[podcast]https://www.wmbriggs.com/audio/wmbriggs_com_lecture_0001.mp3[/podcast]
First lecture:
I’m away to a conference this week, and so thought it would be fun to start a lecture series Understanding Statistics and Probability (this is posting automatically—I’ll be answering all questions by Monday, 12 October). They’ll be a series of brief chats about the meaning of statistical concepts. We’ll try and keep away from any formulas and concentrate on ideas. I’m aiming for 12-15 minutes for each lecture. Look for these every Friday.
I’ll be roughly following my class notes, which are linked to the left—the book Breaking the Law of Averages.
Feel free to ask questions. I never use canned examples—nobody ever remembers a canned example—so I’ll be relying on you, my faithful readers, to supply situations, and maybe even data.
All knowledge is conditional on evidence which follows a chain that least ultimately back to our intuitions. There are many things we know are true based on no evidence except that of our intuition. Another way to say this, is that our beliefs, all of them, are eventually grounded in faith. This is true for everybody.
While all knowledge is conditional, not all the information leads to certainty. Some propositions are thus known uncertainly. We use probability to quantify this uncertainty. Because of this, all probability, like all knowledge, is conditional. Probability, therefore, is a matter of logic, and we have to understand some basic steps in logic before we can go further. We’ll do that next week.
Much appreciated, this is growing to be my favorite podcast ;). I’ll be one of the listeners only.
Regarding your acute observations on agnosticism, I almost thought it was written or thought out with me in mind, until I slapped me out of my own solipsism. I think it’s spot on, and I’d like only to add one thing to consider.
You say, all knowledge is tentative. All our knowledge is based, ultimately, on faith. Couldn’t agree more, and I’d say: quite a relativistic diatribe you did there, showing how our truths are only true “given X”, and it’s turtles all way down! I wasn’t expecting such an admission on Hume’s induction problem point (given your admiration for Stove).
Now, you say, we can’t decide whether god exists or not, we have to rely on faith. Well then, if all knowledge is tentative, then this was already a given, was it not? How is our knowledge of god not existing any different from any other knowledge, a priori? You say, if we want to get our of this connundrum, agnosticism is the way.
Agreed. But again, following your own rationale, we should be agnostics on everything! That is plainly absurd. If we are to be agnostics on everything (even tautologies, they might be wrong!), then it’s an useless word. Or, to put it another way, I say that the knowledge that there is no God can be as justified as saying that the sun is up right now.
So in your own podcast, your contradictory statements undermine the very denomination of “agnosticism”.
I’m not starting the god debate all over again, but I thought it was a good point worthy of mention, regarding the topic of epistemology that was brought up.
Thanks for the podcast!
Briggs,
This is fantastic,and generous. I’ve heard it twice because I didn’t concentrate first time
Luis!
I’ve forgotten what I was going to say. Start without me, I’m going to drink coffee and eat a cupcake.
You say these are linked to the left. Really, let’s keep ideology out of statistics and probability.
Luis,
“The knowledge that there is no God†is a type of prediction or an imagined condition. Nobody knows, that’s reality.
So, if one starts from an imagined state, then one can justify anything from that point.
The second statement, is one of empirical fact:
The sun is out (in England) because it’s experienced seen with our primary sense.
The ball in the sky is what we all agree is called the sun, we can see it and feel it’s affect so it is out, or present and accounted for. I expect it’s shining brighter in Portugal and you can feel it on your skin.
Luis,
Joy has it right. Propositions such as “there is a God” are ultimately untestable. Presumably, the “given X” is not “given any X we imagine” and ‘X’ is tentative.
Briggs,
Not supplying examples is sneaky. Can you give an example of something you know is true based solely upon your intuition? I’m having trouble coming up with an example of something I believe true by itself. Even “there is a God” can be traced back to “there is a Universe; something must have made it; let’s call it God”. Poor evidence perhaps but evidence nonetheless.
While you’re at it, define “intuition”. My intuitive definition is “informal/subconscious reasoning based upon prior reasoning which can’t be readily listed or described.” Like one of your examples.
It’s quite probably one of our automatic processes heavily refined by experience. Using some of those things “no one remembers”: a cop’s instincts; a mother’s intuition, etc.
I finally listened to the podcast. I disagree that axioms are intuitively true. Instead, I maintain they are definitions. “X=Y and Y=X” means the left and right of “=” are interchangeable. It’s the definition of “=” .
I think you continuously confused logical truth (in which “truth” means “traceable to the axioms”) and truth of reality. Truth of reality is established by model (presumably a logical model) while logical truths are literally “known” given the initial definitions. Axioms are initial definitions for a logic chain. They needn’t have any correspondence to reality and belief in their validity is irrelevant. It is only when a logic system is being applied to reality that the validity of the axioms becomes of concern.
On the side: interesting conundrums occur in mathematics, and in particular logic, when self-referential (e.g., “I never tell the truth”). Kurt Gödel had a few things to say about this.
“I disagree that axioms are intuitively true.â€
DAV, yeah, “We are all stupid, just on different subjects.” 🙂
Here are some differences between definition and axiom that you might already know.
A definition may not be intuitive at all, for example, 0! (zero factorial) is defined to be 1. If we are using the symbol “=” to indicate equality, then the symmetric property (if x = y, then y = x) is intuitive/self-evident (at least to me, anyway) and is an axiom.
I don’t think that the definition of, for example, the conditional probability of an event A given B is an axiom, but a set of probability axioms is required in the definition of the conditional probability.
Isn’t it an interesting feature of modern physics that not only are there things we do not know, ther are things we can not know? Shrodeinger’s Cat; the location of a particle with known energy — or the energy of a particle of known location …
I would like discussion, at some appropriate point in this series, of “bias”.
Consider two nickles — one a fair unaltered coin; one with a bevel of 45 degrees on the edge such that the “tails” side is significantly smaller in diameter than the “heads” side. If both coins were flipped 100 times and the results presented, the biased coin could, I think, be identified from the results record. One would not need to examine the coin. But if the bias is reduced — if the bevel is only five degrees or if only 1/12th the diameter is beveled — how would the “random” resulted be reliably distinguished from the biased ones?
If the effect is very subtle and the math required to detect the bias very exotic, when does it become meaningful to assert the effect does not actually “exist”?
Real world example:
http://forums.randi.org/archive/index.php/t-29763.html
(which I hope we can discuss with the political overtones than might arise from other famous questions of subtle bias such as global warming )
“1/12th the diameter” of course should be read as “1/12th the CIRCUMFERENCE”
Joy, DAV, I’m perfectly aware of the lack of any epistemological tool that will make us answer that question accordingly, and the difference between a statement about the noumena and a statement about phenomena.
I have two things to say about that.
1st, that it wasn’t always like that. Statements about God were always “testable” in principle, with god interfering in every situation possible. So, firestorms happened when god was mad because we didn’t sacrifice our own sons. Of course, back then, we didn’t have any Mr Briggs and his tests againts this kind of bull, so that we could see the mistake of that kind of thought. But, in principle, because god’s “humour” was something that could be “dealt with”, in principle it was a testable proposition. Such Gods were falsified by experience long ago.
The only ones which remain are the untestable ones. If they are untestable, though, what can or should we say about them? Any proposition we make about them, it’s make believe, we have no possible means to know what we believe, except our profound wishful thinking. If we did, then such knowledge should be testable, a priori. I’ve yet to find such “testable” principle that hasn’t been discredited yet.
2nd, every knowledge is tentative. So, to say that the sun “is up there”, while being in a dark room with no windows, and merely knowing it’s 10 o clock, how is this, in principle, different from knowing that there is or not god? Remember, I am saying all this, given what Briggs said, which was, every statement is tentative, and dependent on another axiom (and because we aren’t infinite beings, we must assume some axioms, the “faith” part).
It’s this thing that makes both beliefs equal. Because in the sun example, we trust what people around us tell us, and our own empirical experience, that no matter where we are, if it’s 10 o clock, the sun “is up there in the sky”. Even if covered by clouds. How do people know that there “is” a God? By trusting those who told them there is one, and a few “empirical” findings, more vague and obtuse (dealing with false positives and all the psychological tricks), but nevertheless important. We may discuss the strenght of different kinds of evidence, but in principle, such beliefs are of the same kind.
Why I press on this matter? Because I do not believe that the distinction between noumena and phenomena is relevant. I think that Kant made a hell of a trouble with that, and by placing god in the noumena category, the “thing in itself”, which is oh-so-distant from us, and discarded of any empirical evidence whatsoever, god indeed “died”.
“X” is only tentative if there’s some work we can do about it. Otherwise, it’s complete fantasy.
Actually, I am sitting on my house with temperatures surrounding 25 degrees celcius. The sun is very hot down here, guess it must be El Nino. Or is it Global
Warming? 😀
JH,
Briggs used x=y;y=x as his (apparently only) example of something we know to be true without evidence. but there is no ‘truth’ here — it’s what the operation means. You probably think it intuitive because you assimilated the definition long ago. The only time the definition could be “not true” is when x and y are things in the real world which are not necessarily equal. IOW, an indication that any reasoning based upon it doesn’t apply (i.e., “false”). Or, pf course, if I was lying about what I wanted it to mean.
The idea of calling axioms (postulates) truths likely started with Euclid who, despite protestation, was actually applying his theorems to the real world or at least thinking in terms of real world examples. That he stated them as constructions instead of simple definitions muddied things. But in fact, he was merely defining terms such as straight line stretching to infinity and what it means when two of them are parallel. Without the definitions it would be impossible to follow his reasoning if only because the follower might supply different definitions.
An example, Euclidean geometry doesn’t apply to the surface of spheres. What does that say about the “truth” of his postulates? My answer: nothing.
Dear Matt, thank you for another wonderful podcast. I am eager to learn more about statistics and probability.
Luis,
The difference is how the “truth” was arrived at. Yes, ultimately what I call reality could just be a bad dream but at least there are things in the dream that support my supposition that it is there. Using your dark room example, I “know” the sun out there because it has been every time I looked (presumably it’s noon). Using “God” as a term for the cause of thunder is perfectly OK but one is making other suppositions about “God”, such as “He is appeasable”, is wandering into the area about how one’s beliefs should be applied and that means questioning the belief. Still, the idea that there is a supernatural god is a projection of experience that happenings have causes and therefore not an unfounded belief. And if the cause appears humanly capricious the supposition that the cause possesses many of the properties of humans also has foundation.
IOW: “intuitive” does not equal “unfounded”.
Briggs said he wanted to talk about the philosophical concept of “truth”. I don’t think the current thread is staying on the track he had in mind. He didn’t define what he meant by “truth” so it’s left up to his listener/reader to do so. Something I mentioned might happen in my previous post addressed to JH (and apparently with the definition of “intuitive” as well).
His lone example indicated to me that he was confusing “logical truth” and “real world truth”. Since he’s going to talk about statistics, I assume he wants to concentrate on the “real word” thing because real world application is at statistics roots. Unfortunately, the use of statistics in finding “real world truth” can entangle that truth with mathematical truth. I wanted to point out that he needed to specifically state the difference but that seems inadvertently to have started a discussion of the definition of “definition”.
Luis,
Sounds lovely, enjoying the Autumn here.
“Such Gods were falsified by experience years ago�
First, you must define God, Therein lies the problem with all religions, in my view, definitions, some so elaborate as to seem silly. Without wanting to labour the point, your own position is no more a state of knowledge about God than anybody else on earth, Chief Rabi, Archbishop of Canterbury, Pope XXXXXX, the orange ‘fella’ with the bald head. (All of them very wise men, and you would not be out of place in their company on that score). However, it’s just a fact that nobody knows. When somebody does, for real, one way or the other, things will be very different whoever’s right or wrong.
I do know where you’re coming from but try as I might I can’t disbelieve. That doesn’t mean I can define what I believe because it’s not an intellectual point, it’s a spiritual one. (That thing that puts you in you and me in me) I’m definitely in here! (Sorry, I’m not being facetious but how else can one put it? That’s what I’m trying to describe and so far nobody has done this and I can’t either. My only intellectual contribution would be that the universe had to start for a reason. The ‘why’. My father’s opinion is ‘there doesn’t need to be a reason’. This isn’t good enough as a logical argument unless the premise is that not everything needs a reason or a cause. This would require me to accept a fact that is in direct conflict with all my experience of the world so far. Everything we have ever experienced to now has a reason or a cause. It’s a huge leap of faith to believe in a big bang from nowhere and nothing. I can’t make that leap. We’ve been here before, Luis, but that’s the problem.
I do find your perspective fascinating if a bit scary.
I’m looking forward to following this series of podcasts very much indeed.
Thanks very much for undertaking it.
DAV, gotcha. I obviously cannot answer for Mr. Briggs.
I don’t know the history or ideology of the word “axiomâ€, but my experience tells me that there are differences between “definition†and “axiom†(if x ≠ y for all cases), and (then) using them interchangeably is incorrect in many contexts.
Regarding the Euclidean and non-Euclidean geometry systems, in my view, they demonstrate the fact that an axiom or postulate is a statement which is accepted as true (without proof, by faith?) for a given branch of mathematics.
Ah… an axiom may not be intuitively clear to everyone, for example, the axioms in the aforementioned geometry systems. To further validate your point, we know that knowledge, experience and intelligence build our intuition. In this case, without proper intuition, a layman may have to accept an axiom by faith. Have I twisted things around (my specialty, so says Mr. JH) enough?!
DAV,
Such as? There is nothing in this thing you call “reality” that couldn’t be false. That is what mr Briggs alludes to in his “everything you believe is only true “given x””. But I am being a liiitle bit facetious. There is one thing that support your supposition that we are not living a dream: the lack of evidence of that hypothesis, the lack of explanation value, the lack of testability of it, all in all, an hypothesis that will get you nowhere, that is unworkable. The only end of that would be an increase paranoia and severly compromised social interactions ;).
It’s “OK” in the sense of not being “false”, but it is also irrelevant and unnecessary. Might as well say that the Pink Unicorn did it. Equally ok. Equally vapid.
There’s a difference between a philosophical foundation and an historical foundation. I’m not disputing the latter. Of course people pondered about this and concluded that the cosmological argument is strong. However, philosophy managed to get out of this mess way back ago, because it’s an untestable assertion that incorporates a lot of beliefs. One, that every happening has a cause. Not only it’s an inference, one could even say it has been disproven by quantum mechanics. Two, that such “cause” should be called “God”, which is an even greater jump of logic. Call it macaroni on cheese instead, will ya? At least I know what we are talking about.
It only confirms the paranoia of men, the ability of recognising false positives more than missing true negatives. Perhaps people do think that when it rains, it’s a sign of god. That when the train came too early and you missed it, that is god telling you something. That when you met the girl of your dreams in that beautiful afternoon it was god’s gift. It just shows how superstitious we all are, and notice, I’m making no personal criticism here: I’ve dwelled in these feelings too, long ago. It’s the human condition.
Joy,
I was specifically mentioning the Gods that were appeasable by human sacrifices. Do you consider those Gods yet to be valid, Joy? I believe we can agree on this.
About the point of agnosticism, I already addressed this, of course my atheism is a belief. Many atheists would be angry at me for making this concession, but I recognise this point. I am atheist wrt many kinds of gods, including the catholic, the protestant, the muslim, the greek, the mayan, etc., etc. I may simply call me an “agnostic” on the most extreme deistic god, which is so vague and obtuse, so completely voyd of definitions (always “up there” the guy, “infallible”, “ineffable”, etc., etc.), that I see no point of debating such “entity” at all. For me, it’s discussing angels on top of pin heads.
But I see the evidence against such entities so very well defined as “Yawhe” or “Allah” so great, that even the people that defer to these religions tend to think of their God as something “different” than “Yawhe” or “Allah”, and tend to formulate something more akin to a deistic god, albeit it is rare for a person to stop being paranoid about the meddling of his own life by an active god.
About spirituality, Joy, I fully respect your feelings, and I sometimes go on just to say what I think about stuff, while my intentions are not really to “deconvert” people at all. Mostly, I talk about this stuff in the internet with the hope of someone smarter than me pointing out mistakes on my diatribes. I also think that a kind of a belief is indeed required in order not to panic. But you see, I’m inclined for the arts, and I am a little romantic (in the aesthetic sense), so I kinda harvest this feeling of vertigo, which is very productive! Now that I’m used to it, I am more resistant at it. All the while knowing that not everyone loves mountertops. But whoah, just witness the view!
All,
Sorry for being slow to respond. Been out and away. I’m back now, but I’m sure to miss somebody’s question, or, worse, answer it incompletely or inattentively. Please feel free to re-ask or email.
DAV,
Suppose I tell you it is the definition that the symbol = means “always add 2 to the right and clap twice.” Now what? And why did I select this odd “definition”? You cannot—are not allowed—to criticize it without recourse to something. What? Tradition? Does tradition guarantee truth? Certainly not. But why reject my “definition”? What can you point to that invalidates it?
The axiom I chose is one of Peano’s of course. Another is that we assume that truth is conveyed—or preserved—by saying: x = x (where x , and all the other lowercase Latin letters I use stand for natural numbers). My axiom was: if x = y then y = x. Another is: if x = y and y = z then x = z. Lastly, if x is a number and x = y, then y is a natural number, too.
With those, all of mathematics can be developed. You can try to escape asserting their obvious truth (as many have done) by saying, “These are just definitions” but this always fails because you are always unable to answer the question, “Ok, but why these and not others?” Saying an axiom is a “definition” and not a “intuitive truth” merely pushes the inconvenient question about “What is true?” back one level. It does not evade it.
I also might devote a lecture to cover this in more depth. Stove’s proof, for example, of the non-empiricity of deductive logic.
Luis,
(Your first comment) But you are right! For all contingent (I haven’t defined this formally yet, but essentially all questions of matter of fact and existence) propositions, we should be agnostic. That is, we can not know with certainty the answers to contingent questions, even though we can know them almost certainly. We’ll measure this amount of certainty with probability.
Your decision to be atheistic is, of course, in no way inconsistent with this, conditional on some unstated suppositions about what you’re being atheistic against. One of the largest, well, the largest, problems in this perpetual debate is that hardly anybody bothers to define their terms. Don’t believe/believe in God. OK. What do you mean by God? We can talk about that separately.
Kevin,
Nobody likes a bad pun better than I. Thanks.
pouncer,
You can bet on it. We’ll talk lots about this.
masmit, Candy, Joy,
Thanks!
JH,
Conceptually there is no difference between axiom and definition except as perhaps a status thing. You are effectively arguing fine semantic distinction like that which exists between “laws” and “rules” while ignoring the fact that “laws” are just special “rules”.
As for intuitively obvious, an axiom doesn’t have to be obvious to anyone. The parallel postulate wasn’t particularly obvious to the Greeks. Euclid apparently had trouble with it himself. In fact, he avoided it for his first 28 propositions but eventually found it necessary.
There is an alternative to Euclid’s parallel postulate known as Playfair’s Axiom which states:
Personally, I find that less than obvious.
Luis,
Oddly, I think we agree. My original post regarding this was perhaps muddled. The major difference between hypotheses is whether they can be eventually proven empirically. If the existence of X can never be proven because no empirical test for it could be designed then it is effectively irrelevant to me — excepting, of course, the ever-present people with a belief in X which they think gives them some right to bully others.
My other point which I merged into my posts to you is that there is no such thing as a truth which exists by itself. That seems to me to be what Briggs meant when he said “intuitive”. This concept is irrelevant to what you are trying to discuss. The merging was a mistake on my part. Sorry about that.
Speaking of diatribes, you really don’t want to get me started on the people who insist on forcing upon me might-be’s that are nearly impossible to prove: such as will shorten one’s life. Or do you?
Briggs,
It’s no different than starting a argument with the arguably given “All me are created equal”. For the purpose of the argument, it’s veracity has to be assumed and can only be determined outside of the argument. Kant pointed out some apparent antinomies in a few things although he later “re-Kanted” (* ahem *). None of the antinomies are detectable within their respective systems.
As for “Why this and not others?” the answer is: you could use others but why bother? See Playfair’s alternate to Euclid’s parallel postulate. And in your “… clap twice” example I might eventually be able to point out that at least the “clap twice” part was never necessary in formulating a proof using your system so it might be more elegant to omit it.
What truth is being preserved by “=”? In “3=3” what is the truth of “3”? I read the “x=y” thing as meaning I can use either interchangeably because by definition they are the same thing. It’s included in the axiom list because it is a necessary requirement to all proofs that follow.
I’m actually taking issue with your “intuitively true” as if “intuitive” means “unfounded”. I believe that for the most part (like, all of it) , what is taken to be “true” is really a resultant of a logical argument starting with assumptions taken to be true vs. known, of themselves, to be true.
That said, the “x=y” thing comes the closest to a self-contained truth. It comes from experience which says similar things can be used as if they were the same so it follows that two things not dissimilar at all will as well. The problem is — and it is one in AI too — what exactly does “similar” mean? Another AI problem related to this thread is: what does know mean?
The problem is much deeper than I think you want to go into just to talk about the “truth” of statistics.
DAV,
One more try. Let me try to explain the difference between definition and axiom by reiterating that a set of axioms is required in the definition of a certain kind of structure. Sorry, reiteration is rude. There might be situations that they mean the same thing.
For example, Axioms (a), (b) and (c) below define the relation > on R, and they are accepted as logically and intuitively true assuming we accept the definition of the “plus” sign +.
There is a relation > on R. It satisfies the axioms: (The Order Axioms)
a) Trichotomy: For any a in R exactly one of a > 0, a = 0, 0 > a is true.
b) If a, b > 0 then a + b > 0 and a.b > 0
c) If a > b then a + c > b + c for any c
Would you say that the statement “If a > b then a + c > b + c for any c†is a “definition”?
Another definition: (not an if-then statement)
Something satisfying the algebraic axioms and the order axioms is called an ordered field.
DAV,
No, sir, not quite. To say in answer my question, “you could use others but why bother?” merely begs the question.
You’re on a better line when you say that arguments start with assumptions taken to be true. But nobody I know also says that the four axioms I presented are not true. Everybody believes they are. And nobody can prove it. The thing “taken” that starts the argument in this case is your intuition.
You’re also right that it is more complicated to demonstrate. Incidentally, the steps in any logical argument, those connectives, are also assumed to be true. I’ll try and do something deeper on this.
You were after other examples: there are many. Any moral, for example. If you say, “This moral is good”, you have only your faith to back it up. No empirical data can support it ultimately.
JH,
Definition of “definition” stolen from somebody or another’s dictionary (1a no less): A statement conveying fundamental character. In your example, there are three components constituting the fundamental character of “>”.
Briggs,
By “why bother” I thought it was apparent that there could be more than one set of starting points although some sets may be better than others (i.e., they shouldn’t be self-contradictory; there’s that elegance concept; etc). Their validity only matters if you wish to apply the resultant proofs to something outside of the system so it helps if there is agreement on the starting point or else one gets a resounding “So What?”
Anyway, what started all of this was my objection to what appeared to be an equation of “intuition” with some self-standing “truths”. I maintain those “truths” are obtained by induction which can never “prove” in the sense deduction can.
I find the subject fascinating. I’ve studied Cognitive Psychology and have more than a passing interest in AI. One of the things that got me started was asking what it is that people are doing when they say they are “thinking”. This topic is almost fundamental to that. I’d like to talk more about this though perhaps at a later time. It’s going to be a busy week.
Don’t worry, DAV, I’m not here trying to troll you ;). Just havin’ fun making connections and confronting ideas.
Well, everybody’s doing exactly that, but I agree that most of the “debate” revolves a lot around implicit ideas “you know what we’re talking about”, when we actually don’t (ain’t that the start of every “Let’s go to Alabama” story?). To put it complex, I’m an atheist regarding what I see “God” as described by catholics, by protestants, by muslims, and so on. Regarding a more vapid deity, I’m probably agnostic, but you are right of course, and one should regard oneself as an ignostic on such occasions.
To put it simple, I’m a skeptic. I’m a passive observer watching the doorstep and for every godly denomination that has ever crossed the doorstep of my mind, I found it lacking. Perhaps we should call “God” as “the failed hypothesis”, and be done with it. Not forever, of course, but economically, pragmatically, for the time being. A kind of “if you have something more interesting, call me” thing. I’m not exactly “waiting”, though.
Nice chat about truth, but don’t start on morals. That’s a can of worms.
Luis,
Sweet!
Thoughtful reply, thank you, I knew you were a softie. Whilst I’m happy leaping off real mountaintops, I’m not able to make my mind leap anywhere. I’m happy looking at the view. You answered my question when you say that a belief is essential to prevent panic. It’s not my own pannic I worry about, it’s others. That in a nutshell is why you might call me a defender of faith. It’s bad enough dealing with people who think they’re going to hell in a hand-basket without having to suffer the primitive nothing really matters crew. What a horrible world that would be. The human spirit won’t go anywhere, John Lenin’s imagination was a nightmare, actually.
The motive’s not even altruistic, it’s selfish with my own dire prediction of impending apocalypse at its core. Imagine six billion people withbroken spirits all trying to ‘share’!
I’m leaving on the next rocket, they’d better have F&M’s Royal Blend.
Damn, I wish that I could understand math vocabulary and concepts by looking up dictionary definitions for words.
Leo,
I know what you mean. I have tried to define a lot of the terms in the class notes/book. A crude copy of it is available on line. Use the search function for “Chapter” and you’ll see most of the chapters, in abbreviated form, on-line.