An argument against using pattern sequences in IQ tests.

An argument against using pattern sequences in IQ tests. — jgill

I'm reading this elementary textbook on math. The only way one can determine the persistence of a pattern i.e. know that a pattern you discovered will continue (is the correct one) is if you can explain it.

What means this?

Are patterns exactly equal to rules?

Notice the difference between simply listing the array: 2, 4, 8 ...

and requiring such array to: start from 2, multiply the previous number by 2 to get the next number in order.

Well, you might as well argue that we understand each word from observing patterns. Such as the way we follow the rule of the word "multiply" by doing what gives us results that accord with what we observe: we observe that multiplying 1 by 2 is 1+1; 2 by 3 is 2 + 2 + 2, etc, so when we follow my more specific rule above, we just follow the pattern to get 2+2 = 4; 4+4 = 8... So the rule is still just patterns packed together, each allowing some discrepancy of understanding.

Btw, I think at some point in Philosophical Investigations Wittgenstein actually discussed the margin where the rule learner turns from observing patterns to understanding rules - his conclusion if I remember correctly is just as obscure. I'll take a closer look at what he was trying to elaborate on.

Are patterns exactly equal to rules? — D2OTSSUMMERBUG

More or less. If we figure out a pattern, we have a rule that helps you find the next term in a sequence. If we work out a word's usage pattern, we have in our hands a rule that determines where it should be used.

.

Wittgenstein out of context is problematic. There are a bunch of paragraphs leading up to this statement, that are very helpful in understanding what Wittgenstein is on about. Most importantly the paragraph just before that sentence, which gives a practical example:

200. It is, of course, imaginable that two people belonging to a
tribe unacquainted with games should sit at a chess-board and go
through the moves of a game of chess; and even with all the appropriate
mental accompaniments. And if n>e were to see it we should say they
were playing chess. But now imagine a game of chess translated
according to certain rules into a series of actions which we do not
ordinarily associate with a game—say into yells and stamping of feet.
And now suppose those two people to yell and stamp instead of playing the form of chess that we are used to; and this in such a way
that their procedure is translatable by suitable rules into a game of
chess. Should we still be inclined to say they were playing a game?
What right would one have to say so? — Ludwig Wittgenstein

201. This was our paradox: no course of action could be determined by a rule, because any course of action can be made out to accord with the rule. — Ludwig Wittgenstein

Do note that Wittgenstein employs the term "was" not "is". This is by no means a coincidence or mistake. The paradox was but is no more, Wittgenstein resolves it in his very next sentence.

The answer was: if everything can be made out
to accord with the rule, then it can also be made out to conflict with it.
And so there would be neither accord nor conflict here.
It can be seen that there is a misunderstanding here from the mere fact
that in the course of our argument we give one interpretation after
another; as if each one contented us at least for a moment, until we
thought of yet another standing behind it. What this shews is that
there is a way of grasping a rule which is not an interpretation, but which
is exhibited in what we call "obeying the rule" and "going against it"
in actual cases. — Ludwig Wittgenstein

The entire point of Wittgensteins argumentation was that "interpreting a rule" and "obeying a rule" are two completely different things.

202. And hence also 'obeying a rule' is a practice. And to think one
is obeying a rule is not to obey a rule. Hence it is not possible to obey
a rule 'privately': otherwise thinking one was obeying a rule would be
the same thing as obeying it. — Ludwig Wittgenstein

Now, let's take your numerical example:
From a 2, 4, 8 sequence we could interpret all kinds of pattern (rule) that this sequence follows - but as it has been established, our interpretation of the rule has nothing to do wether we're obeying it or not.

But what exactly is our rule then? Where does it come from? How can we confirm it?
The rule, in the case of such a sequence, is determined by the author. They are the ones that write down the numbers partaining to a pattern. The only way for us to know wether our interpretation of such a sequence is truthful to it's rules is to ask the one who made the rules if we are correct or not.

In other simplified words:
The only way to know if you're playing chess or if you're only doing something that looks like playing chess is consulting the rulebook of chess.

Consider the propositions

"The sequence x1, x2, ... is determined by the function f(x)"

"The function f(x) is determined by the sequence x1, x2, ... "

There are two permissible interpretations of the word 'determined' in the above propositions:

A) As an imperative when for instance normatively insisting that " f(x) means the sequence x1,x2,..

B) As a descriptive hypothesis when for instance alleging that a given sequence x1,x2,... obeys f(x)

Classically, the sign 'x1,x2...' is interpreted as an abbreviation for a particular sequence of infinite extension for which there isn't time to write the whole sequence down. Under this identification, interpretations of the form A leads to the identification of f(x) as also denoting a particular domain and image of infinite extension. Thinking of functions extensionally in this way leads to scepticism whenever it is asked if some unbounded sequence S obeys a given f(x), given that only a finite prefix x1,x2,..xn of S can be observed. In conclusion, hypothesis of the sort (B) aren't verifiable when thinking this way.

On the hand, in the Russian school of constructive mathematics, functions aren't directly interpreted as representing entities of infinite extension, but as being finitely describable computable maps whose domain is unbounded. But this can lead to the same impression of such functions as having actually infinite and precise extensions if one thinks of such functions as denoting ideal and physically infallible computation. Thus the same platonistic skepticism about rule-following arises as in the classical interpretation.

The alternative to adopt Brouwer's philosophy of Intuitionism, in which ' x1,x2,... ' is interpreted as referring to partially defined finite sequence of unstated finite length, rather than as referring to an exactly defined sequence of actually infinite length. In other words, x1,x2,... is interpreted as referring to a potentially infinite sequence whose length is unbounded a priori, but whose length is eventually finitely bounded a posteriori at some unknown future date. Likewise, the domains and images of functions are also interpreted as being potentially infinite rather than as being actually infinite. Relative to this philosophy, rule-following scepticism is avoided due to the fact that the meaning of potentially infinite sequences and functions are both understood to be semantically under-determined a priori.

The problem of induction (Hume-Wittgenstein):

There's no necessity to a law of nature - things might've been a certain way, are that way, but there's absolutely no reason why it should be that way in times to come.

Crucial difference: Hume's insight implies a law of nature's violated. Wittgenstein's rule following paradox doesn't mean a law (rule) is broken; au contraire, Wittgenstein is saying is that all observations (word usage) are compatible with any conceivable law.

:chin:

The entire point of Wittgensteins argumentation was that "interpreting a rule" and "obeying a rule" are two completely different things. — Hermeticus

I believe this is a good interpretation, and the difference here amounts to the difference between a descriptive rule and a prescriptive rule. When we produce a descriptive rule, we come up with 'this is the way things behave'. That's a conclusion of inductive reasoning. There is nothing within the rule itself, to compel that a thing will behave like that in the next instance, so what causes the thing to behave like that, is a completely different issue.

If we say that being caused to behave according to a rule is what "obeying a rule" means, then we see the difference between "interpreting a rule" (as in understanding the described behaviour), and "obeying a rule", (as in causing oneself to act according to the described behaviour). On the other hand though, we can say that "obeying a rule" is to act in a way which can be judged as being consistent with the described rule.

The difference between these two interpretations of "obeying a rule" is the difference between judging the cause, and judging the effect. Wittgenstein opts for the latter, making "obeying a rule" something which is observed after the fact, rather than something decided prior to the act, in the sense of interpreting a prescriptive rule, and acting accordingly. So the prescriptive rule is not relevant to Wittgenstein's position on rule following, and we must be careful when reading him not to misunderstand.

au contraire, Wittgenstein is saying is that all observations (word usage) are compatible with any conceivable law. — Agent Smith

Not quite. Wittgenstein only criticised logical conceptions of meaning, especially in relation to the view that the meaning of a proposition is static and a priori decidable . He didn't criticise individuals for their idiosyncratic interpretations of rules and language, which generally don't invoke theoretical interpretations of meaning.

There is a world of difference between speculating that an event E must logically follow from the a priori definition of a law L, versus recognising for oneself post-hoc, that E follows from L.

For example, often when you judge for yourself that two colours are the same, (which you usually do without any external guidance), your recognition wasn't contingent upon you invoking a priori definitions of the colours involved and calculating a truth value.

This is the reason why Wittgenstein wasn't a verificationist - meaning doesn't normally involve processes of verification - ergo Wittgenstein wasn't against the idea of private meaning.

Not quite. — sime

In other words,

some observations are not compatible with any law. That means...

Right, very interesting. It implies that the patterns that you point out are actually commensurate with your specific thought process going into the problem. Not that you've necessarily discovered a pattern that is actually there, but that your brain discovered only that pattern that makes sense to your brain. Either because of how we perceive symmetry, or some other pattern processing function. But, then again, a good test as to whether that was the case, would be to experiment with the usefulness of the specific pattern in question. Which is where I think things like sacred geometry come from out of history. Very cool topic, Smith.

-G

Very cool topic, Smith. — Garrett Travers

:up: You're :cool: too, sir/madam as the case may be.

Sir, will do just fine. :cool:

the difference here amounts to the difference between a descriptive rule and a prescriptive rule. — Metaphysician Undercover

The differentiation between "descriptive rule" and "prescriptive rule" is fantastic. :ok:

We could also say the difference boils down to epistemological demand as well, which is why one method raises no questions, while another only raises questions.

A descriptive rule faces the typical scrutiny of any epistemological consideration along the lines of subject/object matters. That's the interpretation part. Descriptions require relations, a context in which they can be established. We could say a truly complete description of anything would require us to describe the entire world as context alongside with it.

A prescriptive rule on the other hand doesn't make any demand to knowledge at all. It is a dictation of how things must be. The context - as far as I can think - is always already implied by engaging with the rule itself (context of life, math, games, society, law, language etc)

The difference between these two interpretations of "obeying a rule" is the difference between judging the cause, and judging the effect. Wittgenstein opts for the latter, making "obeying a rule" something which is observed after the fact, rather than something decided prior to the act, in the sense of interpreting a prescriptive rule, and acting accordingly. So the prescriptive rule is not relevant to Wittgenstein's position on rule following, and we must be careful when reading him not to misunderstand. — Metaphysician Undercover

I actually think that's exactly what Wittgenstein himself is trying to get across. To not misunderstand one for the other.

It can be seen that there is a misunderstanding here from the mere fact
that in the course of our argument we give one interpretation after
another;(descriptive rule) but that there is a way of grasping a rule which is not an interpretation, but which is exhibited in what we call "obeying the rule" (prescriptive rule) — Ludwig Wittgenstein

Sir, will do just fine. :cool: — Garrett Travers

:up: I knew you were a knight! Sir!

Now, let's take your numerical example:
From a 2, 4, 8 sequence we could interpret all kinds of pattern (rule) that this sequence follows - but as it has been established, our interpretation of the rule has nothing to do wether we're obeying it or not.

But what exactly is our rule then? Where does it come from? How can we confirm it?
The rule, in the case of such a sequence, is determined by the author. — Hermeticus

I find this confusing.

If there genuinely is a pattern with 2,4,8,... then that pattern will describe the number chain or series to infinitum or otherwise it's a wrong pattern or the series of numbers is basically without a pattern, patternless. Here to talk about rules it would be better to talk about algorithms in the general sense. And either you have an algorithm that correctly tells you how the series 2,4,8,... goes or either you have the wrong algorithm or the series is non-algorithmic.

Nothing to do with the author, the subject. Our understanding or incorrect understanding about the series doesn't brake this logic.

If there genuinely is a pattern with 2,4,8,... then that pattern will describe the number chain or series to infinitum or otherwise it's a wrong pattern or the series of numbers is basically without a pattern, patternless. Here to talk about rules it would be better to talk about algorithms in the general sense. And either you have an algorithm that correctly tells you how the series 2,4,8,... goes or either you have the wrong algorithm or the series is non-algorithmic. — ssu

yes, in the case of a potentially infinite sequence of numbers, it is meaningless to consider any particular function, let alone algorithm, as being descriptive of the sequence unless and until the sequence comes to an end. Until then, one cannot even decide whether or not the sequence is computable. Nevertheless it is meaningful in the meantime to speak of falsified hypotheses in relation to the sequence.

However, the problem goes further than that, because on Kripke's interpretation, the skepticism is calling into question the very meaning of "algorithm", and hence the distinction between algorithmic versus non-algorithmic processes, which computing and constructive mathematics take for granted. Such philosophies treat the definition of an algorithm to be isomorphic with the input-output pairs generated by it's execution, which presupposes the existence of ideal calculators. But as we know practically, physical implementations of algorithms have finite capacity and finite reliability, making the intensional definitions of functions misleading with respect to their implemented behaviour. Kripke is asking how it can be decided that a sequence corresponds to a given total function, given the irreparable inability to define what the 'correct' outputs of the function are for most of it's inputs.

yes, in the case of a potentially infinite sequence of numbers — sime

That what is generally depicted with 2,4,8,...
Not that it ends sometimes, which would be 2,4,8,.....,a.

it is meaningless to consider any particular function, let alone algorithm, as being descriptive of the sequence unless and until the sequence comes to an end. — sime

?

if the sequence is for example N, then the correct algorithm is "list all natural numbers". And natural numbers don't come to an end.

Either there is the correct description or their isn't. Those are your choices, they aren't meaningless.

if the sequence is for example N, then the correct algorithm is "list all natural numbers". And natural numbers don't come to an end. — ssu

To understand the paradox using your example, you have to distinguish the intensional definition of a function, such as one reproducing the natural numbers

i.e. f(n) = n for all n in Nat

from a potentially infinite list of elements such as

S = { 1, 2 , 3, ... }

that looks like it might be a prefix of Nat.

By 'potentially infinite' I mean that S is finite but of unknown size and whose elements are only partially defined a priori (in the above case, only the first three numbers). it's remaining elements are denoted by the dots "..." and are a priori unknown and decided when S is instantiated.

Obviously, until S is fully instantiated it cannot be decided as to whether S is a prefix of N or some other function. The paradox concerns the fact that one cannot know in advance what the prefixes of N are.

One can try to define the prefixes of Nat intensionally as a function, namely

P(n) = {0,1,....n} for all n

But in order to know what all of the prefixes are as implied by this definition, we would need to define what it means to treat n as a free-variable that can be substituted for any natural number. But if this definition of a free-variable is also intensionally defined, we will have gotten nowhere. So the only way to decide what the prefixes are, is to resort to writing them out extensionally for some random number of terms, which we can represent as the potentially infinite set

{ {0}, {0,1}, {0,1,2} , ...}

This set will be instantiated with a random number of prefixes, relative to which it will be decided, in a spur-of-the-moment bespoke fashion, as to whether or not S is a prefix of Nat.

I'm reading this elementary textbook on math. The only way one can determine the persistence of a pattern i.e. know that a pattern you discovered will continue (is the correct one) is if you can explain it. — Agent Smith

Prove it continues. Mathematical induction is one way. Providing an algorithm is another. For example, I am working on a theorem now that has the product

$\prod\limits_{k=1}^{n}{\left( 1+\frac{k}{{{n}^{2}}} \right)}$

And its fairly simple to prove that, for all positive integers n,


$\prod\limits_{k=1}^{n}{\left( 1+\frac{k}{{{n}^{2}}} \right)}\le e$

But here you predetermined the pattern of the sequence. Suppose I saw the ten first members of the pattern of the outcomes of your products. All smaller then e. Are we sure that only your prescription for generating the numbers (2, etc.) is unique?

This was our paradox: no course of action could be determined by a rule, because any course of action can be made out to accord with the rule.
— Ludwig Wittgenstein — Agent Smith

Given some finite set of numbers, how many formulae are there that fit?

Given some finite set of actions, how many rules are there that fit?

In both cases, an indeterminate number.

Wittgenstein's response is found further into the same remark:

...there is a way of grasping a rule which is not an interpretation, but which, from case to case of application, is exhibited in what we call “following the rule” and “going against it”.

One shows one has understood a rule not by stating it, but by following it.

Following or going against a rule is using the rule. So he is making the point that stating the rule is not grasping the rule; using the rule is grasping the rule.

Hence, don't look to the meaning, look to the use.

Suppose I saw the ten first members of the pattern of the outcomes of your products. All smaller then e. Are we sure that only your prescription for generating the numbers (2, etc.) is unique? — Cornwell1

Good point. Fortunately that's not an issue normally in a mathematical discussion. Of course there may be other "rules" generating that finite sequence. For the most part in math research one generates a sequence of numbers according to a given process, and then tries to ascertain whether an observed outcome is true for all numbers so generated. I'm happy it doesn't go the other way around!

I'm happy it doesn't go the other way around! — jgill

Haha! You could make it your opus magnus. "Okay, for n=56545434566, is the product smaller than e... yes! Next n..." :joke:

I actually think that's exactly what Wittgenstein himself is trying to get across. To not misunderstand one for the other.

It can be seen that there is a misunderstanding here from the mere fact
that in the course of our argument we give one interpretation after
another;(descriptive rule) but that there is a way of grasping a rule which is not an interpretation, but which is exhibited in what we call "obeying the rule" (prescriptive rule)
— Ludwig Wittgenstein — Hermeticus

I don't exactly agree. For Wittgenstein, "obeying a rule" is to be observed and judged to be acting in a way which is consistent with the rule, hence his use of "exhibited". The need for a prescriptive rule really disappears for him. For a person to obey a prescriptive rule, in the sense of 'I should respect the rule and do what I ought to do', this requires that the person interpret the rule, then move to act according to one's interpretation of it. Notice that the interpretive part is what he is trying to avoid. So for him, "obeying a rule" is to be described as acting in a way consistent with the rule. And the means by which the person comes to act that way becomes sort of irrelevant. The person might just be copying the actions of others, or whatever.

The way a word is being used (following a rule) let's us know what its definition (comprehension of the rule) is in a particular language game. The definition (rule) changes with the language game one is playing e.g. the word "god" means different things in theism proper, deism, and pantheism, these being distinct language games, each with its own unique rule (definition) for the word "god". In that sense, we could say that the word "god" lacks an essence, a common thread that runs through all of the aforementioned domains.

Questions:

1. True that, sticking to the example above viz. "god", words lack an essence that's cross-domain (the pantheistic god is different from the deistic god and both have no essential connection with the theistic god). However, within a given (one) language game (say deism), how does one pin down meaning? Doesn't the word "god" in deism have an essence? I suppose what I mean to inquire is whether there's any difference at all between essence (of a word) and rule (how a word is supposed to be used)?

2. Indeed, as regards a word, there's a difference between stating the rule and following the rule governing that word. Comprehension of a rule is best demonstrated by a person following the rule rather than just being able to state it. Step 1 (A rule) $\rightarrow$ Step 2 (Comprehension of the rule) $\rightarrow$ Step 3 (Following the rule). There's a lot going in step 2.




The way I understand it is that when we're asked to explain a pattern, a necessary step if one is to make the case that one has homed in on the right pattern, we must be able to demonstrate why the pattern, well, makes sense. Consider the pattern in the sum of the series 1, 3, 5, 7, 9...

1 + 3 = 4 (perfect square)
1 + 3 + 5 = 9 (perfect square)
1 + 3 + 5 + 7 = 16 (perfect square)
.
.
.
The sum of the odd integer series above is always a perfect square.

Now how do I explain it?

I use tiles/polyominos to show that when I use a monomino and a tromino, I can construct a square. Add a pentomino and another square can be constructed, so on and so forth.

Now how do I explain it? — Agent Smith

$\sum\limits_{k=1}^{n}{\left( 2k-1 \right)}=2\sum\limits_{k=1}^{n}{k}-n=2\frac{n(n+1)}{2}-n={{n}^{2}}$

Nighty nite, my friend

G'nite.

Language games are neither discrete nor inviolable. Deism, theism and pantheism are not constituted by different uses of "God".

To your question, is there a difference between a rule and an essence... Well, I think the notion of an essence cannot be made clear without being wrong. If oyu think otherwise, have a go for us.SO rules and essences are not he same sort of thing.

IN your second point you seem to think that the rule proceeds the use. More usually, it would go the other way. Unless one is Tolkien, does not construct a dictionary and then use the language. One examines the language and deduces the rules. The rules are usually post hoc.

More interestingly, use overrides the rules, in cases such as nice derangements of epitaphs.