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Saying that 2+2=4 is a tautology in a certain axiomatic system defined with '+' means that you couldn't have anything but 2+2=4 in that system. It's simply mandatory, and a rational person could not wake up one day and be convinced that 2+2=3 within a self-consistent system that deduces 2+2=4.

While tautological truth is independent of observation (let's call it mathematical truth), it is dependent upon context (i.e., a self-consistent axiomatic system). Some mathematical truths in one axiomatic system are false in another. When we talk about whether a a mathematical statement is true, we need to specify the context, and, in my opinion, in the most demanding definition of truth, the context is the real, actual, empirical world. So I agree with Eliezer that a mathematical tautology must be observed in order to be true.

When we humans talk about "2+2=4", it is because we have chosen arithmetic from an infinite number of possible axiomatic systems and given it a name and a set of agreed-upon symbols. Why did we do that? Because we observed arithmetic empirically. Obviously, addition is just one operation of infinitely many operations. The ones we have defined (multiplication, subtraction, addition mod n, taking the cardinality of subsets of, etc.) usually have some empirical relevance. While we don't feel very comfortable thinking of those that don't (and this says somethng about the way we think), I have faith that if we were presented with a very strange set of observations, it would take a pretty short amount of time to train ourselves to think of the new operation as a "natural" one.

... I idly wonder if there is a such thing as a mathematical truth that could not be realized empirically, in any context, and if there would be any way of deducing it's non-feasability.

Jack16y10

Is saying "we could have a different axiomatic system" different from saying "2, 4, +, and = could all mean different things? Of course we've only defined the operations and terms that are useful to us. I don't care about the naturalness of '+' only that once I know the meaning of the operations and terms the answer is obvious and indisputable.

Math isn't my field, so my all means show me how I'm wrong.

See in context

41 "What Is Wrong With Our Thoughts"

by Eliezer Yudkowsky

17th May 2009

1 min read

109

41

"But let us never forget, either, as all conventional history of philosophy conspires to make us forget, what the 'great thinkers' really are: proper objects, indeed, of pity, but even more, of horror."

David Stove's "What Is Wrong With Our Thoughts" is a critique of philosophy that I can only call epic.

The astute reader will of course find themselves objecting to Stove's notion that we should be catologuing every possible way to do philosophy wrong. It's not like there's some originally pure mode of thought, being tainted by only a small library of poisons. It's just that there are exponentially more possible crazy thoughts than sane thoughts, c.f. entropy.

But Stove's list of 39 different classic crazinesses applied to the number three is absolute pure epic gold. (Scroll down about halfway through if you want to jump there directly.)

I especially like #8: "There is an integer between two and four, but it is not three, and its true name and nature are not to be revealed."

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[-][anonymous]16y90

Stove himself concludes that his "nosology" is probably not worth compiling. I think he's actually just using it to make the same point you've made by mentioning entropy. He considers it in order to justify rejecting it.

He then does something similar with the possibility of figuring out individual cases, rejecting it because the findings won't be generalizable.

Then he gets to what seems like his main point: getting rid of almost all philosophy because it's crazy.

(I thought the piece as whole was much funnier than the list. It's a tongue-in-cheek version of bending over backwards to avoid accusations of dismissing something crazy out of hand.)

[-]RolfAndreassen16y70

Possibly Stove intended this only as an extended Take That to philosophers he dislikes; but it seems to me that he is a bit too dismissive of his own project, the 'nosology'. Without wanting a Fully General Counterargument, I think it might be useful to have a set of, say, five or six different classes of erroneous statements; and I also think Stove is too eager to insist on the singularity of each of his examples. For example, he states that the objection "not verifiable" cannot be applied to his example 8; I don't see why not. Anything whose &q... (read more)

0cousin_it16y

I'd unify your "So What" with "meaningless" into a single category "does not constrain observations". Math passes the test inasmuch as it constrains observations about outcomes of proof checking. But now some people will complain (are already complaining) that we reject the majority of humanity's thought.

0RolfAndreassen16y

Again, it does seem observable that nobody has explained why three is prime and four isn't. (I'm not sure you can actually use 'why' in an intelligible way here; possibly I'm being confused by non-mathematical language applied to math.) It's not an observation I would expect anyone to care about, and possibly it may be the equivalent of nobody having seen something invisible; but it does seem to make a statement that could in principle have gone the other way.

0thomblake16y

I agree that I'm not sure how you're intending to use 'why' here, and I'm pretty sure there's a good answer for any particular meaning. To answer the question in a possibly unsatisfactory way, 3 is prime because it is a natural number which has exactly two distinct natural number factors, whereas 4 is not prime because it has more than two distinct natural number factors.

0Annoyance16y

What humanity does isn't "thought", by and large. Not in any meaningful sense. It's mostly the expression of prejudices combined with associational triggers and repeating what others say. Part of becoming an effective thinker is recognizing that unpleasant realities need to be acknowledged even when we'd prefer they weren't the case. For people living in this time, in this place, one of those truths is that we're surrounded by blatant stupidity. Even worse, we're blatantly stupid a lot of the time. Deriving those conclusions from the evidence, and then acknowledging their validity, is one of the basic necessary steps to becoming better. No problem can be (expected to be) solved if we deny its reality.

[-]Matt_Simpson16y60

I have to say that the positivist critique that "it's all meaningless" is seductive and it may well be correct - it feels like the words have meaning, but when you try to parse the sentence the feeling quickly disappears.

The problem is, this isn't very useful for talking about specific errors and how to avoid them. Many of the statements on that list looked rather meaningless to me, but to someone who believes in one of these statements, there are some underlying beliefs or confusions that need to be addressed before the "meaningless critique" will have any effect. At this point, pointing out the meaninglessness of their pet statement becomes entirely superfluous.

[-]Jack16y100

There is a pretty innocent reason for why those passages look meaningless– they're all jargon filled when you don't know what the jargon means you will likely fail to understand what the passages mean. A paper on quantum chromodynamics is going to look meaningless to someone who doesn't know what quarks, quanta, flavor symmetry, gluons, hadrons, chirality etc. refer to. Similarly, I assume most people here have no idea what Plotinus means by "Being", "Essence", "Intellectual-Principle", "form" etc. I've done course work on Neo-Platonism and I don't remember what all of that was about. The same goes for the other passages.

Now Plotinus is particular might still be meaningless since some of that jargon is actually meant to refer to real things that he thinks exist. And insofar as he is referring to non-existentials whether or not the passage is meaningful depends on your philosophy of language (it is either false, meaningless or non-propositional).

Occasionally you find an analytically trained philosopher working on continental subject matter and they tend to assure me that the jargon and unconventional usage actually DO mean things. What does... (read more)

1jimrandomh16y

No, the passages given in the article have much deeper problems than just the jargon. The jargon only serves to defend these texts from criticism; because they're difficult to understand, anyone who says that these passages are wrong or mere gibberish can be accused of not understanding them. This defense works even if the critic understands the text perfectly.

4Jack16y

Uh, maybe. I'm willing to hear arguments to that effect. But you didn't give one. I think Plotinus is definitely wrong, I don't know enough about Hegel to form an opinion, and I disagree with what I know of Foucault. But that doesn't make what they wrote meaningless.

0jimrandomh16y

Arguments to what effect? Are you objecting to my claim that "you don't understand" is used inappropriately to defend bad philosophy, to the claim that jargon makes it easier to do so, or to my claim that the passages have deeper problems?

6Jack16y

Sorry, I should be specific. I don't think the passages, or the writing of these philosophers and the well-know continental philosophers, generally, are gibberish. I think the reason people think they are gibberish is because of the jargon. I would like to see an argument for why I should consider them gibberish for reasons other than jargon I don't understand. And since I hold that the jargon is meaningful, I don't think that the jargon "only" serves to defend the texts from criticism (did you really mean "only")? I also, deny that a critic who understands the text perfectly would argue that the text is meaningless– but that issue will be addressed by the argument I ask for above. (Note: Of course there are deeper problems to these passages. But those problems don't have anything to do with the syntactic rules for sentence formation or semantic rules for word usage. In other words, the problem isn't that their gibberish.)

1jimrandomh16y

I define "gibberish" to mean "difficult to understand and entirely or almost entirely false or meaningless". Since you have said you think Plotinus and Foucault are wrong, and I think we can agree that they're at least somewhat obfuscated, then we must have different definitions. What's yours?

5Jack16y

I define gibberish as "difficult to understand and entirely or almost entirely meaningless". I think Plotinus and Foucault are "difficult to understand and entirely or almost entirely false". A statement is meaningless if it either fails to follow rules of syntax, i.e. "Running the the snacks on quickly!" or semantics, i.e. "Green ideas sleep furiously." The distinction is actually pretty important. If you know something is meaningless then you can move on, but you can't decide something is false without first considering the argument, obfuscated or not. There is some middle ground when it comes to arguments about things that don't exist. The trinity argument (and probably Plotinus) appeals to something that doesn't exist and so it says things that would be meaningful if the holy trinity was real but can't really be evaluated since there is no such thing. Obviously there is no reason for you to care much about this argument. But I don't think Hegel, Foucault or Heidegger and the other usual suspects are talking about things that don't exist.

7saturn16y

Syntax does rules necessarily broken imply meaninglessness not.

2Jack16y

Semantic rules aren't holding knives to the throat of meaning either. So yeah, it is more complicated than what I said before because our brain is pretty good at fixing broken sentences with context. Rules for context and pragmatics should also be included in requirements of meaningfulness. My bad for missing that.

3cousin_it16y

The word "exist" confuses you. Does three exist? Maybe yes, maybe no; what real-world consequences would arise from three existing or not? If a tree falls in the forest, etc. Humanity to date knows two families of statements that appear to possess truth values independent of the listener's psychology: 1) Experimental results, objectively verifiable by repeating the experiment. 2) Axiom-based mathematics, objectively verifiable e.g. by proof checking software. Of course people can make personally or culturally meaningful statements that don't fall into type 1 or 2. Just don't delude yourself about their universal applicability or call them "science".

6Jack16y

First, the word exist does not confuse me anymore than it confuses anyone else. If you think it does you should say why, since it wasn't explained in the previous post. The ontological status of numbers is a classic and ongoing philosophical dispute, whether there are real-world consequences to the question, I don' t know but even if there aren't it does not follow that the question has no truth value. Experimental results don't verify anything, they either falsify or fail to falsify huge sets of different scientific propositions. When an experimental test of a hypothesis comes up false one can dismiss the hypothesis or one can dismiss any number of auxiliary assumptions that you had when you made your hypothesis. It is the job of scientists to find the best interpretation of experimental results according to criteria such as parsimony, consistency, usefulness, etc. But scientific theories are better understood as best working interpretations not objectively verified truths that exist independent of human interpretation. Metaphysics uses the exact same criteria to try and figure out the best interpretations with regard to other issues for which experiments are sometimes relevant but often not. Also, axiom-based math can't really be addressed by proof checking software since you can't program proof-checking software before discovering some axiom based mathematics. Plus it isn't like we started believing math was true 60 years ago. We figured it out because our vulnerable, biased, human brains happen to have considerable abilities for ascertaining the truth. Anyway, we also know things based on non-experimental observation and data gathering. This includes non-scientific things like whether or not there is a car on the street as well as the less experimental sciences like, astronomy, linguistics and economics. Knowledge in linguistics and economics is certainly somewhat more precarious than in physics since in the former fields it is by turns often impossible or un

1cousin_it16y

You might have missed my emphasis on well-transferable truth value. Even if the "ontological status of numbers" question has a well-defined truth value to you, or non-experimental economics, or linguistics... how do you transfer the answer between individuals? I've indicated two methods of independent verification that correspond to science and math; is there a third one? Persuasive-sounding literature doesn't cut it, because it can be used for religion just as well. About your final question, what makes philosophy distinct from literature?

4Jack16y

Fine. My point was that experimental results aren't perfectly transferable either. They require interpretation. Hypotheses are proven and disproven in the context of hundreds of other assumptions. We construct models and theories that we think best explain experimental data. Linguists and economists construct models and theories that best explain observed economic behavior and language use. Linguists and economists, granting some exceptions, agree with each other on lots of issues! There is not as much agreement as there is in physics since there is more room for biases and preconceptions when you can't repeat an experiment over and over again to prove someone wrong. But that doesn't mean linguists and economists aren't getting less and less wrong. Philosophy is a broad subject. A lot of it (definitely most of it in the English speaking world) isn't just culturally meaningful but scientifically meaningful. In epistemology they try and form models and theories that best explain our understanding of what counts as knowledge. In ethics they do the same for our understanding of what counts as moral. In philosophy of science they try and explain our understanding of what counts as science. There are also fields such as philosophy of physics and philosophy of biology which address philosophical and interpretive issues within those fields like interpreting quantum mechanics (where the philosophers new the Copenhagen interpretation was bunk long before the physicists began realizing it) and the ontological status of biological categories like 'species'. There is Philosophy of Mind, Philosophy of Time, Philosophy of Mathematics etc. That doesn't even include the fields that were once part of philosophy but expanded to become separate subjects (logic, linguistics, cognitive science).There is even something called experimental philosophy where they, you know, experiment. I don't know what your experience with philosophy is. Maybe you just had someone make you read Plato. Now

0jimrandomh16y

So you maintain that anything which follows a few syntactic and semantic laws cannot be gibberish? I disagree; text can have meaning and still be gibberish. Consider a sequence of words drawn uniformly at random from a dictionary, then slotted into a repeating template like (noun) (verb) (article) (adjective) (noun). The template ensures that no rules of syntax are violated. A few constraints on the vocabulary can ensure there are no egregious violations of semantic rules, like green ideas and furious sleeping. Restrict the vocabulary to a few hundred concrete words and you can even ensure that every sentence makes a testable prediction. But it's definitely gibberish.

2Jack16y

Well there are a lot of semantic rules and plenty that we've haven't formalized. So I'm not convinced anyone now alive could write such a program. But I'm not a programmer so maybe someone has proved me wrong. However,iIf they were successful I don't think I would consider the result gibberish- especially if each sentence made a testable prediction. In this case wouldn't some of the predictions be true? If so then it is clear that your definition is not broad enough. Thats troubling since I had already concluded your definition was too broad because it seemed to include important but complex and falsified scientific claims,

0[anonymous]16y

If you narrow it down to one passage, I will. The reason I didn't before was because you referred to all the passages collectively, and I don't have time to analyze all of them.

[-]PhilGoetz16y30

The history of philosophy can't really have been one of thousands of years of nearly unrelenting adoration of stupidity. What probably happened is that philosophers became popular only if their ideas were simple enough and appealing enough. There is a bandpass filter on philosophy, and it has both a low and a high cutoff.

We propagate knowledge by collective judgements about it. In fields where we can't eliminate bad ideas by experiment, both the very worst and the very best ideas must be rejected. The requirement that an influential philosopher appeal ... (read more)

7jimrandomh16y

On the contrary, philosophers became popular only if their ideas were complicated enough to fill a book. The ideas that were simple enough to be true were also too short to publish.

1PhilGoetz16y

An interesting possibility. (Nitpick: "Simple enough to be true" implies that complex ideas can't be true. This is wrong.) Can you give an example of a simple but non-obvious truth that was available but passed over in philosophy?

0AllanCrossman16y

What do you mean by "available"?

0PhilGoetz16y

Eg., I'm not interested in hearing that medieval philosophers ignored the idea that the motion of the planets are governed by the same laws that govern the motion of bodies on earth.

1AllanCrossman16y

So, are we looking for something which is: * Simple, * True, * Not obvious, * Was claimed as true by someone or other, * But mostly ignored? Perhaps Aristarchus and his heliocentrism would fit the bill (while not strictly true, it was truer than the alternative).

4ChrisG16y

I for one would be interested in hearing these sentences, and also which fields you feel are being held back by simple errors of logic. The margins here are quite large ;).

7PhilGoetz16y

Some examples off the top of my head: Rodney Brooks and others published many papers in the 1980s on reactive robotics. (Yes, reactive robotics are useful for some tasks; but the claims being made around 1990 were that non-symbolic, non-representational AI was better than representational AI at just about everything and could now replace it.) Psychologists and linguists could immediately see that the reactive behavior literature was chock-full of all the same mistakes that were pointed out with behavioral psychology in the decade after 1956 (see eg. Noam Chomsky's article on Skinner's Verbal Behavior). To be fair, I'll give an example involving Chomsky on the receiving end: Chomsky prominently and repeatedly claims that children are not exposed to enough language to get enough information to learn a grammar. This claim is the basis of an entire school of linguistic thought that says there must be a universal human grammar built into the human brain at birth. It is trivial to demonstrate that it is wrong, by taking a large grammar, such as one used by any NLP program (and, yes, they can handle most of the grammar of a 6-year-old), and computing the amount of information needed to specify that grammar; and also computing the amount of information present in, say, a book. Even before you adjust your estimate of the information needed to specify a grammar by dividing by the number of adequate, nearly-equivalent grammars (which reduces the information needed by orders of magnitude), you find you only need a few books-worth of information. But linguists don't know information theory very well. Chomsky also claims that, based on the number of words children learn per day, they must be able to learn a word on a single exposure to it. This assumes that a child can work on only one word at a time, and not remember anything about any other words it hears until it learns that word. As far as I know, no linguist has yet noticed this assumption. In the field of sciencology?,

5PhilGoetz16y

Though you also see cases where people from the outside do get their message across, repeatedly, and fail to make an impact. Something more is going wrong then. The FDA, in its decision whether to allow a drug on the market, doesn't do an expected-value computation. They would much rather avoid one person dying from a reaction than save one person's life. They know this. It's been pointed out many times, sometimes by people in the FDA. Yet nothing changes. EDIT: Probably a bad example. The FDA's motivational structure is usually claimed to be the cause of this. Maybe when one particular stupidity thrives in a field, it's because it's a really robust meme for reasons other than accuracy. There are false memes that can't be killed, because they're so appealing to some people. For example, "Al Gore said he invented the Internet" - a lie repeated 3 times by Wired that simply can't be killed, because Republicans love it. "You only use 1/10th of your brain" - people love to imagine they have tremendous untapped potential. "Einstein was bad at math" - reassures people that being good at math isn't important for physics, so it's probably not important for much. So, for example, NASA keeps trying to get ET's attention, not because it's rational, but because they read too many 1950s science fiction novels. The people behind project Hindsight and Factors in the Transfer of Technology wanted to conclude that basic research was ineffective, because they were all about making research efficient and productive, and undirected exploratory research was the enemy of everything they stood for. Saying that humans have a universal grammar is a reassuring story about the unity of humanity, and also about how special and different humans are. And the FDA doesn't picture themselves as bureaucrats optimizing expected outcome; they picture themselves as knights in armor defending Americans from menacing drugs.

3Nick_Tarleton16y

This, and your comment below, should be top-level posts IMO.

1Douglas_Knight16y

These are interesting examples, but they're not what I envisioned from your original comment. (The Brooks example might be, but it's the vaguest.) A problem is that people gain status in high-level fights, so there is a lot of screening of who is allowed to make them. But the screening is pretty lousy and, I think, most high-level fights are fake. Are Chomsky's followers so different from other linguists? Similarly, Brooks may have been full of bluster for status reasons that were not going to affect how the actual robots. It may be hard for outsiders to tell what's really going on. But the bluster may have tricked insiders, too. Also, "You don't understand information theory," while one sentence, is not a very effective one.

1steven046116y

People are still doing it, not NASA though. Their rationalizations can get pretty funny. It seems stupid but rather harmless; it's hard to find a set of assumptions under which there's a nontrivial probability that it matters.

3Richard_Kennaway16y

TED.

3Douglas_Knight16y

I think that you're saying that the outsiders can't be published without learning the jargon and doing experiments. But publication is not the only avenue. If it really only takes a single sentence, the outsider should be able to find an insider who will look past jargon and data and listen to the sentence. Then the insider can tell other insiders, or tack it onto a publication, or do the new experiments. If jargon is not just a barrier to publication, but also to communication it's a lot harder to find a sympathetic insider, but it hardly seems impossible. Also, in that situation, how can outsiders be sure they understand? These situations sound like there is a much bigger problem than the elementary error, perhaps that the people involved just don't care about seeking truth, only about having a routine.

0MrShaggy16y

"These situations sound like there is a much bigger problem than the elementary error, perhaps that the people involved just don't care about seeking truth, only about having a routine." Well, a large part of it is funding/bureaucracy/grants. I tend to thing that's the main part in many of these fields. Look at Taubes's Good Calories, Bad Calories for a largely correct history of how the field of nutrition went wrong and is still going at it pretty badly. You do have a growing number of insiders doing research not on the "wrong" path and you did all along, but they never got strong enough to challenge the "consensus" and it's due not just to the field but the forces outside the field (think tanks, government agencies, media reports). So even being published and well-known isn't enough to change a field.

2CannibalSmith16y

It's just you.

2PhilGoetz16y

I was talking about the content of artificial intelligence books published in the 1980s. None of the examples you gave involved anything from the BOFAI school of artificial intelligence; nothing that would have been in those books.

[-]ShardPhoenix16y20

While I mostly agree with the article, I don't think the Foucalt example given at the start is entirely bad - it just seems like a long-winded warning against confusing the map with the territory (or more specifically against trying to hammer a square territory into a pre-conceived round map).

[-]PhilGoetz16y20

Asking the big question!

[-]Annoyance16y20

The history of philosophy can't really have been one of thousands of years of nearly unrelenting adoration of stupidity.

I often see statements like that. "This couldn't possibly be the case", "that can't really happen", etc.

The first question we should ask ourselves when we see such statements: Why?

Usually, the person speaking is dismissing possibilities and potentialities out of hand for one of a variety of reasons, rather than having a valid and justifiable reason for discarding the contingency.

And even when there are good reason... (read more)

[-]hrishimittal16y20

Genetic engineering aside, given a large aggregation of human beings, and a long time, you cannot reasonably expect rational thought to win. You could as reasonably expect a thousand unbiased dice, all tossed at once, all to come down 'five,' say. There are simply far too many ways, and easy ways, in which human thought can go wrong. Or, put it the other way round: anthropocentrism cannot lose.

That's the same argument against rationalist winning that has been seen many times on LW. However, it is based on hopelessness and fear, rather than on knowledge... (read more)

0[anonymous]16y

Hi, "first time, long time." :-> The way I read that, I thought he was talking about even larger, longer term societal structures. Like, imagine many generations of atheist eudaimonia that doesn't collapse on itself -- creating ridiculous new philosophy-religions, over generations. The author's future history seems to involve static human nature at play for a long, long time. Someone needs to give this guy a hug. Or, even better, a copy of "Engines of Creation". And, according to wikipedia, he died in 1994...

[-]cousin_it16y10

I've long loved this piece, but today would file most of its examples simply under "getting carried away".

Items on the list that reminded me of Eliezer's writings: #19, #22, #32, #35. Indictment not intended.

0[anonymous]16y

Sir, you can bet that if I ever used a phrase like "transcendental unity" I would be able to tell you exactly what it meant. It would probably have math in back of it. Because if I wanted to talk about something I couldn't really define, I would call it something much less impressive and as close to ordinary English as possible, for fear of being called on it. The above is not applicable to Eliezer_2003 and below.

[-]phane16y10

I don't like this paper. It's wholly scathing for no reason other than to justify ignoring all of philosophy. Some philosophy is valuable and some is not, and of his 40 statements about three, I'd say 6 of them are claims I would take seriously and would hear arguments for, were I interested in the nature of three.

Generally, continental philosophy is trash, but I wouldn't throw out the baby with the bathwater.

7PhilGoetz16y

Analytical philosophy is a quest for truth; continental philosophy is a way to get laid. (I hear it works better in France.)

1Tyrrell_McAllister16y

But it bears noting explicitly that many of his examples represent positions from within analytic philosophy. For example, "23 The proposition that 3 is the fifth root of 243 is a tautology, just like 'An oculist is an eye-doctor.'"

2Eliezer Yudkowsky16y

I'd agree with that, actually, I'd just note that tautologies have to be empirically observed somehow and also that the case of the oculist and the eye-doctor is nowhere near as clear-cut.

3Jack16y

The piece about tautologies having to be empirically observed is one of the most bizarre posts I've ever read by you. It is so strange that I'm not really sure if there is anything I can say that would change you mind if you really think you could be convinced that 2+2=3 in that way. I can't even tell where you went wrong. Do you also hold that that the identity relation has to be empirically observed? Could you be convinced that 4=3? That 3 doesn't = 3? Do you believe you could be convinced that triangles on Euclidean planes are round? Do you not trust modus ponens and modus tollens? How does one even empirically observe tautologies in symbolic logic?

2JGWeissman16y

That 2+2=4 is a fact about a mathematical system that exists independently of the physical universe, including us humans that decided to use those symbols to express that fact. That fact is in the territory. But, in order to interact with the physical universe, it has to be discovered by some physical system that explores logical conclusions, such as our brains. This exploration builds our map of the territory. Our uncertainty about the tautological statement does not reflect some vagueness in the territory of logic, but our uncertainty about the workings of our physical brains, and their ability to build maps that reflect the territory. Problems of logic have 100% correct answers, but our physical brains cannot become 100% entangled with those correct answers. It is observation, which can be abstract observations of our own logical reasoning, which give us increasing entanglement which approaches, but never reaches, 100%.

1Vladimir_Nesov16y

Whatever you could possibly know and value about reality can only exist independently of the physical universe. (Huh?) If your uncertainty about math doesn't indicate uncertainty of the math, and it's an argument for math being otherworldly, it's also an argument for the territory being otherworldly, which is clearly a confusion of terms. And so you should bring the math back where it belongs, an aspect of the territory.

0JGWeissman16y

That is not what I am saying. I mean that things that we think of as tautologies, or purely logical truths, which are true no matter what universe we are in, exist independently of the physical universe. Facts about the physical universe are not in this class. Indeed, the entanglement of our physical brains with these logical truths is an example of a fact about the physical universe that, of course, depends on the the universe. You have my argument backwards. I first make the point that facts about math are not facts about the physical universe to support that the uncertainty we have about math, which exists in our heads, in our physical universe, does not exist in math itself. The argument does not work the other way, there are plenty of instances of uncertainty in our minds that are not uncertainty in the things elsewhere in the physical universe that they are about. My comment was an attempt to explain why we need observation to believe things that are objectively true regardless of the world we exists in. Basically, we need evidence that our brains, existing in the physical worlds, are suitable for representing the logical truths.

0Jack16y

This is really helpful and I think I agree with all of it. I've just never understood "observation" to include my logical reasoning. If your position is that we know 2+2=4 by virtue of observing our own reasoning and not by virtue of any sensory data (information about the outside world) then I don't think that position is any different from the one I already hold. But is this Eliezer's position? His OB post made it sound like he could be swayed to think 2+2=3 as a result of external events mediated by his sensory perception of those events. That is what I objected to.

1JGWeissman16y

Well, I think that observations can be both our reasoning and sensory data. Suppose you have a model* of your own accuracy at addition of integers, which is that you are 95% likely to get the correct answer, 2% to be one high, 2% to be one low, and with the remaining 1% divided somehow amongst other possibilities. Then, when you actually observe that when adding 2 + 2 you get 4, this is Bayesian evidence that gives a likelihood ratio of 42.5 : 1 in favor of the theory that 2 + 2 = 4 compared to the theory that 2 + 2 = 3. Now suppose you have a collection of pebbles, and your model of the pebbles claims that if you count out 2 distinct collections of pebbles, and then combine them and count the total, that the sum of the counts of the distinct collections is 90% likely to be the count of the combined collection, and is 4% likely to be one high, 4% to be one low, and 2% to be something else. And then you actually count out a collection of 2 pebbles, and another collection of 2 pebbles, and combine them, and when you count the combined collection you count 4 pebbles. This is Bayesian evidence with a likelihood ratio of 22.5 : 1 in favor of 2 + 2 = 4 as opposed to 2 + 2 = 3. In both cases, belief in a logical proposition results from our belief that an observable system has some probability of reflecting logical truth. If, as in the example numbers that I made up just now, we believe that our reasoning process is more likely than observations of our environment, then the results of our reasoning is stronger evidence, but it is still the same class of evidence. * I have neglected the harder problem of simultaneously updating propositions about additions and propositions about a given system's probability of representing addition. That is, I have not explained where the models I asked you suppose you have really should come from.

2komponisto16y

It may be worth noting that Quine had a view similar to Eliezer's -- which Stove alludes to (dismissively) in the essay.

1Jack16y

Thanks. That is worth noting. My recollection is that Quine denies the existence of analytic statements but doesn't go as far as to hold that tautological statements are just like regular empirical statements. Logical truths still have some kind of special status for Quine. Plus, I think his reasons for denying analytic truths had very little to do with actually being able to imagine a series of experiences that could change his mind about them– it is one thing to claim that such experiences are possible. Its another thing to claim you have just described that set of experiences. Finally, I remember thinking Quine was being silly, but it has been a while so I'm going to go read and come back.

0randallsquared16y

I don't think I'd read Eliezer's piece about tautologies having to be observed before, but it matches my pre-existing beliefs about its topic, and it seems so obvious that I'm left wondering how you think you got the understanding that 2+2=4, or that triangles on Euclidean planes are not round. Given that you got that understanding somehow, couldn't the same process give you the new understanding, assuming (for this argument) it was true?

0Jack16y

This is certainly a strange divergence of intuitions. I think the story of how I came to know 2+2=4 goes like this: Someone taught me that 2 meant -oo- and 4 meant -oooo-. Then someone probably be told me that 2+2=4 but I don't think they would have needed to. I think I could easily have come to the conclusion myself since given -oo- and -oo- I can count four dots. If pushing four objects together meant one of the objects disappeared I would probably just stop pushing objects together and count in my head. If counting the objects made one of them disappear I would be pretty damn frustrated but I'm pretty confident I could realize that reality was changing as a result of a mental operation and not that I was counting wrong. Aside from being tortured with rats or Cardassian pain sticks I don't see what would make me think that 2+2 didn't =4. I'm not sure how to explain my thinking any better except to say that it is the same thinking that lead generations of philosophers and mathematicians to conclude that mathematical knowledge was a different kind of knowledge than knowledge of our surrounding and the natural world. My reason is the reason Kant distinguished the analytic from the synthetic- a sense that a rational mind could figure these things out without sensory input.

5orthonormal16y

The trouble there is the claim of a rational mind, in my opinion. It's not logically necessary that our evolved brains, hacked by culture, are going to mirror reality in their most basic perceptions and intuitions. The space of all possible minds includes some which have a notion of number and counting and an intuitive mental arithmetic, but for which 2 and 2 really do seem to make 3 when they think of it. These minds, of course, would notice empirical contradictions everywhere: they would put two objects together with two more, count them, and count four instead of three, when it's obvious by visualizing in their heads that two and two make three instead. Eventually, a sufficiently reflective mind of this type would entertain the possibility that maybe two and two do actually make four, and that its system of visualization and mental arithmetic are in fact wrong, as obvious as they seem from the inside. Switching "three" and "four" in this paragraph just illustrates how difficult accepting that hypothesis might actually be for such a mind. The thing is, we ourselves are in this situation, not with arithmetic (fortunately, we receive constant empirical reinforcement that 2+2=4 and that our mental faculties for arithmetic work properly) but with our biases of thought. Things like our preferences and valuations seem to be rational and coherent, in that we can usually defend them all with arguments that look solid and persuasive to us. But occasionally this fiction becomes untenable, as when we are shown to have circular preferences in situations of risk and reward. As Eliezer put it in Zut Allais: Or, in this case, "Don't start by assuming that our minds work rationally whenever they see something as obvious! If this is true, it is an empirical fact; and you should be able to see the alternative as possible!"

0byrnema16y

Indeed, 2+2=4 is only true in some contexts. For example, sometimes 1+1=1 -- in contexts where separate objects lose their distinct identity as soon as they are grouped. (Think of a particular object several times. How many times did you think of it? But how many objects did you think of?) Later edit: It is interesting that such a benign comment would get 4 down votes. Perhaps I understand this group well enough to guess why: the experiment I suggested is an entirely "internal" one, it provides no external proof of what I am suggesting. I think that a common reader here feels dismissive of, if not entirely antagonistically towards, knowledge that is internally generated. Personally, I have a preference for the knowledge that arises from internal experience.

6komponisto16y

I agree that the downvoting of this comment was overly harsh. My theory on why it occurred is different, and best illustrated by an example: if someone posted a comment saying "2+2=4 is only true in some contexts; in arithmetic modulo 3, 2+2=1", that comment would have been similarly downvoted. However, let me be so bold as to say a word in defense of even that hypothetical commenter. Anyone mathematically sophisticated (including our downvoters) will agree that it is possible to construct a mathematical system in which 2+2 equals anything you like -- or, more precisely, for any symbol x, a system can be constructed in which the formula (string of symbols) "2+2 = x" is given the label "TRUE". Mod 3 arithmetic is an example for x = "1". Now, it is at this point that the downvoters protest: "But this is not the same thing as saying 2+2=1! All you've done is change the meaning of the symbols in the formula, such as '2' and '1'. Two plus two is still four, for the original meaning of those words. You're confusing the map and the territory. Downvoted!" Well, the downvoters do have a point. But, at the same time, let me suggest that they're also making the same mistake as our poor beleaguered commenter! What they've done, you see, is to make a leap from "Ordinary (i.e. non mod-3, etc.) Arithmetic accurately models certain physical phenomena" to something like "Ordinary Arithmetic is true in (or of) the physical world". Instead of saying what they mean, which is "the physical world is best modeled by a system that has '2+2=4' as a 'TRUE' formula", they say "2+2 is in fact equal to 4". Small wonder that confusion arises about whether mathematical statements are "emprical" or not! "The physical world is best modeled by a system that has '2+2=4' as a 'TRUE' formula" is clearly an empirical claim. But what about 2+2 = 4, all by itself? When a mathematician at a blackboard proves that 2+2=4 in Ordinary Arithmetic (or, for Eliezer's benefit, that infinite sets exist in stan

1byrnema16y

In the OB post tautologies have to be empirically observed somehow, Eliezer writes about waking up one day and discovering all sorts of evidence that 2+2=3. This wouldn't be evidence that 2+2=3 in Peano arithmetic, it would be evidence that Peano arithmetic just doesn't apply for some reason. In my down-voted comment, I was just giving an example of how there can be different kinds of arithmetic if you are willing to be flexible about what arithmetic is. (If you are not willing to be flexible, then you are not willing to allow the observation that 2+2=3 as an observation about arithmetic, because this is not possibly true in standard arithmetic. Well, the observations are possible but you'd have to account for it as some kind of grand delusion.) My point is that 2+2=4 in Peano Arithmetic independent of observation, but observation tells you if Peano arithmetic applies or not.

3komponisto16y

Exactly. It is worth emphasizing that to claim that "2+2=4 in Peano Arithmetic independent of observation" is not to claim that our knowledge of this fact about Peano Arithmetic is independent of observation. (The former claim is about our map of the territory; the latter is about our map of our map of the territory.)

0[anonymous]16y

Could you elaborate? It sounds to me like the former claim is about the territory, and the latter is just hard for me to parse.

-1[anonymous]16y

I'll emphasize with the following analogy: you need to observe the sun to know of it. However, you can nevertheless be certain -- as certain as you are of anything at all -- that the sun exists independently of observation. You need to define the Peano axioms and observe the deductions that lead to the tautologies to know of them, but they are mathematically true independent of your observation.

1Alicorn16y

How many words are in this list? * Duck * Duck * Goose

0JGWeissman16y

The list contains 3 word instances and represents 2 word prototypes.

1Alicorn16y

The jargon I was looking for was 3 tokens of 2 types, but close enough :)

0JGWeissman16y

Heh, with the right jargon, you can accomplish anything. ;)

1steven046116y

Doesn't that just mean that grouping doesn't always correspond to addition?

1byrnema16y

Yes. If 1+1 is anything other than 2, then it's not addition. I gave an example of how combining two things together doesn't always yield addition.

1byrnema16y

Saying that 2+2=4 is a tautology in a certain axiomatic system defined with '+' means that you couldn't have anything but 2+2=4 in that system. It's simply mandatory, and a rational person could not wake up one day and be convinced that 2+2=3 within a self-consistent system that deduces 2+2=4. While tautological truth is independent of observation (let's call it mathematical truth), it is dependent upon context (i.e., a self-consistent axiomatic system). Some mathematical truths in one axiomatic system are false in another. When we talk about whether a a mathematical statement is true, we need to specify the context, and, in my opinion, in the most demanding definition of truth, the context is the real, actual, empirical world. So I agree with Eliezer that a mathematical tautology must be observed in order to be true. When we humans talk about "2+2=4", it is because we have chosen arithmetic from an infinite number of possible axiomatic systems and given it a name and a set of agreed-upon symbols. Why did we do that? Because we observed arithmetic empirically. Obviously, addition is just one operation of infinitely many operations. The ones we have defined (multiplication, subtraction, addition mod n, taking the cardinality of subsets of, etc.) usually have some empirical relevance. While we don't feel very comfortable thinking of those that don't (and this says somethng about the way we think), I have faith that if we were presented with a very strange set of observations, it would take a pretty short amount of time to train ourselves to think of the new operation as a "natural" one. ... I idly wonder if there is a such thing as a mathematical truth that could not be realized empirically, in any context, and if there would be any way of deducing it's non-feasability.

1Jack16y

2Richard_Kennaway16y

Can you give some examples of valuable philosophy, and why you judge it valuable? I incline to the view that ignoring all of philosophy is, to a first approximation, the right thing to do, and that there are very few exceptions worth making.

0Vladimir_Golovin16y

Off the top of my head: Karl Popper, due to his influence on the scientific method. (Perhaps it's not as valuable today as it was back then, due to bayesianism, but still.) Edit: Also, epistemology is a branch of philosophy.

5Richard_Kennaway16y

Yes, historically, Popper performed a valuable service, by showing (imperfectly) what distinguishes science from nonsense. (Stove characterises Popper as someone who overreacted to the fact that scientists sometimes make mistakes, but that is less than his due.) But it's interesting that one can say that about Popper, and a few other philosophers -- that they were at least partly right, and where they were wrong, they were at least wrong, rather than "not even wrong". They created something to be corrected and improved on, not trash to be thrown out. A colleague in theoretical computer science once showed me a Ph.D. thesis that a logician of his acquaintance had sent him. He found it rather strange in form, compared with the sort of mathematical thesis he was accustomed to reading. I looked at it and laughed. It followed precisely the standard form for a thesis in philosophy. (I think Pirsig describes this in "Zen and the Art...") In chapter 1, the author states the subject he is going to address. In chapters 2 to 8 he writes a detailed history of everything of significance that has ever been written on the subject. In chapter 9 he introduces his own modest contribution, and in chapters 10 to 12 indicates how it relates to the history. A scientific thesis, on the other hand, begins with a similar chapter 1, surveys the previous literature in chapter 2, going back only far enough to establish the context for his work, and the remainder is all about the author's own work. No subject is worth anything whose entry qualification is a thesis of the first form. It would be interesting to write press-release style digests of current papers in philosophy, summarising their findings in bite-sized chunks: * What we studied. * What we discovered. * How we discovered it. * Why it matters. I don't think it could be done other than as a work of satire. Maybe it should be. Any iconoclastic grad students in philosophy want to give it a go?

[-]pdf23ds16y00

Your link is now broken. Is there some other web archive of the chapter? I've saved a copy from the google cache, in case it matters to anyone.

1Vladimir_Nesov16y

The Internet Archive has a copy.

[-]nazgulnarsil16y00

does #23 have a quick explanation or does it require a serious delve into abstract math?

0cousin_it16y

One is a provable theorem in an axiomatic system, the other isn't.

[-]Drahflow16y-10

Regarding most of the lengthy examples of "philosophy" given by Stove:

Reading a text takes time, time can be spent acquiring utilions. Hence reading a text is only worth if the expected utilion win due to additional knowledge is grater than the expected utilions when using the time differently. This approach kills most of his examples dead in their tracks for me. This also implies positivism, if a text does not either generate utilions directly, i.e. fun reading fiction, then it needs to provide knowledge (in form of testable statements about the... (read more)

[-]kim016y-30

Interesting, but too verbose.

The author is clearly not aware of the value of the K.I.S.S. principle, or Ockhams razor, in this context.

[-]CannibalSmith16y-30

David Stove's "What Is Wrong With Our Thoughts" is a critique of philosophy that I can only call TL;DR.

-4PhilGoetz16y

Voted down for being deliberately obscure. TL;DR?

8Nominull16y

Voting something down as deliberately obscure because you don't understand it strikes me as an egregious case of the mind projection fallacy.

3PhilGoetz16y

If you can take the time to write a comment, but conceal all of its meanings in an acronym, with no hint from context as to the meaning of the acronym, then you deserve to be voted down, because you save 2 seconds of your time but waste many seconds of the time of everyone who reads it and doesn't know what the acronym means. We need a catchy name for the fallacy of being over-eager to accuse people of fallacies that you have catchy names for. (I could be wrong about TL;DR. It's a judgement call as to whether your readers should know an acronym or not. I'd expect people here to know what LW or OB mean.)

3JGWeissman16y

How about "Catchy Fallacy Name Fallacy"? And I agree that it applies here.

1John_Maxwell16y

Use urban dictionary.

1hirvinen16y

Thanks to fast internet connections, good web search and online dictionaries, failing to expand an acronym only increases the cost from 5 seconds to 5 seconds per reader...

0Nominull16y

lol

4mattnewport16y

Too long; didn't read.

1PhilGoetz16y

Thank you.

[+]Nominull16y-60

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