by without speaking of the existence of numbers, any more than I know how to get by without talking about the existence of physical things.
Could we never have empirical support for a Tegmark Level I multiverse? More to the point, isn’t it at least meaningful to pose the possibility of such a multiverse, even though it amounts to suggesting the existence of many things with which you never can have and never could have had any causal interaction?
Past interaction were ruled out in my scenario (“while you cannot because you (let’s say) believe in a universe in which it is and was always impossible for you to interact causally with anything that far away.”).
I spoke sloppily. I meant that I would use ‘exist’ about a species I had interacted with in the past, not one I could in-principle interact with by breaking known laws of physics.
This gives me a new idea, actually. If you assert the existence of such a multiverse, you are saying that things like us exist. They have consciousness, they interact with objects, in short they have all the hallmarks of the existence of physical things. When I say that such a thing exists, I’m using the word in the same sense as when I speak of a rock. With what does a number interact? Nothing. If you allow interventions from outside the Matrix, I could interact with humans that are causally separated from my past and future. But even that power will not allow me to interact with the number 2; I cannot affect it, or it me, in any sense.
Taboo ‘interact’.
The decision-theory people on LW talk about agents controlling abstract mathematical entities in a way that I cannot so easily dismiss.
The post you link to talks about controlling a computer program by making decisions, in particular the decision to one-box. I think that this would sound rather less impressive if it were converted to talking about controlling an FPS by moving the mouse and pressing the space bar, which is functionally equivalent.
Those devices take in inputs. The post is talking about controlling programs that take no inputs. In other words, they are fixed mathematical structures.
Just because the input is a fixed simulation of the agent’s decision doesn’t mean the calculation as a whole has no inputs! In particular, it has whatever inputs are decisive for the agent itself. An agent that gets whacked with a stick every time it one-boxes is quite likely to make a different decision from one with the same algorithm but working on data that doesn’t include stick-whackings. It’s not sufficient to specify the laws of physics, you have to know the boundary conditions as well.
This is a universal counterargument against saying that anything besides what we are currently observing exists.
Yes and the number 2 exists within the natural numbers, but not within the model your system describes.
I would argue that the same argument MinibearRex uses here to justify his belief on a reality underlying our physical observations. Specifically, if there is no real system underlying our models how do you account for their seeming consistency?
What consistency? You just acknowledged that there are formal systems in which 2 doesn’t exist.
The fact that the formal systems are consistent.
And there are planets on which humans don’t exist. I don’t see how this is inconsistent.
Formal system A: The number 2 exists. Formal system B: The number 2 does not exist.
I cannot fathom how you can call these systems consistent. Each has a theorem whose negation is a theorem in the other. What possible meaning of ‘consistency’ describes this situation?
Map of America: Washington, D.C. exists. Map of Europe: Washington, D.C., doesn’t exist.
Each is a consistent map of its part of the territory. I never said they describe the same part.
As far consistency, would you say PA is as likely to be inconsistent as consistent, because if you believe that PA is just a game of symbols that doesn’t describe anything there seems to be no reason for it to be consistent.
Consistent with what? I believe it is consistent with itself, yes; but then again so is my toy variant with the mod-two arithmetic. If they’re describing a single reality they should be consistent with each other.
Your analogy fails, because PA and the toy system both agree in describing 0 and 1 as next to each other. But PA asserts that 2 is next to 1, while the toy system explicitly denies that it is so. The map of Europe doesn’t in fact make a claim about the existence of Washington; it just says that if it exists, it’s outside the map. But the toy system makes an explicit claim about the number 2. It’s not that it’s outside the range of the system; the system aggressively asserts that it covers the place where 2 would be if it existed, and also that there ain’t no number there.
Can you tell me your basis for this belief?
Answered here.
It sounds like what you’re looking for is a (large) corpus of material containing the majority of the community’s foundational insights and a ground-level-up course of rationality instruction. Fortunately, we have just such a thing! It’s called the Sequences. It’s a series of posts, mostly by Eliezer Yudkowsy, from 2007 or so onward, and is divided into several sections by topic. You can find it by clicking the “sequences” link on the upper right hand corner of this page, or just click here. A warning: the Sequences are a lot of material—more than the entire Lord of the Rings trilogy. It’s probably best to start with the “core sequences”, the ones titled “map and territory”, “mysterious answers to mysterious questions”, and “how to change your mind”. There’s a more detailed explanation of reading order and summaries of what topics various sequences cover on the above page, and you can read whichever ones seem most likely to be useful to you.
Dear Normal_Anomaly, I thank you for the kindness and tone of your answer. Could I upvote it I would. I’m aware of the existence of the sequences but it’s still not quite what I mean. The sheer size of them detracts a lot from their usefulness and there seems to be no organization.
What I mean was some kind of page where one could self or externally assess and then based on his shortcomings be directed to adequate pages.
So something like: To Win you must:
-Add mindware -Fix corrupted mindware -Fix cognitive miserliness
Then adequate assessment of the state of these elements and links based on this organization (so, adding mindware would link to probability theory, logic, the virtue of scholarship; fixing corrupted mindware would link to debiasing, dissolving the question; and so forth) [based on lukeprog’s “A cognitive Science of Rationality”].
This is just a model of how it could be, just a way of organizing it. Which is what appears to be missing, organization.
Cheers
It’s true that LessWrong could have more focus on catering to newcomers as opposed to catering to people who’ve been here for months; which is to be expected when most content (posts and wiki) are made by long-timers.
I guess the best approach for newcomers for now is just to show up and talk to people (like what you’re doing now!) - not as smoothly organized as having a formal system, but at least it’s a self-correcting process that adapts to the quirks and interests of people better than a pre-programmed self-assessment system could!
User interfaces are hard, especially if they’re for the general public. Empirical testing helps.
Maybe we should start with what sort of things you personally would like to learn.
Just my opinion, but I think Eliezer’s posts are what they are because he doesn’t just want to say “here is how to be rational”, he wants to give convincing arguments for why rationality makes sense. Not only does this show more respect for his readers’ minds, it improves the odds that their understanding of rationality will resemble his. I’m hoping that there will be good ways of explaining rationality to people of average intelligence, but (at least in this community) this isn’t close to being developed yet.
For the fun of it, is mathematics invented or discovered? If discovered, what sort of things are being explored? If invented, why is there so much commonality of results?
If you want to see somewhat about why PUA is such an ambiguous thing, check out Clarisse Thorn’s Confessions of a Pickup Artist Chaser.
Thanks for the shoutout!
/Adds blog to bookmarks
On reading the first paragraph, I hoped this post would be an argument in favor of mathematical realism. Somewhat disappointed that it’s not. Ever since reading Tegmark’s paper on the Mathematical Universe Hypothesis, math realism vs math formalism has been at the back of my mind. The MUH fascinates me, but my prior for math realism is pretty low. (Possibly lower than 0.3).
I don’t get the impression that mathematical platonism is especially looked-down-upon around here. (I’m not one myself; maybe the actual mathematical platonists feel more oppressed than they seem to me to be.) There are some lines in Eliezer’s own posts that seem to me to be expressing a flavor of mathematical platonism. I’ll try to track some down if anyone is interested.
I am very interested.
I was thinking of this post, which has lines like these:
(Bolding, but not italics, added.)
Thanks for pulling that up. I tried reading through the entire post, but I became confused at several points. Might be because I haven’t read the previous posts in that particular series. Or maybe I’m just overly tired. I’ll take another crack at it tomorrow.
From what you quoted, I do have to positively update my degree of belief in the “Eliezer is a mathematical Platonist” hypothesis. It’s weak-medium evidence, but still evidence. I think much stronger evidence would be if he actually identified as a math realist. If he said, “Hey guys, I believe the natural numbers exist in a mind-independent, non-spatiotemporal way.” and proceeded to explain how to meshes well with reductionism, naturalism, etc.
If someone at in contact with him sees this and could ask him for me, that’d be awesome.
This quote from the post Beautiful Math seems to take a position of “math is in part pre-existing (at least those that involve physics) but Platonia write large is illusionary.” It’s written in 2008 and EY doesn’t seem to have changed his mind in the interim.
Stars are mine. EDIT: Erm Italics. Damn sorta reddit markup while also having some differences!
http://lesswrong.com/lw/mq/beautiful_math/:
“The joy of mathematics is inventing mathematical objects, and then noticing that the mathematical objects that you just created have all sorts of wonderful properties that you never intentionally built into them. It is like building a toaster and then realizing that your invention also, for some unexplained reason, acts as a rocket jetpack and MP3 player.
Numbers, according to our best guess at history, have been invented and reinvented over the course of time. (Apparently some artifacts from 30,000 BC have marks cut that look suspiciously like tally marks.) But I doubt that a single one of the human beings who invented counting visualized the employment they would provide to generations of mathematicians. Or the excitement that would someday surround Fermat’s Last Theorem, or the factoring problem in RSA cryptography… and yet these are as implicit in the definition of the natural numbers, as are the first and second difference tables implicit in the sequence of squares.
This is what creates the impression of a mathematical universe that is “out there” in Platonia, a universe which humans are exploring rather than creating. Our definitions teleport us to various locations in Platonia, but we don’t create the surrounding environment. It seems this way, at least, because we don’t remember creating all the wonderful things we find. The inventors of the natural numbers teleported to Countingland, but did not create it, and later mathematicians spent centuries exploring Countingland and discovering all sorts of things no one in 30,000 BC could begin to imagine.
To say that human beings “invented numbers”—or invented the structure implicit in numbers—seems like claiming that Neil Armstrong hand-crafted the Moon. The universe existed before there were any sentient beings to observe it, which implies that physics preceded physicists. This is a puzzle, I know; but if you claim the physicists came first, it is even more confusing because instantiating a physicist takes quite a lot of physics. Physics involves math, so math—or at least that portion of math which is contained in physics—must have preceded mathematicians. Otherwise, there would have no structured universe running long enough for innumerate organisms to evolve for the billions of years required to produce mathematicians.”
I don’t think that physics existed before humans; physical law did. Physics is the study of physical law, just as math is the study of mathematical objects and systems of laws that are obeyed by those objects. Thus I wouldn’t say that math existed before intelligent life, but nathenatical objects did.
On first blush that seems to be a semantic argument. It doesn’t seem you actually disagree with EY, but rather you seem to object to the use of the Physics and put in its place “Physical law” and put “mathematical objects” in place of “mathematics.”
Is this an accurate description of what you are trying to say?
yep. I figure we should be more specific with our words if we’re going to be understood properly.
Very informative! Thank you kindly for pulling this up for me.
Eliezer also wrote multiple times that he’s an “infinite set atheist”. I’m not sure that’s actually compatible with mathematical Platonism. (The way I understand it, at least.)
Also, see How to Convince Me That 2 + 2 = 3.
I don’t see why such a roadmap should exist.
Rationality isn’t something one ought to do for its own sake and hence calling its practitioners artists seems misguided.
Just because someone doesn’t have a ‘cause’ to shout about from the rooftops, it doesn’t mean that that same person has no reason to want to be more rational.
I never really agreed with that post; it seems simpler to say that because it is easier to judge instrumental rationality’s effectiveness, it is less likely to become corrupted with idiosyncrasies or blind spots. This is not a sufficient condition for saying that rationality for its own sake (or just without an overriding, save-the-world ‘purpose’) is doomed to failure
This seems contrary to LessWrong’s best interests. Communities live or die by how they treat new members and discouraging newcomers will lead to stagnation, marginalization, and eventual irrelevance. There is admittedly a large inferential gap between a theist and your typical LW member, but what would you say to someone who had just come in from reading HPMOR and wanted to know more about LessWrong? “Sorry, come back when you have a cause like an anime character.”?
Sorry for the rant, but this really rubbed me the wrong way.
I don’t disagree, but becoming a PUA is the end result of studying whatever it is the field is called, so in this twisted analogy presumably RA is the end result of studying rationality. My point is that if you develop rationality in a vacuum and never significantly confront anything with it, there’s no way to know if your rationality is actually effective, and no reason to improve its flaws—and this is basically what you say in the second paragraph, so surely we agree here.
It sounds like you were turned off by EY’s illustrations with anime references, and not the actual conclusion of the article. In any case, I suspect you would have agreed with it had it been written about e.g. HPMoR!Harry’s obsession with science-ifying magic.
EDIT: If by some chance that question wasn’t rhetorical, of course I wouldn’t say that.
It took me a (comparatively, compared to some other strangeness in the Sequences) short time to get past all of the anime references. I don’t think that Harry wanting to science-ify magic would have been enough to bring me around, as what I don’t like about the ‘something to protect’ post is that it seems to say that wanting to be more rational for small, mundane, and more importantly, common reasons aren’t enough.
Not wanting to be ripped off at the car dealership, trying to find the best way to make economic profits, out-competing rivals, etc., are not sufficient for rationality, only a grand purpose like FAI or cryogenics or curing cancer or designing more efficient wheat yields like Borlaug are enough, otherwise you’re just wasting everyone’s time and should be content being a mortal.
From the article:
You took the analogy too far.
There’s no real way you could have known this if you just got here, but references to pickup artistry tend to rub people the wrong way. There was a flame war a couple years ago (before I joined the site) between people who thought studying PUA would be useful for romantic success, and people who thought PUA presents romance/dating as overly adversarial and therefore immoral or more harm than good. Other analogies that get your points across equivalently will be better received.
Ah thanks. I didn’t know that back story. I always kind of inferred it was a touchy subject around these parts, but I wasn’t around for that. Was there ever any real conclusion to that? It seems to me like the consensus has been towards the former, rather than the latter, but I could be wrong.
Your guess may be as good as mine. As far as I can tell, the consensus was that we shouldn’t talk about it. This probably came from both the people who believed it was wrong and the people who just wanted to get on with life.
It’s a bad analogy.