Definitions and math (split from one-syllable words)

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Definitions and math (split from one-syllable words)

Post by KathTheDragon »

Salmoneus wrote:Kripke uses it as a concrete demonstration of (his interpretation of) Wittgenstein's rule-following considerations, framing them as a skeptical paradox. The problem is, if you have never added together numbers higher than 50, all your additions are compatible both with taking "add" to mean "mathematical plus" and with taking "add" to mean "quus". It may be that everyone asking you to add things together has really been meaning you to quus them all along, and you've been plussing them. It's only when we deal with numbers over 50 that we start to see that plus and quus are different, and the question arises of which rule we are meant to follow when we are told to "add", and we discover whether we have been following the same rule as everyone else. But more than that, in what way is it true that we have been quusing when they have been plussing, because in what have they following the plus rule rather than the quus rule? Not because of what they did, since what they did was compatible with both rules. And not because of what they thought, because it's possible that they never even considered what they would do when dealing with numbers over 50 (have you ever consciously considered how you would answer 547+789? Maybe you would answer '5'. You probably wouldn't, but that fact doesn't come from what you have consciously thought before about what you would do in this situation). And of course you can't try to explain by using other rules (like 'x+y means give the yth successor of x'), because those rules are themselves subject to the same ambiguities, and indeed in the case of mathematical rules are just restatements of the problem in other words ('the successor of x' is no less ambiguous than 'x+1').
So how are we ever able to learn, and use correctly and in the same way as everybody else, rules that cover an infinite number of different circumstances, when we can only learn from a finite number of circumstances, and when an infinite number of eventually-conflicting rules are compatible both with our finite experiences and with any attempt to describe the rule in language?

"Quus" isn't a very commonly used word in philosophy (nowhere near as common as 'grue', I'd wager), but people do still talk about it.
I'm not sure I understand what the point of all that is, but I think it can be summarised as philosophy (which I don't understand the point of either). I'll just go back to my well-defined mathematics.

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Re: One-syllable words with specific technical or rare meani

Post by Salmoneus »

The point is that mathematics is not really well-defined at all, and nor is 'well-defined'.

Well, that's not the point really. I think Wittgenstein's point was actually about the nature of meaning, and his point that meaning something isn't the same as thinking something: we may genuinely mean 'plus' rather than 'quus', even though everything we're thinking when we say it is equally applicable to quus as to plus, and so 'what we mean' is not a matter of 'what we are thinking when we say something'. I think Wittgenstein's conclusion is that there is no fact of the matter about what our words would mean in certain situations until we have arrived in those situations and actually decided what to say.

But the fact that our mathematical symbols and operations are not well-defined, but rather radically underspecified (either the functions themselves are not defined, or else our reference to the functions in any bit of mathematical reason are not defined), follows from it.
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Re: One-syllable words with specific technical or rare meani

Post by KathTheDragon »

Salmoneus wrote:The point is that mathematics is not really well-defined at all, and nor is 'well-defined'.
I think that's just a misunderstanding of what it means to define something in mathematics.

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Re: One-syllable words with specific technical or rare meani

Post by Salmoneus »

KathTheDragon wrote:
Salmoneus wrote:The point is that mathematics is not really well-defined at all, and nor is 'well-defined'.
I think that's just a misunderstanding of what it means to define something in mathematics.
I don't think that it is. Although that I think that the idea that defining something in mathematics means something may be a misunderstanding of what it means to define something in mathematics, or of what it means to mean something...
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Re: One-syllable words with specific technical or rare meani

Post by KathTheDragon »

Salmoneus wrote:
KathTheDragon wrote:
Salmoneus wrote:The point is that mathematics is not really well-defined at all, and nor is 'well-defined'.
I think that's just a misunderstanding of what it means to define something in mathematics.
I don't think that it is. Although that I think that the idea that defining something in mathematics means something may be a misunderstanding of what it means to define something in mathematics, or of what it means to mean something...
Ok, what do you understand to be meant by defining something in mathematics?

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Re: One-syllable words with specific technical or rare meani

Post by Salmoneus »

I don't think the surface details of the definition of definition matter all that much if they rely on an insecure foundation. What definition of definition would you like to use that does not rely on either a posteriori extensional definition or a priori intensional definition?

I should probably say that I've been trying to look up some precise answer to your question, and can't find one. I can, however, find a bunch of qualified mathematicians insisting that 'definition' is not a mathematical term but only a linguistic/philosophical one.

Closest I can get to something specific is some random guy saying "I would define a definition to be a finitely generated formula... of set theory that is legitimate according to the grammar whose quantifiers range over previously known results." This obviously falls within the sphere of the Wittgensteinian considerations, as it proposes a term and an application of the term. Besides, that guy is relying on set theory, which I guess as an anti-philosopher you'd find illegitimate.
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Re: One-syllable words with specific technical or rare meani

Post by KathTheDragon »

Could you write that in English?

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Re: One-syllable words with specific technical or rare meani

Post by Salmoneus »

...so you pretty much admit you're just trolling, then?
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Re: One-syllable words with specific technical or rare meani

Post by KathTheDragon »

No, I have absolutely no idea what it was supposed to mean.

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Re: One-syllable words with specific technical or rare meani

Post by Thry »

I'm not sure I understand what the point of all that is, but I think it can be summarised as philosophy (which I don't understand the point of either). I'll just go back to my well-defined mathematics.
It's futile. The neatness of logic-mathematics are an illusion, everything is subject to philosophical considerations, and so ultimately the abyss. Do you think definitions of any kind - linguistical or logical - amount to anything other than self-reference, which ultimately rests on intuition?

What does it even mean you don't understand the point of philosophy, or do you mean philosophy of mathematics? What point?

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Re: One-syllable words with specific technical or rare meani

Post by Salmoneus »

KathTheDragon wrote:No, I have absolutely no idea what it was supposed to mean.
The words and syntax are all very straightfoward. Combined with your opening gambit of "what does that philosophy mean? oh right, it's philosophy, it's pointless", I think it's pretty clear that you're trolling.
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Re: One-syllable words with specific technical or rare meani

Post by KathTheDragon »

The only thing I can get is that you think that mathematics is meaningless, which makes even less sense. I simply do not understand what you are trying to say.

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Re: One-syllable words with specific technical or rare meani

Post by Salmoneus »

KathTheDragon wrote:The only thing I can get is that you think that mathematics is meaningless, which makes even less sense. I simply do not understand what you are trying to say.
I think most mathematicians (all?) would agree that mathematics is meaningless in the sense of not containing meanings, which are instead a linguistic phenomenon. But that's not what I was saying.

If you're being serious, do you have any specific difficulties with the words or syntax I used that I could clarify?
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Re: One-syllable words with specific technical or rare meani

Post by KathTheDragon »

You know what, forget it.

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Re: One-syllable words with specific technical or rare meani

Post by Salmoneus »

...aaand trolling it is, gotcha.

EDIT: but because I'm bad at not engaging... OK, I can see that 'intensional' and 'extensional' are maybe a bit technical (so are a priori and a posteriori, but conlangers should know those already).

Intensional means to do with intension, the 'meaning' of a thing in terms of... I don't know, a concept or idea or something, or a dictionary definition, or a connotation. The intensions are the properties or qualities that the thing in question has. An extension, on the other hand, is something in the real world that the word actually applies to.

So to define intensionally is to define by describing the properties, and to define extensionally is to point to the things the word applies to. So intensionally a 'key' can be something a user pressed to input data into a computer program, and extensionally a key is any one of these things here.

Language and mathematics are both rule-following endeavours, and both rely on terms within the game helping to determine what are legal moves. So in something of the form "x+y=z", the 'meaning', in one sense at least, of '+', is that there are certain combinations of values for x, y and z that are not permitted, that are against the rules of how that symbol is used. So if we say '1+1=3', we have misunderstood the meaning of '+' (or of '1' or of '=' or of '3'... or at least, we are using one of these symbols in an unusual way!)

If we tell a child to add 534 and 1, and they give the answer '5', we have the sense that they have given the wrong answer. They have not understood (or have pretended not to understand) what we have asked them to do. They have not understood adding 1 to 534.

So, for instance, going back to intension/extension, we have addition, or more specifically 'adding one'. We can define this intensionally by saying, for instance, that 'adding one' is the operation we perform to transform one number into its successor. Or extensionally we can point to lots of instances of adding one to a number and say 'look, it's what we're doing there'.

But if we're just defining one term by another, we're not getting anywhere, are we? Because that other term still needs to be defined somehow. In fact, mathematical definitions like that are meant to be EXACT identities, so saying "adding one means replacing with its successor" means no more than "adding one means adding one". That doesn't teach the child who has not understood.

So at some point we need to show what the word means by showing examples of what we mean by it. So we show the child examples of 'adding one' to various numbers. But the child still gives the answer '5' to the specific question of 534+1. We show the child that it is the wrong answer, that it should be 535. But then the child gets the question 598+1 and says '5'. The child says she has realised that 534+1=535, but that surely 598+1=5? And then the day after, the child says that 534+1=5 again. I'm sorry, she says, I thought you meant that 534+1=535 on Tuesdays. We keep showing her examples, but she keeps getting the questions wrong - but she keeps getting the questions wrong in ways that are perfectly compatible with her learning all of the examples that we give her. And after study, we find that her answers are not random, but do seem to follow some mathematical function that she has come to believe is what is meant by '+1'.

Is there such a function? Of course! For any number of examples we give, there are an infinite number of functions that could have given those examples. So how do we teach the child which function we mean? Which rule to follow? How do we define that rule? Through other terms? That relies on her understanding THOSE terms, and how do we define THOSE terms for her? Or through examples? In which case, we're back to the same problem of examples underdetermining the rule.

We want to say something like "just do the same thing every time we tell you to add one!" - but of course she can't. Because adding 4 to 1 is not the same as adding 5 to 1. For one thing, they have different correct answers! We want to say that some part is different (4 vs 3) but some part is the same (the 'adding one' part). But what is that part? Adding 1 is a function, a procedure, something we do correctly or incorrectly. But how can we state that rule in a way that is not begging the question, that does not rely on the other person already knowing what we mean?

And then we have to stop asking "does the child understand adding one?" and start asking "do WE understand adding one?" - how can we tell? We can know that we have never added one 'wrongly' - our answers have always agreed with everyone else's. But what if one day we add one to 65784 and we say 65785 and everyone looks at us as though we were mad and they tell us the answer is 5? We cannot be sure that this situation may not arise - and not because everyone might go mad, but because they may be following a different rule from us all along (they my be interpreting the instruction differently... they may be linking the words to a different mathematical function.... etc, however you want to put it). A perfectly valid function they may be following, just not the same as ours. After all, we have never been taught the answer to 65784+1.

Of course, this is an extreme example. But it applies to all our use of symbols, when we use a finite number of symbols, learned through finite examples and finite explanation, which we then expect to somehow be able to unambiguously describe all possible situations, an infinity of possible situations. How can we ever learn which rule we are meant to be following in such cases? An infinite rule - one of an infinite number of possible infinite rules - ...how can that be unambiguously labelled by a finite 'definition'?

One approach is to say: this idea that there's this infinite rule we're following is a myth. Symbols have meanings in situations where they have been assigned meanings, and outside that their meaning is yet to be determined. So 'understanding a rule' stops being some theological act where the finite mind 'comprehends' the infinite rule that exists in abstraction, and starts being a sociological observation that our behaviour is not wildly out of keeping with those of other people, within our culture. And where our culture ends, our rules become indeterminate, subject to future determinations.

And mathematical rules, of course, are infinite abstract rules labelled by finite labels and learnt through finite teaching, in the same way as linguistic (or etiquette) rules are.

So the radical approach would be to say that the application of mathematical rules is itself only a sociological phenomenon, a matter of convention. And conventions change. So once upon a time, a mathematician writing "i*i=-1" would have been wrong - would, indeed, have been writing nonsense. The square root of minus one did not exist - there was no correct answer to the question 'what is the square root of minus one?'

Or a halfway approach would be to say that abstract mathematical rules DO exist in some Platonic realm of perfectness... but that our descriptions of them were a matter of convention, and that we can never be sure we 'understand' any mathematics, or that any of our mathematical conclusions are correct (because we may fail to understand the convention, and the convention may fail to represent the True Mathematical Forms). This solution may make people feel happier, because believing in God seems to make many people feel more comfortable somehow even if God is only plus signs and minus signs, but it's not clear that it is significantly different from the radical approach when it comes to the status of actual mathematical practice.


Of course, the really fascinating thing about addition - about most of language, really, is that despite everything being radically underdetermined, most of the time we still learn it. Which is to say: despite there being no rational way to 'teach' these rules that would ensure that people all followed them in the same way, nonetheless people do, an awful lot of the time, follow them in the same way, or at least they do so enough of the time to get along in life. ['plus one' is something we master quite well' 'point only at the green objects' is much harder, and 'point only at the descriptions of moral acts' is downright diabolically difficult in some of the more complicated cases]

The conclusion most people come to is: there must be elements of symbolic behaviour that are innate. That is, we all do the same thing note because we perfectly convey our rules to one another, but because we all share many of the most fundamental rules already. This is where we come to Chomsky's "universal grammar". UG and quus, closely related....

I'm surprised that a mathematician wouldn't have thought of these issues already. I mean, playing with the symbols is all very well, but isn't there a temptation to wonder what they mean, what the significance of the 'replacing symbols with other symbols' game is in a wider sense? I guess that's why so many mathematicians become philosophers and vice versa.
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Re: One-syllable words with specific technical or rare meani

Post by KathTheDragon »

You know what, Sal. Fuck you and your mile-high horse.

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Re: One-syllable words with specific technical or rare meani

Post by Matrix »

Sal, I think you're being uncharitable with these claims of trolling, but Kath, I also think you're being kind of obtuse?

I think what Sal means to say is that according to Wittgenstein, nothing -including mathematics - can be well defined because we can only ever define things with words, and words are inherently not well-defined.
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Re: One-syllable words with specific technical or rare meani

Post by KathTheDragon »

Thankyou, Matrix. That's literally all I wanted to know.

I still don't believe it, though.

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Re: One-syllable words with specific technical or rare meani

Post by Salmoneus »

It's easy to not believe things when you refuse to consider anything other than gross simplifications and misrepresentations (sorry matrix).
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Re: One-syllable words with specific technical or rare meani

Post by Matrix »

Well, I'm not a Philosophy major, and all I've read on Wittgenstein are two Wikipedia pages, so yeah. Figures I wouldn't get it right.
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Re: One-syllable words with specific technical or rare meani

Post by Salmoneus »

Matrix wrote:Well, I'm not a Philosophy major, and all I've read on Wittgenstein are two Wikipedia pages, so yeah. Figures I wouldn't get it right.
Sorry, I didn't mean to be so sharp. Kath's baiting was getting to me. My objection wasn't so much to your description, as to the way she didn't bother reading my longer description because she'd already disagreed with your version and all she was looking for was an excuse to be contemptuous.

But I think the important issue I'd take with your description, for the record, is that it's not so much that W's considerations apply to maths because they apply to words, but that they apply to all rules, and symbols that invoke rules, whether those symbols are found in mathematical language or in other language.
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Re: Definitions and math (split from one-syllable words)

Post by zompist »

Salmoneus wrote:Of course, the really fascinating thing about addition - about most of language, really, is that despite everything being radically underdetermined, most of the time we still learn it. Which is to say: despite there being no rational way to 'teach' these rules that would ensure that people all followed them in the same way, nonetheless people do, an awful lot of the time, follow them in the same way, or at least they do so enough of the time to get along in life. ['plus one' is something we master quite well' 'point only at the green objects' is much harder, and 'point only at the descriptions of moral acts' is downright diabolically difficult in some of the more complicated cases]

The conclusion most people come to is: there must be elements of symbolic behaviour that are innate. That is, we all do the same thing note because we perfectly convey our rules to one another, but because we all share many of the most fundamental rules already.
I kind of agree, except that I think "innate" isn't a good word... some folks (Fodor and Chomsky especially) make innate abilities/beliefs into a sort of magic box that solves problems by fiat. I think the question should be not so much "What comes encoded in our DNA?" as "What minimal principles do we need to make these abilities work well enough for our life as overeducated primates?"

E.g. Quine made a big deal of how ostension doesn't produce clear definitions. If I point at a rabbit and call it a rabbit, you don't know that I'm indicating the rabbit; I might mean "the rabbit's spleen" or "a bag of rabbit parts" or "the end of my finger". And he's absolutely right: ostension isn't foolproof! But ostension actually works pretty well— you can teach a toddler a new term by pointing at things. Why?

Language acquisition specialists like Eve Clark have pointed to a few principles the child may be using. E.g., assume that new words are basic-level categories (so, "rabbit" is far more likely than "bag of rabbit parts"— also more likely than "mammal" or "Eastern cottontail"); assume that new words are different from old words; assume that adults are consistent in their word usage. Michael Tomasello talks about "joint attentional frames", meaning that children learn most language in an intense session with adults or older children, where their understanding of what's going on in front of them aids their understanding the words spoken at the same time. (Thus, watching a car ad on TV is less effective for teaching "car" than playing with cars with your parent who is also talking about cars.)

These assumptions aren't foolproof either, but they work, given the situations young children find themselves in. Once you have the basic vocabulary, you can worry about more complicated words, including those for which the basic assumptions fail.

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Re: Definitions and math (split from one-syllable words)

Post by M Mira »

I believe (subjective, yeah) human can and do distinguish discreet knowledge and inductive reasoning, so "If I don't eat, I starve" doesn't lead to "If I don't drink, I thirst", but "I didn't eat on the 21st, 15th, and 9th days before today, and I starved on the 20th, 14th, and 8th days, and they have nothing in common beside that I didn't eat." can lead to "If I don't eat, I starve" given enough samples to eliminate the unrelated factors.

Similarily, if "plus" has been known to depend entirely on its two inputs, and it's communicative, then for a person who does know "these two, and only these two, matters" and "this side and that side doesn't matter" and can handle inductive reasoning, then there shouldn't be problems.

The quus seems to be a case where the function is actually a collection of string lookup table and 5 fills the missing entries, and the person lacks inductive reasoning and probably treat the "plus" as an integral part to the expression instead of a function on its own. The infant example is the lack of delimitation of inputs to two (Tuesdays) or inductions (534+1).

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Re: Definitions and math (split from one-syllable words)

Post by rotting bones »

(I'm not feeling too well, so I may misjudge my sense-making status. Prod me if it dips too low.)

Philosophers say the reason Occam's Razor does not apply to this objection is that formalizations of Occam's Razor such as Shortest Message Length already rely on a background of presupposed "fundamental" concepts. But if these fundamental concepts were themselves gruesome properties (love that adjective BTW) like grue and bleen, then you would have to translate your seemingly simpler notions like blue and green in those terms. Eg. Blue is more complex than bleen because something is blue if it is bleen until the day you were born and grue afterwards.

Or in terms of a lookup table you are using to mark the days on which starvation is a factor, the entries in it already seem to be evenly divided and assigned to sequential days. Who says you get to do that? What if the lookup table mapped to days in such a fashion that in order to represent "If I don't eat, I starve", the entries had to look like a tangled mess rather than a uniform sequence of marks in every slot because of the "fundamental" concepts you started out with? A discontinuous function may look more complicated in your chosen representation, but it's still a function.

I do have my own objection to this skeptical position, but I'm not sure how to formalize it in the general case. My objection is based on a special application of Rawls' notion of reflective equilibrium. There are many formalized models of it that are deployed in practice. A simple one is called the Hidden Markov Model, which looks like this:

Code: Select all

X1->X2->X3->...->Xt->...->Xn
|   |   |        |        |
V   V   V        V        V   (<- these are arrows BTW)
Y1  Y2  Y3       Yt       Yn
This is a Bayesian network where each arrow represents a conditional probability, like X1->Y1 represents P(Y1|X1) or probability that variable Y1 has <some value> given that variable X1 has <some value>. Xt is a series of unobservable (i.e. "hidden") factors at times (or locations or whatever) t. Each Xt depends on Xt-1. Yt is an observable factor that depends on Xt. For example, Xt can represent someone's stress at time t and Yt can represent a combination of variables like heart rate, sweat, etc at time t.

(That example has actually been solved using this model and gives correct answers >90% of the time. The problem is that it somewhat predictably tends to register any form of excitement such as laughter as "stress".)

What can we use this model to do? First we are going to have to pass a decent amount of training data to learn certain conditional probabilities. If Xt can take 2 values Xa/Xb and Yt can take 2 values Ya/Yb, then we need the probabilities for the horizontal "transitions" Xa->Xa, Xa->Xb, Xb->Xb, Xb->Xa, and vertical "emissions" Xa->Ya, Xa->Yb, Xb->Ya and Xb->Yb. Once we have the values for those arrows, we can feed it a stream of Yt and get an estimate for Xt out of it.

The amazing thing about this model is that in order to train it, you don't need to feed it any hidden X values at all! There exists an algorithm that only needs to know how many values X can take and a stream of Y values to learn those conditional probabilities.

The way it works is that it assumes some initial values for X1, and calculates everything forwards to the end. Once there, it begins calculating backwards until it reaches the front and gets a new value for X1, when it begins calculating forwards again and so on until it reaches equilibrium. It's like water with waves in it trying to reach an even level. But to get to the even level, we had to begin with an incorrect value and then approximate our way closer to the equilibrium!

This approach is nicknamed the forward-backward algorithm, and it actually works in real life for those cases where this model is applicable at all. (The best way to do it in this simple case is to fix some end goal, and then hold tournament-like elimination matches at every t to find the path most conducive to reaching the fixed end goal, but that's harder to explain and the end result is the same. If you're going to Google the full details, also see the Baum–Welch algorithm.) The model does well in cases where Xt really does depend on Xt-1, i.e. transitions are smooth.

I don't know how to formally apply such techniques to concepts in general, but if there are certain areas where they can be applied, then it seems to me that the magisteria where such techniques are applicable are those where it makes sense to speak of having "reasonable opinions", because some kind of equilibrium of this nature could be a precondition for having a stable relationship among the concepts, hidden and observable, in that domain based on the observed values derived from experience.

The way this gets around the Occam's Razor objection is that the concepts used in the universal grammar don't need to have some metaphysical connection with the hidden variables constituting reality. As long as they are used consistently as a background, we can find their pattern of correspondence with those hidden variables.

(You may not even have to start with the universal grammar. As long as you use your own concepts consistently, you can find their pattern of correspondence with whatever "universal" grammar most people use. Also note how everything we're doing here is probabilistic. (But yeah, it's a good idea to wait until I understand this stuff well enough before declaring that consistency is sanity. I mean, I think I understand what I'm saying, but how much of it is me repeating what my professor said in class?)

This stuff is so fascinating. I'm considering a PhD in Data Science.)
If you hold a cat by the tail you learn things you cannot learn any other way. - Mark Twain

In reality, our greatest blessings come to us by way of madness, which indeed is a divine gift. - Socrates

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