"Every time you post... you keep me up at night thinking about what you just said. Shame on you." - rjo__
Ha!
I liked the suggestion that something as "simple" as f=ma might not be the same as we know it a trillion years from now, nor was it the same as we know it a trillion years ago, but it with everything else is in a state of constant change, and right now the representation that we use to describe it just happens to work.
It sort of reminds me of what we have been running into with Moore's Law ... many of the "dominant" traits or characteristics that we have come to understand are becoming more and more "recessive" and what was once considered a "recessive" trait is now exploited as a "dominant" feature. This phenomenon is very peculiar to me and makes me wonder as technology does advance, what other recessive physical law traits are just waiting to become the dominant standard.
On a side note. I believe in something I call "ordered chaos" for lack of a better definition or accepted terminology. It's when a series of unrelated random events come together in alignment. A phase shift if you will, and somewhat related to the "dominant" / "recessive" flip I mention above. This phase shift causes a sudden spike in the signal to noise ratio one might observe. Enough so that even subconsciously you are made aware of it and without realizing it you become a part of it. I don't see why this could not happen with so called gravity waves. The ever increasing earthquakes ... 5.1 just the other day ... who knows...
I don't believe in a single big bang theory, but rather multiple big bangs at different points in time and location analogous to something like boiling water on a universal scale. The perpetual "engine" for something like this could stem from a black hole, either through some finite critical mass event or some hyper dimensional energy/mass transfer event that our brains can't fathom. These events happen so far apart in time from one another that it is difficult for us to see the complete scope. But there is that funny word again ... "time" ... which is believed to be a byproduct of mass moving through space time... but as you increase the mass and acceleration what happens to time in a black hole? does it stop? or more accurately, does it approach an infinite parallel tangent? .... funny thing with a tangent curve is that at some point it "snaps" to the opposite polarity. Consequently a tangent curve can also be used to express population growth .... take a look above the zero line of the x-axis and you might recognize where "we" are on the curve ... at some point there will be a collapse or a "snap" to the opposite polarity. It has happened over and over again throughout history, and I dare say it will happen again. It's "natures" way of keeping up with maintenance.
Edit: Most people will contemplate what I just said unitarily and that is the wrong thought mode... this is analogous to the boiling water big bang, in that within the tangent curve there are multiple overlaps. i.e. cultural overlaps, etc.
Language is always a barrier... but it is the thought that is important.
I think my views are probably identical in the important aspects.
I don't have an issue with gravity waves... if they exist, fine, if they don't exist... fine. I fully support scientists looking for them. Even if the experiment had turned out to be a dud, the scientists would have learned a lot, just from the effort. If next year, another team on the other side of the globe tries to replicate the experiment and finds a flaw... even better. We can look again:) If it feeds scientists, I'm all for it. They are mostly good people and mostly have to jump through hoops just to get the right to try to advance our knowledge.
My question about gravity doesn't really have anything to do with black holes, etc. I have always been curious about what goes on at the very center of a gravitational field and if that isn't where all of our elements actually come from.
If you remember the crises at Los Alamos during the middle years of the Manhattan Project... when the guys finally came to the conclusion that they needed a perfectly spherical implosion... keep that thought in mind.
Now consider what is happening at the center of the Earth. We know it gets hot down there, but do we really know why?
I have always thought that one reason might be that the elements down there are being pulled in all directions by gravity... and that what we see in the heat is at least partially the result of the loss of space between atoms... causing more collisions. So, we have atoms being symmetrically pulled all scruntched together... sub-atomic particles flying around. I think you can catch my drift here:)
When carbon13 makes its way up to us... it starts falling apart immediately. Just saying.
Ok... this sounds like a joke, but it is an actual question. So, before even asking, one more short story.
Hitachi... best company in the entire world, maybe the best ever. My company decided to build the first mobile MRI based on Hitachi's permanent magnet design. It outperformed every fixed site high field system on the market... at that time. One day we are scanning in the middle of a corn field, and I get a frantic call that the unit is down... well, not down but not working the way it should. The guy from Hitachi shows up five minutes later, picks a bushel of corn and tells me that we have a tiny hole in our RF shield and that radio signals were getting in and bouncing around.
I was floored... how in the world could a radio wave get through a hole that was much smaller than it was... and then it occurred to me... That is the length of the wave...
So what is the width?
ps the only thing not true in the above is that we were deep in the sticks, but we were not actually in a corn field. No corn was injured in the process of fixing our unit.
I don't believe in a single big bang theory, but rather multiple big bangs at different points in time and location analogous to something like boiling water on a universal scale.
Yes! I totally agree. If you look at the distribution of galaxies in the universe, they seem to congregate on the ridges of a sponge or foam. It's as if expansive forces keep percolating everywhere, driving matter to the interstices of neighboring expansion zones.
There was a brief discussion about papers. You won't find my name on any published papers.
Valerie Lednev once begged me to allow him to put my name on a paper. I was afraid that if I did, someone would have eventually asked me to answer some question about it. The paper was about using the theory of parametric resonance to explain the reversal of gravitational field effects on plant metabolism... I wasn't sure I even believed it and I sure couldn't explain it. I have a better handle on it now and understand what he was doing and trying to accomplish by publishing the paper, but please don't ask:) It was solid work.
I did self-publish (in those days, self-publishing amounted to taking the paper to a place like Kinkos). I wrote the paper for an upcoming fellowship, but I decided to publish it out of pure spite after Jose B. withdrew my fellowship offer... just because I had managed to get myself fired for calling my mentor's replacement a liar... which he was. While I was at it, I should have called him an intellectual midget. Missed my chance. Water under the bridge.
I distributed my little paper at a summer seminal at Stanford and at big meeting on Maui. I thought my little outburst would convince someone to give me a second chance if I poke a few holes in Jose B.'s math... It didn't work:)
I personally believe that our souls are tied to Earth so long as anyone remembers us. So, after we depart it is best if people just forget us. I don't have a problem with Earth, it is a real hoot. I just want to keep my options open. I am planning a tour for after my final repose, and I want to start it within a few decades at the latest.
I'm planning to give Seti, the first, the cosmic finger on my way out:)
I don't like the idea of ignoring basic abstractions. Points, lines, planes, circles, surfaces are really, really useful.
The making of things is close to my heart.
And as far as measuring something... the only number you won't get is the one you ask or design for. (no joke) In fact, you just can't have that one at all. So, why bother with it in the first place?
Teach 'em that, and have the conversation that absolutely will happen the moment they get introduced to that idea, and suddenly a whole lot of real world manufacturing makes a ton of sense. Plus, those abstractions all suddenly can be useful, like bounding the set of numbers they do find acceptable just by way of one example.
Equal? Only atoms consistently measure up and perform to spec. Everything else? Not equal. Start there. (In the context of making things)
We can keep points, lines, and planes, etc. But we won't tell the kids about them until later.
There is no reason to burden children with moral dilemmas. Truth is really important to some kids... if you ask them to accept something that they view to be wrong or impossible... you lose them, until later, when you call the mother to let her know that her child is retarded... no joke. That actually happened. My mom informed the lady that she didn't know she was talking about and walked out:)
When my wife worked at a local church I conducted an experiment utilizing the relatively long stretches of hallways late at night to measure the speed of light using ahem) a Propeller. Since the clock speed of a Propeller is 12.5ns at 80MHz and light travels about 1 foot per ns, my "measurable" resolution was roughly 12.5 feet per clock cycle. With a few carefully positioned mirrors(Hard disk platters) and a laser I was able to "fold" about 2000 feet in the hallways. I created an optical feedback loop oscillator where the components themselves used in the oscillator had a base line propagation delay, but I was able to observe and predict different countable delays based on increasing or decreasing the distance the light path took. Now thinking about it, if I were to do something like that again, I would dynamically allow the optical feedback oscillator to become the clock source to the processor thus eliminating the 12.5ns resolution barrier. Instead of folding the light multiple time physically in the hallways, I could fold it in software thousands, millions, or even billions of times to observe any compounding error fluctuations.... These days we get a lot of Earthquakes <- that might be interesting to see that effect on the proposed experiment. When I first did the original experiment, earthquakes in our area were unheard of, now we have them every day, increasing exponentially every year.
I believe you and suspected this about you a long time ago. If you could limit yourself to one photon at a time, I would deeply appreciate it:)
just to follow up on the of time of flight issue. I don't know if this will spark anything, but it is interesting and it might. So here goes. I read a paper a long time ago... probably in Nature or Science. Someone, somewhere learned how to control a laser... or the output in such a way that if they fire what they claimed to be a single photon of red light ... a person saw red light. But if they fired two photons, with a known time difference... the person didn't see red... I think they saw blue... but it could have been green.
So, when you are looking around and cogitating about this... consider that you might get more info out of two well placed pulses than you do with one... depending upon the nature of the sensor... which is in your area of expertise... not mine:) And it might help to figure out which photons went in a straight line and back... and which ones bounced around a little and then came back. Right now, I think it is an unsolved problem.
I forgot to say... in the eye, there are red cones, blue cones and green cones... so when there were the two red photons... they some how got linked up and didn't hit a red cone and stop. They didn't stop until they hit a blue... or green cone... can't remember which.
At the biologic level this is roughly equivalent to the physics experiments about on quantum pairing... and supports your idea that there might be all kinds of things that are causally related, but which are commonly dismissed by purely statistical arguments.
What is the harm in calling something "circular" or "linear" rather than a "circle" or a "line"... Those kids know about adjectives. And then fully
explaining why we don't call them circles or lines any more. What is the harm?
I'm not seeing the issue here. When you give a young kid a ruler and a pencil he soon gets the hang of drawing straight lines. Then with the ruler or better, with a pair compasses he soon gets the hang of marking off points on the line and measuring distance with the equal spacing the compasses give. Drawing circles comes naturally with compasses. Soon you have him creating lines at right angles to some other line. Constructing triangles, squares, rectangles, hexagons. And soon we are onto counting out squares to get the area of things. And so on and on.
In all of this, no one is ever getting fussy about the zero width of a line, or the zero diameter of a point or the not quite perfectly circular circles. We
are just playing pencils, rulers and compasses. Counting out steps to get lengths, counting squares to get areas.
We have been doing this since ancient Greek times. It works fine.
The minute you do that, you can then define Pi as a real number
but, but, π has always been defined as a real number, it's π = C/D.
...and Pi actually can be computed.
Not to a perfect accuracy. There are many ways to compute it. Many of which are understandable by primary school kids.
Every circular structure is not actually a circle, it actually does have a circumference and a radius(or series of them)
True enough no physical object is actually a circle. But I will argue that physical objects do not have an easily defined boarder length (We can't call it "circumference" because we are not talking about circles now.) For the same reason any real object does not have a radius. Think of all the imperfections in the material surface, and all those bumpy atoms you see there? If you start measuring the perimeter around all those nooks and crannies in can become huge. Besides now you have to talk about what is the size and shape of an atom? Where is it's boundary actually?.
I suggest that the harm is that once you fuzzyfy things like that you no longer have anything to talk about easily. Only we when we abstract these things out into perfect forms can we make progress.
We also should also stop using the equal sign. Nothing ever equals any other
thing exactly.
Wait a minute. Sure things can be exactly equal to other things. If I put two cats in a box and two cats in another box would you not say that "The number of cats in the first box = the number of cats in the second box"? I can do this for sets of all kind of objects, real or imaginary.
What this is of course just numbers, integers, counting, number theory. We have to have absolute equals here else we can not make any progress. Again I suggest that you fuzzyfy the equals, "2 + 2 is about 4", then there is nothing to talk about easily. It just makes things much harder, if not impossible.
If your teacher said "the equals sign meant that whatever was on one side of the equal sign was exactly the same as what was on the other side." Then she was correct. (Unless whatever equation she had on the board at the time was wrong!)
What did you find abhorrent about the idea?
Why risk it... get rid of the equal sign... never talk about equality... always talk about transforms.
I'm not sure I know what you mean.
In maths the equals sign is not a transform. It is a statement of fact.
1 = 1
1 + 1 = 2
1 + 1 + 1 = 3
1 + 1 + 1 + 1 = 4
Therefore substituting the second equation into the fourth:
2 + 2 = 4
Or: All the points on the circumference of a circle, centred at 0,0 satisfy the equation:
x² + y² = r²
Where x and y are the Cartesian coordinates of the point and r is the radius of the circle.
We in computer programming land get confused because we use = for an assignment of one thing to another. Not as a statement of fact. At least in C an many other languages.
There really is no need for the square root of two. No-one ever measures anything and reports back that the measurement is the square root of 2.
That may well be true.
But what about the other way around? What about if I have two beams joined at right angles. Each has a hole some length from the join. The same length in each case. Call it L. Now I want a beam with a hole in each end that I can mate with the holes in the other two beams. Thus making a rigid structure. How long should my beam be? Clearly the holes in my new beam must be the square root of 2 * L between centres.
Engineering of all kinds is full of such problems.
Taking Phil's example... a rightly sided triangular object can only have an hypotenuse that measures the square root of two if you have chosen the wrong unit:) So, in True Math we could have a rule... If your answer involves the square root of two, you have either measured the wrong side or used the wrong unit.
As pointed out before, changing the units, or number base, to get rid of the root 2 problem only moves the problem elsewhere. It does not help.
Getting rid of root 2 gives me a bit of a headache. If you ban root 2 you now have a hole in the number line where root 2 used to be. The number line is no longer continuous. There is a point missing. It's been broken into two halves.
Worse still if you want to get rid of root 2 you probably want to get rid of all the other pesky irrational numbers. Which is is kind of drastic as there are an infinite number of them. In fact infinitely more than the infinite number of integers and rationals. So now our number line is no longer a line at all. It's sparsely populated row of integers and rationals with gaps in between!
My sense of justice, equality and democracy inclines me to say that I think the root of 2 is a point on the number line and it deserves it's place there as much a 1 or 2 or any integer or rational. As do all the other irrationals. They hold the number line together after all. We should be grateful to them for that
I don't know about "True Math" but Professor Norman Wildberger does not like to use root 2 or other irrationals in his maths. To do that he does not talk about "length", he talks about "quadrance". Where quadrance is the area of the square that sits on the line joining two points. So, for example, our right angle triangle with hypotenuse of root 2 and the two sides of length 1 he would say is actually a triangle with two sides quadrance 1, and one side of quadrance 2. If you work with the squares like that, quadrances, you never have to take square roots and you never have a root 2 problem. Magic!
Norman has a brilliant series of lectures about "Rational Trigonometry" where he can show you how to do all that geometry and trig you did in school without ever using a root or other irrational number.
I read that back when it was first published (when I was not nearly so mathematically-challenged as I am today), and it's always stuck in my mind. I do think the abstractions have their place, however.
[edit] Or maybe you wouldn't. FWIW, it's an article titled "A Viewpoint on Calculus". The author suggests teaching most students a finite-difference calculus instead of teaching classic (what he calls "infinitesimal") calculus, since so many real world problems require that approach anyway. It seemed relevant to your argument regarding the sloppiness of reality compared to the perfection of mathematical abstractions.
I worked through a nice chunk of the rational trig series. I'll have to finish it at some point.
It's useful. What I like the most is one can use linear measurement to establish a lot of things and do so with both good precision, without transcendental, and also to understand, based on fractional results, just what that precision actually is.
IMHO, this should be taught in HS. It would be a nice addition to algebra, sort of a pre-trig type of thing very useful for those who will likely find comparing and contrasting the two a good check. For those who don't continue, they would get a very life scenario useful set of math they can apply practically. Seems a win-win, for not a lot of effort.
*a general life observation of mine is ratio and proportion type math is very broadly applied by most adults I know. Algebra is second in line. Trig, calc, various matrics, etc... are niche things, and for good reason. They aren't necessary for most people. It's my observation that most people could apply this kind of thing nearly as generally as they can ratio and proportion, maybe needing a bit of algebra to arrange an unknown. In terms of education value, this is pretty high.
The more general case of spherical or circular trig is still very important. Lots of use cases in my life at least. But, there are a ton of problems more easily solved and understood using this method. One super easy example would be sheet metal parts and the unfolding of them. I've got good mastery of this, and it's filled with trancendentals. I suspect I could rewrite that math and it's going to boil down to percentages / ratios and the occasional root to get "the size of it for realz" results, with the added bonus of linear measurement being the only check and sample required for either making, reverse engineering, and unfolding most parts. NICE!
On a side note, I've had some difficult to paramerize CAD problems in the past. For many people, just using math to derive things makes a lot of sense. Because I have a long history with CAD and geometry, I often find it useful to express things geometrically. The software, as well as real world measure, check, etc... cycles align favorably with design intent expressed in this way. It's also visible to others where not everyone can parse a set of equations to infer both design intent and the dynamics of a given system easily.
A little bit of geometry that can move in software, or that can be discussed on a white board, gets right to those things.
This method can make some constraints and dynamics perform better, and not demonstrate erratic behavior on edge cases. (0-180, etc... degrees, precision errors)
Some people I'm currently working with will get this right away. It's more accessible to people with a more basic math background. Some common scenarios can be done in your head too. Nice bonus. Definitely will show this off next chance I get. Sometimes math is fun. This is definitely one of those times. And this guy has a nice, unassuming, enthusiastic style. I appreciate it.
I read that back when it was first published (when I was not nearly so mathematically-challenged as I am today), and it's always stuck in my mind. I do think the abstractions have their place, however.
[edit] Or maybe you wouldn't. FWIW, it's an article titled "A Viewpoint on Calculus". The author suggests teaching most students a finite-difference calculus instead of teaching classic (what he calls "infinitesimal") calculus, since so many real world problems require that approach anyway. It seemed relevant to your argument regarding the sloppiness of reality compared to the perfection of mathematical abstractions.
Thanks
Pretty smart guy and he is exactly right. I think computers have largely solved this problem. Today, I don't think we have to worry about trying to teach kids any kind of calculus. Teach them programming and they invariably end up using discrete calculus, almost without trying and without knowing what it is called.
During college, I was right up there at the head of the class mathematically until I hit calculus (or it hit me)... not sure which. I remember taking entry testing for counseling... and my counselor told me I had about a 1 in three chance of making it out of my first year.
I passed calculus, but boy what a waste of my effort. Didn't understand a thing:) That translated to really struggling with physics and chemistry courses that absolutely required it. I only passed them because I had a little feel for differentials... but it was not fun. Today most of that could be done on a computer, without any calculus and I would be just fine:)
The only reason I survived elementary school is that right about the time things were coming to a boil in my head, I got a disease... don't want to name it because I think they got the diagnosis wrong. But it forced me out of school for a couple of years. And it restricted me to bed. My parents got a beautiful set of the Encyclopedia Britannica and I pretty much read it from cover to cover. It was inspiration and salvation.
I recovered fully... except that my taste for oranges and the lack of exercise gave me the weight of an NFL lineman... but not the height:)
"Therefore substituting the second equation into the fourth:"
That wasn't my problem... my problem was that on one side of the equation you have an operation and on the other side of the equation you have the result of that operation. So to say that what is on the left is exactly the same as what is on the right offended me. If she had stated it differently, I probably wouldn't have had a problem.
The other issue is the idea that you have to "teach" math to kids. I think we are born with all the math we are ever going to have. I'll give you a good example... you could try to teach me algebra all day long and it wouldn't work. I could solve the problems... but I didn't do it the way I was taught, because I couldn't understand what in the heck they were talking about. It just sort of came out of my head. When we got to trig, we had a very good textbook... I read the first chapter and then consciously put it all together in my head... figured it all out from A to Z in a couple of days of thinking. The only reason to read the rest of the book was to find out how the book defined various things that I already knew. That's as honest as I can be.
So, when we "teach" kids math... we are forcing at least some of them to use numbers in a way that their brain just aren't wired for... brains can be wired in different ways. Again, I think computers are the answer. Teach kids to program first... then give them problems and see them come up with the answers with no further teaching about math. You do have to teach math at some point to define terms for everyone, but I think that should come later.
I myself wasn't that great in calculus but survived through Calc II ... prior to that, throughout high school, almost everything math and science related, Chemistry especially, I did on my computer. Cheating? No, I didn't see it that way. I used the computer as a tool and wrote software that would work out the problem as if I were doing it long hand. <-- After all, I needed to show my work. :-) ... I justified this as not cheating, because I had to have a deep enough understanding of the problem to write the software in the first place in the same way I would solve the problem long hand. Using a computer to help facilitate my work just made getting it done much easier. Keep in mind this was way before there was an internet (a BBS if you were lucky), so there was no Googling an answer. ... Meanwhile when I had started Calculus, I had figured out a way of solving complex Inverse Matrices in my head. At the time I thought it was so dumb to waste 3 sheets of paper to work out the solution, when I could just look at the problem and write the answer down. I confronted the Professor and showed him what I was doing, and he allowed me to turn in my work using a third of the paper. ... It was all downhill from there ... NO, just kidding... I enjoyed various other aspects of Calc if not for anything else to cause me to think a little more about certain circumstances in ways I might not have considered before.
Consider Fermat's last theorem. The reason that it hasn't been proven(or has only lately been proven... I can't tell) stems directly from the structure of the math being used.
If you change the structure of the math, add a pragma, it is still somewhat difficult to prove the theorem, but it does become possible.
Here is the pragma in bad English: something is mathematically proven, when after the correct use of the other pragma the question is put into a state, where no other possibility is known to exist and no other possibility could reasonably be expected to exist. Some would argue that if we accept this kind of pragma, then Fermat's last theorem requires no proof... but it is a conjecture, which requires some further proof, the establishment of some mathematical condition, which clearly and unambiguously demonstrates why it is true.
This means that some proofs will be eventually overturned because something which was thought to be impossible is shown to exist... but it would also allow relatively easy proofs for theorems such as Fermat's.
Comments
Ha!
I liked the suggestion that something as "simple" as f=ma might not be the same as we know it a trillion years from now, nor was it the same as we know it a trillion years ago, but it with everything else is in a state of constant change, and right now the representation that we use to describe it just happens to work.
It sort of reminds me of what we have been running into with Moore's Law ... many of the "dominant" traits or characteristics that we have come to understand are becoming more and more "recessive" and what was once considered a "recessive" trait is now exploited as a "dominant" feature. This phenomenon is very peculiar to me and makes me wonder as technology does advance, what other recessive physical law traits are just waiting to become the dominant standard.
On a side note. I believe in something I call "ordered chaos" for lack of a better definition or accepted terminology. It's when a series of unrelated random events come together in alignment. A phase shift if you will, and somewhat related to the "dominant" / "recessive" flip I mention above. This phase shift causes a sudden spike in the signal to noise ratio one might observe. Enough so that even subconsciously you are made aware of it and without realizing it you become a part of it. I don't see why this could not happen with so called gravity waves. The ever increasing earthquakes ... 5.1 just the other day ... who knows...
I don't believe in a single big bang theory, but rather multiple big bangs at different points in time and location analogous to something like boiling water on a universal scale. The perpetual "engine" for something like this could stem from a black hole, either through some finite critical mass event or some hyper dimensional energy/mass transfer event that our brains can't fathom. These events happen so far apart in time from one another that it is difficult for us to see the complete scope. But there is that funny word again ... "time" ... which is believed to be a byproduct of mass moving through space time... but as you increase the mass and acceleration what happens to time in a black hole? does it stop? or more accurately, does it approach an infinite parallel tangent? .... funny thing with a tangent curve is that at some point it "snaps" to the opposite polarity. Consequently a tangent curve can also be used to express population growth .... take a look above the zero line of the x-axis and you might recognize where "we" are on the curve ... at some point there will be a collapse or a "snap" to the opposite polarity. It has happened over and over again throughout history, and I dare say it will happen again. It's "natures" way of keeping up with maintenance.
Edit: Most people will contemplate what I just said unitarily and that is the wrong thought mode... this is analogous to the boiling water big bang, in that within the tangent curve there are multiple overlaps. i.e. cultural overlaps, etc.
Language is always a barrier... but it is the thought that is important.
I think my views are probably identical in the important aspects.
I don't have an issue with gravity waves... if they exist, fine, if they don't exist... fine. I fully support scientists looking for them. Even if the experiment had turned out to be a dud, the scientists would have learned a lot, just from the effort. If next year, another team on the other side of the globe tries to replicate the experiment and finds a flaw... even better. We can look again:) If it feeds scientists, I'm all for it. They are mostly good people and mostly have to jump through hoops just to get the right to try to advance our knowledge.
My question about gravity doesn't really have anything to do with black holes, etc. I have always been curious about what goes on at the very center of a gravitational field and if that isn't where all of our elements actually come from.
If you remember the crises at Los Alamos during the middle years of the Manhattan Project... when the guys finally came to the conclusion that they needed a perfectly spherical implosion... keep that thought in mind.
Now consider what is happening at the center of the Earth. We know it gets hot down there, but do we really know why?
I have always thought that one reason might be that the elements down there are being pulled in all directions by gravity... and that what we see in the heat is at least partially the result of the loss of space between atoms... causing more collisions. So, we have atoms being symmetrically pulled all scruntched together... sub-atomic particles flying around. I think you can catch my drift here:)
When carbon13 makes its way up to us... it starts falling apart immediately. Just saying.
Hitachi... best company in the entire world, maybe the best ever. My company decided to build the first mobile MRI based on Hitachi's permanent magnet design. It outperformed every fixed site high field system on the market... at that time. One day we are scanning in the middle of a corn field, and I get a frantic call that the unit is down... well, not down but not working the way it should. The guy from Hitachi shows up five minutes later, picks a bushel of corn and tells me that we have a tiny hole in our RF shield and that radio signals were getting in and bouncing around.
I was floored... how in the world could a radio wave get through a hole that was much smaller than it was... and then it occurred to me... That is the length of the wave...
So what is the width?
ps the only thing not true in the above is that we were deep in the sticks, but we were not actually in a corn field. No corn was injured in the process of fixing our unit.
Yes! I totally agree. If you look at the distribution of galaxies in the universe, they seem to congregate on the ridges of a sponge or foam. It's as if expansive forces keep percolating everywhere, driving matter to the interstices of neighboring expansion zones.
-Phil
Valerie Lednev once begged me to allow him to put my name on a paper. I was afraid that if I did, someone would have eventually asked me to answer some question about it. The paper was about using the theory of parametric resonance to explain the reversal of gravitational field effects on plant metabolism... I wasn't sure I even believed it and I sure couldn't explain it. I have a better handle on it now and understand what he was doing and trying to accomplish by publishing the paper, but please don't ask:) It was solid work.
I did self-publish (in those days, self-publishing amounted to taking the paper to a place like Kinkos). I wrote the paper for an upcoming fellowship, but I decided to publish it out of pure spite after Jose B. withdrew my fellowship offer... just because I had managed to get myself fired for calling my mentor's replacement a liar... which he was. While I was at it, I should have called him an intellectual midget. Missed my chance. Water under the bridge.
I distributed my little paper at a summer seminal at Stanford and at big meeting on Maui. I thought my little outburst would convince someone to give me a second chance if I poke a few holes in Jose B.'s math... It didn't work:)
I personally believe that our souls are tied to Earth so long as anyone remembers us. So, after we depart it is best if people just forget us. I don't have a problem with Earth, it is a real hoot. I just want to keep my options open. I am planning a tour for after my final repose, and I want to start it within a few decades at the latest.
I'm planning to give Seti, the first, the cosmic finger on my way out:)
I don't like the idea of ignoring basic abstractions. Points, lines, planes, circles, surfaces are really, really useful.
The making of things is close to my heart.
And as far as measuring something... the only number you won't get is the one you ask or design for. (no joke) In fact, you just can't have that one at all. So, why bother with it in the first place?
Teach 'em that, and have the conversation that absolutely will happen the moment they get introduced to that idea, and suddenly a whole lot of real world manufacturing makes a ton of sense. Plus, those abstractions all suddenly can be useful, like bounding the set of numbers they do find acceptable just by way of one example.
Equal? Only atoms consistently measure up and perform to spec. Everything else? Not equal. Start there. (In the context of making things)
We can keep points, lines, and planes, etc. But we won't tell the kids about them until later.
There is no reason to burden children with moral dilemmas. Truth is really important to some kids... if you ask them to accept something that they view to be wrong or impossible... you lose them, until later, when you call the mother to let her know that her child is retarded... no joke. That actually happened. My mom informed the lady that she didn't know she was talking about and walked out:)
I believe you and suspected this about you a long time ago. If you could limit yourself to one photon at a time, I would deeply appreciate it:)
So, when you are looking around and cogitating about this... consider that you might get more info out of two well placed pulses than you do with one... depending upon the nature of the sensor... which is in your area of expertise... not mine:) And it might help to figure out which photons went in a straight line and back... and which ones bounced around a little and then came back. Right now, I think it is an unsolved problem.
At the biologic level this is roughly equivalent to the physics experiments about on quantum pairing... and supports your idea that there might be all kinds of things that are causally related, but which are commonly dismissed by purely statistical arguments.
Some great ideas coming out here to think about.
I'm not seeing the issue here. When you give a young kid a ruler and a pencil he soon gets the hang of drawing straight lines. Then with the ruler or better, with a pair compasses he soon gets the hang of marking off points on the line and measuring distance with the equal spacing the compasses give. Drawing circles comes naturally with compasses. Soon you have him creating lines at right angles to some other line. Constructing triangles, squares, rectangles, hexagons. And soon we are onto counting out squares to get the area of things. And so on and on.
In all of this, no one is ever getting fussy about the zero width of a line, or the zero diameter of a point or the not quite perfectly circular circles. We
are just playing pencils, rulers and compasses. Counting out steps to get lengths, counting squares to get areas.
We have been doing this since ancient Greek times. It works fine.
but, but, π has always been defined as a real number, it's π = C/D. Not to a perfect accuracy. There are many ways to compute it. Many of which are understandable by primary school kids. True enough no physical object is actually a circle. But I will argue that physical objects do not have an easily defined boarder length (We can't call it "circumference" because we are not talking about circles now.) For the same reason any real object does not have a radius. Think of all the imperfections in the material surface, and all those bumpy atoms you see there? If you start measuring the perimeter around all those nooks and crannies in can become huge. Besides now you have to talk about what is the size and shape of an atom? Where is it's boundary actually?.
I suggest that the harm is that once you fuzzyfy things like that you no longer have anything to talk about easily. Only we when we abstract these things out into perfect forms can we make progress. Wait a minute. Sure things can be exactly equal to other things. If I put two cats in a box and two cats in another box would you not say that "The number of cats in the first box = the number of cats in the second box"? I can do this for sets of all kind of objects, real or imaginary.
What this is of course just numbers, integers, counting, number theory. We have to have absolute equals here else we can not make any progress. Again I suggest that you fuzzyfy the equals, "2 + 2 is about 4", then there is nothing to talk about easily. It just makes things much harder, if not impossible.
If your teacher said "the equals sign meant that whatever was on one side of the equal sign was exactly the same as what was on the other side." Then she was correct. (Unless whatever equation she had on the board at the time was wrong!)
What did you find abhorrent about the idea? I'm not sure I know what you mean.
In maths the equals sign is not a transform. It is a statement of fact.
1 = 1
1 + 1 = 2
1 + 1 + 1 = 3
1 + 1 + 1 + 1 = 4
Therefore substituting the second equation into the fourth:
2 + 2 = 4
Or: All the points on the circumference of a circle, centred at 0,0 satisfy the equation:
x² + y² = r²
Where x and y are the Cartesian coordinates of the point and r is the radius of the circle.
We in computer programming land get confused because we use = for an assignment of one thing to another. Not as a statement of fact. At least in C an many other languages.
But what about the other way around? What about if I have two beams joined at right angles. Each has a hole some length from the join. The same length in each case. Call it L. Now I want a beam with a hole in each end that I can mate with the holes in the other two beams. Thus making a rigid structure. How long should my beam be? Clearly the holes in my new beam must be the square root of 2 * L between centres.
Engineering of all kinds is full of such problems. As pointed out before, changing the units, or number base, to get rid of the root 2 problem only moves the problem elsewhere. It does not help.
Getting rid of root 2 gives me a bit of a headache. If you ban root 2 you now have a hole in the number line where root 2 used to be. The number line is no longer continuous. There is a point missing. It's been broken into two halves.
Worse still if you want to get rid of root 2 you probably want to get rid of all the other pesky irrational numbers. Which is is kind of drastic as there are an infinite number of them. In fact infinitely more than the infinite number of integers and rationals. So now our number line is no longer a line at all. It's sparsely populated row of integers and rationals with gaps in between!
My sense of justice, equality and democracy inclines me to say that I think the root of 2 is a point on the number line and it deserves it's place there as much a 1 or 2 or any integer or rational. As do all the other irrationals. They hold the number line together after all. We should be grateful to them for that
I don't know about "True Math" but Professor Norman Wildberger does not like to use root 2 or other irrationals in his maths. To do that he does not talk about "length", he talks about "quadrance". Where quadrance is the area of the square that sits on the line joining two points. So, for example, our right angle triangle with hypotenuse of root 2 and the two sides of length 1 he would say is actually a triangle with two sides quadrance 1, and one side of quadrance 2. If you work with the squares like that, quadrances, you never have to take square roots and you never have a root 2 problem. Magic!
Norman has a brilliant series of lectures about "Rational Trigonometry" where he can show you how to do all that geometry and trig you did in school without ever using a root or other irrational number.
I read that back when it was first published (when I was not nearly so mathematically-challenged as I am today), and it's always stuck in my mind. I do think the abstractions have their place, however.
[edit] Or maybe you wouldn't. FWIW, it's an article titled "A Viewpoint on Calculus". The author suggests teaching most students a finite-difference calculus instead of teaching classic (what he calls "infinitesimal") calculus, since so many real world problems require that approach anyway. It seemed relevant to your argument regarding the sloppiness of reality compared to the perfection of mathematical abstractions.
It's useful. What I like the most is one can use linear measurement to establish a lot of things and do so with both good precision, without transcendental, and also to understand, based on fractional results, just what that precision actually is.
IMHO, this should be taught in HS. It would be a nice addition to algebra, sort of a pre-trig type of thing very useful for those who will likely find comparing and contrasting the two a good check. For those who don't continue, they would get a very life scenario useful set of math they can apply practically. Seems a win-win, for not a lot of effort.
*a general life observation of mine is ratio and proportion type math is very broadly applied by most adults I know. Algebra is second in line. Trig, calc, various matrics, etc... are niche things, and for good reason. They aren't necessary for most people. It's my observation that most people could apply this kind of thing nearly as generally as they can ratio and proportion, maybe needing a bit of algebra to arrange an unknown. In terms of education value, this is pretty high.
The more general case of spherical or circular trig is still very important. Lots of use cases in my life at least. But, there are a ton of problems more easily solved and understood using this method. One super easy example would be sheet metal parts and the unfolding of them. I've got good mastery of this, and it's filled with trancendentals. I suspect I could rewrite that math and it's going to boil down to percentages / ratios and the occasional root to get "the size of it for realz" results, with the added bonus of linear measurement being the only check and sample required for either making, reverse engineering, and unfolding most parts. NICE!
On a side note, I've had some difficult to paramerize CAD problems in the past. For many people, just using math to derive things makes a lot of sense. Because I have a long history with CAD and geometry, I often find it useful to express things geometrically. The software, as well as real world measure, check, etc... cycles align favorably with design intent expressed in this way. It's also visible to others where not everyone can parse a set of equations to infer both design intent and the dynamics of a given system easily.
A little bit of geometry that can move in software, or that can be discussed on a white board, gets right to those things.
This method can make some constraints and dynamics perform better, and not demonstrate erratic behavior on edge cases. (0-180, etc... degrees, precision errors)
Some people I'm currently working with will get this right away. It's more accessible to people with a more basic math background. Some common scenarios can be done in your head too. Nice bonus. Definitely will show this off next chance I get. Sometimes math is fun. This is definitely one of those times. And this guy has a nice, unassuming, enthusiastic style. I appreciate it.
Recommended.
Thanks
Pretty smart guy and he is exactly right. I think computers have largely solved this problem. Today, I don't think we have to worry about trying to teach kids any kind of calculus. Teach them programming and they invariably end up using discrete calculus, almost without trying and without knowing what it is called.
During college, I was right up there at the head of the class mathematically until I hit calculus (or it hit me)... not sure which. I remember taking entry testing for counseling... and my counselor told me I had about a 1 in three chance of making it out of my first year.
I passed calculus, but boy what a waste of my effort. Didn't understand a thing:) That translated to really struggling with physics and chemistry courses that absolutely required it. I only passed them because I had a little feel for differentials... but it was not fun. Today most of that could be done on a computer, without any calculus and I would be just fine:)
I recovered fully... except that my taste for oranges and the lack of exercise gave me the weight of an NFL lineman... but not the height:)
"Therefore substituting the second equation into the fourth:"
That wasn't my problem... my problem was that on one side of the equation you have an operation and on the other side of the equation you have the result of that operation. So to say that what is on the left is exactly the same as what is on the right offended me. If she had stated it differently, I probably wouldn't have had a problem.
The other issue is the idea that you have to "teach" math to kids. I think we are born with all the math we are ever going to have. I'll give you a good example... you could try to teach me algebra all day long and it wouldn't work. I could solve the problems... but I didn't do it the way I was taught, because I couldn't understand what in the heck they were talking about. It just sort of came out of my head. When we got to trig, we had a very good textbook... I read the first chapter and then consciously put it all together in my head... figured it all out from A to Z in a couple of days of thinking. The only reason to read the rest of the book was to find out how the book defined various things that I already knew. That's as honest as I can be.
So, when we "teach" kids math... we are forcing at least some of them to use numbers in a way that their brain just aren't wired for... brains can be wired in different ways. Again, I think computers are the answer. Teach kids to program first... then give them problems and see them come up with the answers with no further teaching about math. You do have to teach math at some point to define terms for everyone, but I think that should come later.
Rich
If you change the structure of the math, add a pragma, it is still somewhat difficult to prove the theorem, but it does become possible.
Here is the pragma in bad English: something is mathematically proven, when after the correct use of the other pragma the question is put into a state, where no other possibility is known to exist and no other possibility could reasonably be expected to exist. Some would argue that if we accept this kind of pragma, then Fermat's last theorem requires no proof... but it is a conjecture, which requires some further proof, the establishment of some mathematical condition, which clearly and unambiguously demonstrates why it is true.
This means that some proofs will be eventually overturned because something which was thought to be impossible is shown to exist... but it would also allow relatively easy proofs for theorems such as Fermat's.