After reading Euler: The Master of Us All and excerpts from Journey Through Genius, I have felt inclined to speak about the incredible mathematician and person, Leonard Euler.
The sheer volume of Euler's work cannot be appreciated, nor comprehended until one gazes at how many shelves the volumes of his life work takes up. The quality and influence of Euler's work is not grasped until one studies many areas in math and realizes how many groundbreaking discoveries bear Euler's name (and many he worked that don't.) The saying goes: "...there is ample precedent for naming laws and theorems for persons other than their discoveries, else half of analysis would be named for Euler."
Euler's creativity withing mathematics is astounding. Take for example his proof to solve for the infinitie sum 1+1/4+1/9+1/16+... where the denominators are all squares. He starts by introducing a infinite sum and showing that it's equal to sin(x)/x. He then solves for the zeros of sin(x), changes the sin(x)/x into root form, and then equates this back to his original sum, getting two infinite sums to equal each other. He then separates terms by their power (x^n) and equates infinite sums to these finite coefficients. This yields not just the sum of squares, but the sum of any even power: an incredible result!.
Never has there been a mathematician as prolific as Leonard Euler, and I daresay there will never be again. Although this is not because someone as talented and hardworking will never exist; they surely will! Euler also had luck on his side: the time period Euler lived still had many discoveries left open. Not that we don't have unanswered questions today but they are seemingly getting more and more difficult to solve. As my advisor Shane Larson tells me: "There is less and less 'low hanging fruit'." None of this is to degrade his achievements; far from it. Just the thought that we could again have someone so prolific is exciting but I just doubt that a modern mathematician could be able to keep up with the inane pace of Euler.
But I digress. Euler will continue to be a revered thinker. His works will be studied for centuries, as there is much any young aspiring mathematician can learn from style, work ethic, and simple curiosity. Truly, Euler is the master.
Sunday, December 6, 2015
Friday, November 20, 2015
Aristotle's Wheel Unraveled
Ahhhh... Aristotle's wheel. How you have confounded us for so long! I mean just look as this animation, something must be misleading here. And oh there is. Let us examine three different cases/views of this animation to shed light on this mystery.
1. Examine how the different points on the wheel move around: obviously by inspection the different points are unfurling at different rates. Also to mention, every point along the radial line have the same angular frequency, also called angular velocity (ω). This means that all the points on the radial line rotate around in a circle at the same rate (same revolutions/second.) However they are moving at different linear velocities. This must be true since points lying farther out have a larger distance to travel around (larger circumference) and therefore must move faster in order to go around in the same time. So, if they have move at different linear speeds within the same time, they cannot possibly have traveled the same arc length, and hence circumference. (Note that arc length s=rθ , and ω=Δθ/Δt so if ω and Δt are the same for both, Δθ is the same for both. This means that since they both have different radii r, they cannot rotated the same arc length s. Furthermore, since v=Δs/Δt , having the same Δt and different velocities v implies that the wheels will again travel different arc lengths.)
2. Zero Radius.
Consider the central point of the circle. Is it rotating? Angular mechanics dictate that the tangential velocity of a point on a circle (in the specific case of rolling without slipping) to be
v=rω .
However the central point has a radius of zero so this mean that it has a tangential velocity of zero and therefore does not rotate. Similarly this point will trace out a circumference of zero as it moves. Yet this point is still displaced exactly as far as a point of any other radius. By inspection we notice points of any radius are all displaced the same. So this must mean that rotation does not have anything to do with horizontal displacement; every point of the radial line is on the same footing regardless of circumference.
Consider the case whether the center of the wheel is fixed (think a wheel suspended that's spun by a hand.) In one revolution of the wheel, what is the displacement of the points on the radial line? Zero. They go back to the same point they were before. Wheel motion is simply this case plus some horizontal displacement.
The "proof for this contradiction is simple.
Assume since that two wheels are rigidly connect with respective radii of r and r' (where r ≠ r') so that they have the same angular velocity ω. For both wheels to travel the same horizontal distance Δd they must have a velocity (v=Δd/Δt) and time (Δt) so that vΔt =v'Δt'. Both wheels having the same ω implies directly that they will have the same period T (time to go around one revolution.) This comes from T=1/f= 2π/ω, where f is frequency. So this means that Δt =Δt'. Assuming that both wheels are rolling without slipping, v=rω and v'=r'ω. But since r ≠ r' , v ≠ v', so therefore vΔt ≠v'Δt' which is a contradiction. We conclude then that both wheels cannot be rolling without slipping.
But what does this mean for the larger problem?
Well for starters, this means that least one wheel is definitely not tracing out its circumference. This matches our intuition about the animation where clearly the wheels seems to be "unraveling" at different rates, where if both traced out their circumference we should see no difference since they complete this task in the same time. If both circles cannot simultaneously trace out their circumference we should see no problem that they can both travel the same horizontal distances despite their different circumferences.
To end, we should clarify the physical difference between rolling without slipping and rolling with some slipping. After all, those from Michigan can attest to driving while slipping on ice/snow. When this occurs, some slipping between the ice and tires is occurring, which causes the wheel to slide on the ice and make the driver lose control (this is because the force changes from static friction to kinetic friction, link to a good explanation here.) Another example of rolling with slipping is getting your car stuck in gravel/dirt/snow. The tires will just rotate indefinitely without catching. Obviously they are not traveling their circumference (the car isn't moving forward!)
So for the example with the tire, replace both wheels with metal tires and drive over sandpaper. Although the larger wheel may get indents because of the unevenness of the sandpaper, the smaller wheel will be covered in scratch marks which indicates that it is "dragging" or "slipping." If you don't believe me, build and test it!
Sunday, October 18, 2015
Response to Euler: The Master of Us All
In a nutshell, the book Euler: The Master of Us All seeks to detail a number of mathematical questions/scenarios left open that Euler either solved or addressed. William Dunham would begin a section explaining the problem and the previous progress made on it, Euler's particular progress detailed in proofs/mathematical work, and then later contributions, as well as open questions left to future researchers.
The book really highlighted the brilliance of Euler, as well as the seeming absurdity that is his style of analysis. On multiple occasions Euler would simply write an object in terms of an infinite series and then take logarithms of each side, and vice versa. Euler techniques were purely for the purpose of analysis, lacking the formalism of proof writing. However, one could argue that this let Euler be more creative as he simply pursued mathematics in terms of reaching results, without regards to crippling cases and full comprehensiveness. On occasion Euler can be faulted for his lack of rigor, but then again one can hardly imagine the number of successes if he restricted his particular style. As the author stated, "One could reasonably ask whether modern mathematics would even exist without him." Euler also loved to tackle the same problem from multiple directions in order to solidify the strange results he derived to his doubters, as well as test his understanding and prowess.
Another observation I made is the sheer volume, as well as variety, of Euler's contributions. His work ranged from geometry, to number theory, to infinite series, and to complex analysis; this is an incredible body of work for one man. Interestingly enough, every area he brought his unprecedented analytical abilities and insight, even areas such as geometry where proofs were usually performed using clever insights rather than analytical techniques. His willingness to examine old topics in new ways (e.g. the relationship between the centroid, circumcenter, and orthocenter of a triangle) as well as work with new topics that others refused to consider (e.g. complex numbers) is simple incredible.
Although not comprehensive, this work gives the reader enough material to come to an appreciation of Euler and his works. Although Euler may not be the master of us all, he was certainly a master of his craft and was a person is not easily equaled.
The book really highlighted the brilliance of Euler, as well as the seeming absurdity that is his style of analysis. On multiple occasions Euler would simply write an object in terms of an infinite series and then take logarithms of each side, and vice versa. Euler techniques were purely for the purpose of analysis, lacking the formalism of proof writing. However, one could argue that this let Euler be more creative as he simply pursued mathematics in terms of reaching results, without regards to crippling cases and full comprehensiveness. On occasion Euler can be faulted for his lack of rigor, but then again one can hardly imagine the number of successes if he restricted his particular style. As the author stated, "One could reasonably ask whether modern mathematics would even exist without him." Euler also loved to tackle the same problem from multiple directions in order to solidify the strange results he derived to his doubters, as well as test his understanding and prowess.
Another observation I made is the sheer volume, as well as variety, of Euler's contributions. His work ranged from geometry, to number theory, to infinite series, and to complex analysis; this is an incredible body of work for one man. Interestingly enough, every area he brought his unprecedented analytical abilities and insight, even areas such as geometry where proofs were usually performed using clever insights rather than analytical techniques. His willingness to examine old topics in new ways (e.g. the relationship between the centroid, circumcenter, and orthocenter of a triangle) as well as work with new topics that others refused to consider (e.g. complex numbers) is simple incredible.
Although not comprehensive, this work gives the reader enough material to come to an appreciation of Euler and his works. Although Euler may not be the master of us all, he was certainly a master of his craft and was a person is not easily equaled.
On the Axiomatic Nature of Logic Systems
Although in a previous blog post I have already addressed the relationship between math and science, both being simply human creations to understand and predict various phenomena, I will take this blog post to instead further highlight the similarities of both as well as respond to some of the criticism (mostly self-inflicted) of the previous post.
To understand the studies of math and science we must start at the core of each. Mathematics must start somewhere, this much we know. Meaning the entire field of mathematics concerns itself with proving new results from old ones, so we must eventually assume that something is intially true. Of course such a statement is called an axiom. An important note about axioms is although we cannot prove them, we do not believe they are true for no good reason. We can perform operations (e.g. A+B) and make observations for what we believe is true (A+B=B+A since the order should not matter). For instance, saying that there is an integer 0 such that a+0=a for all integers a matches physical observation (I can add nothing as much as I want to a pile but the pile remains the same.)
Therefore we should ask if science have axioms, and if so, what are they? To see scientific axioms we must delve briefly into questions that philosophers, not scientists, ask. For example, how do I know the world I perceive is real? Meaning how do I know that this reality is real, not some weird pseudo-reality (think The Matrix) or that reality doesn't just exist in my own head. Despite the "proofs" of one's existence (e.g. I think therefore I am- Descartes) one simple cannot prove that one exists insofar as to be able to reliably trust one's senses to the extent of being able to make accurate scientific conjecture (and I believe that this is an inherently unsolvable problem). This inability to answer the most basic metaphysical question has plagued much of philosophy for millennium. So how do scientists make claims about how the universe functions? The simple answer is that they just assume basic properties of reality: that 1.) the universe is real to point of understanding and measuring it and 2.) we are able to measure physical properties of the universe. Again these are reasonable axioms, since by taking them as true we are able to predict physical phenomena and build neat devices utilizing this knowledge (e.g. we can build the computer/phone/tablet you're using to read this.)
So we come to the conclusion that both math and science are inherently axiomatic (and I conjecture that all logic systems share this property.) Unlike my previous post where I claimed that science and math were inherently different since the physical and mathematical worlds are inherently different, the reality is that the existence of each world is dependent on humans' belief in them. In other words, since I cannot prove the existence of the physical world and therefore simply assume it, it does not exist without this assumption. We then reach the conclusion of the previous post while covering up some of the logical inconsistencies present in the original theory.
Note: We can think to apply the same ideas to other logic systems: religion, music, cooking, etc. For example in the case of religion one assumes the existence/non-existence of a overarching reality (e.g. God) based on observation.
To understand the studies of math and science we must start at the core of each. Mathematics must start somewhere, this much we know. Meaning the entire field of mathematics concerns itself with proving new results from old ones, so we must eventually assume that something is intially true. Of course such a statement is called an axiom. An important note about axioms is although we cannot prove them, we do not believe they are true for no good reason. We can perform operations (e.g. A+B) and make observations for what we believe is true (A+B=B+A since the order should not matter). For instance, saying that there is an integer 0 such that a+0=a for all integers a matches physical observation (I can add nothing as much as I want to a pile but the pile remains the same.)
Therefore we should ask if science have axioms, and if so, what are they? To see scientific axioms we must delve briefly into questions that philosophers, not scientists, ask. For example, how do I know the world I perceive is real? Meaning how do I know that this reality is real, not some weird pseudo-reality (think The Matrix) or that reality doesn't just exist in my own head. Despite the "proofs" of one's existence (e.g. I think therefore I am- Descartes) one simple cannot prove that one exists insofar as to be able to reliably trust one's senses to the extent of being able to make accurate scientific conjecture (and I believe that this is an inherently unsolvable problem). This inability to answer the most basic metaphysical question has plagued much of philosophy for millennium. So how do scientists make claims about how the universe functions? The simple answer is that they just assume basic properties of reality: that 1.) the universe is real to point of understanding and measuring it and 2.) we are able to measure physical properties of the universe. Again these are reasonable axioms, since by taking them as true we are able to predict physical phenomena and build neat devices utilizing this knowledge (e.g. we can build the computer/phone/tablet you're using to read this.)
So we come to the conclusion that both math and science are inherently axiomatic (and I conjecture that all logic systems share this property.) Unlike my previous post where I claimed that science and math were inherently different since the physical and mathematical worlds are inherently different, the reality is that the existence of each world is dependent on humans' belief in them. In other words, since I cannot prove the existence of the physical world and therefore simply assume it, it does not exist without this assumption. We then reach the conclusion of the previous post while covering up some of the logical inconsistencies present in the original theory.
Note: We can think to apply the same ideas to other logic systems: religion, music, cooking, etc. For example in the case of religion one assumes the existence/non-existence of a overarching reality (e.g. God) based on observation.
Sunday, September 27, 2015
0/0- The Definitive Answer
After a quite heated, albeit illuminating class discussion on what 0/0 should be, we can finally put this question (to the best of our ability) to rest. To this aim, we should first discuss each position and see the evidence for each individually before coming to a definite conclusion (and oh boy is the conclusion a doozy.) So the choices supported by class discussion are:
0/0= 0, 1, ∞, or undefined/indeterminate
Examine the function y(x)=0/x. For all x ≠ 0, y(x)=0. Graphically this looks like
So a logical conclusion would to allow the discontinuity to simply equal the limit as x approaches 0
So we would have when x=0, y(0)=0/0=0.
Let us now look the function g(x)=x/x. For all x ≠ 0, g(x)=1. We can see this visually as
Again we should be tempted to fill this discontinuity by the limit as x approaches 0
3. 0/0= ∞
For any a > 0,
When a=1, this looks like
We know this is true for any a > 0, so we could think to simply extend this property to when a=0 as well (there is nothing particularly wrong with this assumption.) Therefore we would have 0/0= ∞
There is similar argument when a < 0, except there the limit goes to -∞,
another logical answer.
another logical answer.
4. 0/0 is undefined/indeterminate
This is nothing inherently wrong with refusing to answer the question. In fact why should we answer a question that seems to have multiple correct answers? All three approaches above seem somewhat logical, so we should be worrying about now about how to pin down a single answer. After all, when we solve something like 2b=5 we don't get several logical non-equivalent answers. So where is the inconsistency? Where was our mistake? The reality is that all of the above answers are correct.
Consider the equation bx=a, where we can see that x is solved for by x=a/b. Now, consider when a=0, or bx=0. From the properties of the 0, we see that either b or x must be zero. Since we don't want to let x be trivially zero, let us say b=0 instead. Then we have
0x=0
Here is the key part: because ANY number times 0 is 0, x can be any number. Again we can see that
x=0/0
follows trivially from this result. Therefore we concluded that 0/0 can be any number x that has the property 0x=0.
Another way to see this is with the following question: how many times can I add up zeros to get zero? Any number of times!
5. Conclusion
If 0/0 can be any number, what is the point in defining it? The concept of having a "solution" to 0/0 seems rather meaningless, since any number we could possible wish to be the answer is. So, we shouldn't feel the need to provide a single answer. Our conclusion is then this: we in fact will refuse to answer the question. In other words, since any 0/0 can be any number, 0/0 is undefined/indeterminate.
Sunday, September 13, 2015
Doing Math da Greek Way- Volume and Surface Area of a Sphere
As modern mathematicians, we are very aware of all the results of classical geometry, especially regarding areas and volumes. Heck, these problems are usually designated to poor freshmen in order to acquaint them with rudimentary calculus. However, my question is a bit different: how in tarnation did mathematicians figure these "simple" results using basic geometry? For the time being let us abandon calculus so that we can stick to our intuition and wits. If you wish to see the "official" calculus way, here is a link to my attempt to explain it. SO without further ado, let us do math the "Greek" way.
Let r be radius. Let us assume we already know the results of basic circles (I can't derive everything, you know!) such as Circumference =2π*r and Area=π*r^2 as well as squares such as Area=(side)^2. We can find the surface area of a sphere first. Since the surface area of a square box is 6(side)^2 (we are just adding 6 square areas up), we should expect our answer for surface area of a sphere to have the form of constant*(side)^2. Our first guess, a dumb guess if you will, would be (Circumference)^2=4π^2*r^2 since like the case of the square we have our length squared. Geometrically we can think of the circumference "sweeping" out area as it rotates. A link to a Mathematica file I made of this phenomena is right here. However, we can see that during the first animation that there is an overlap where area is counted more than one time. Upon measurement we would see that experiment matches this observation. Adjusting our thinking slightly, we can see that we don't need one of the circumferences to sweep out entirely. In fact one of these terms can simply be the diameter (2*r) while we let the other sweep. Physically this means that we can let the sweeping occur around a diameter. The second animation shows this accurately. So now we have
Upon measurement, we can agree that this is the correct expression.
Now onto volume. I can postulate on the volume of sphere by the fact that I know the volume of a cylinder with the length 2r (Volume=Area*2r=2pi*r^3). We know that the sphere could not have a greater volume since it would sit inside the cylinder. A VERY VERY keen observer would notice while looking at the cross section of a cylinder, a sphere, and cone all with the same radius, the cross sectional areas of each three objects would have the relation.
Let r be radius. Let us assume we already know the results of basic circles (I can't derive everything, you know!) such as Circumference =2π*r and Area=π*r^2 as well as squares such as Area=(side)^2. We can find the surface area of a sphere first. Since the surface area of a square box is 6(side)^2 (we are just adding 6 square areas up), we should expect our answer for surface area of a sphere to have the form of constant*(side)^2. Our first guess, a dumb guess if you will, would be (Circumference)^2=4π^2*r^2 since like the case of the square we have our length squared. Geometrically we can think of the circumference "sweeping" out area as it rotates. A link to a Mathematica file I made of this phenomena is right here. However, we can see that during the first animation that there is an overlap where area is counted more than one time. Upon measurement we would see that experiment matches this observation. Adjusting our thinking slightly, we can see that we don't need one of the circumferences to sweep out entirely. In fact one of these terms can simply be the diameter (2*r) while we let the other sweep. Physically this means that we can let the sweeping occur around a diameter. The second animation shows this accurately. So now we have
Surface Area=(2π*r)*(2*r)=4π*(r)^2
Upon measurement, we can agree that this is the correct expression.
Now onto volume. I can postulate on the volume of sphere by the fact that I know the volume of a cylinder with the length 2r (Volume=Area*2r=2pi*r^3). We know that the sphere could not have a greater volume since it would sit inside the cylinder. A VERY VERY keen observer would notice while looking at the cross section of a cylinder, a sphere, and cone all with the same radius, the cross sectional areas of each three objects would have the relation.
Cone+Sphere=Cylinder .
So a dumb guess would be that same relation holds for the volume (not unfounded since the volumes and cross areas differ only from a "sweeping" mechanism), so we would have
Sphere=Cylinder-Cone=4π*r^2-2/3π*r^3
Sphere=4/3π*r^3
And by golly we have it.
Tuesday, September 1, 2015
On the Human Origins of Mathematics
The perceived beauty and interconnectedness of mathematics often leads many to conclude that math is wholly apart from man, and that advances in mathematics are in fact discoveries rather than creations. However, mathematics is simply one of many logic structures used to explain the world around us (i.e. science), meaning its creation is the result of human cooperation and world building (refer to Peter Beger's Sacred Canopy). Let us take a quick thought experiment to evaluate this statement: what is mathematics without humans studying it? Does math still exist then? A quick parallel to help us understand this concept is the difference between science and nature. Barring philosophical questions, we know that nature would exist without humans, but science would not. This is because science is our understanding of reality, it is not reality itself. Thus goes the saying, "Newton's Laws never moved a billiard ball."
Examining a brief history of mathematics we can see that mathematics has always come about as an explanation of the world we see. Imagine the earliest forms of algebra: I have 3 apples, and after picking 2 more, I have 5 now. The question would be: why do I have 5 now? Why not 4 or 6 or something else? The answer we know now is that this a physical manifestation of addition, by adding 2 I must increase my original number by 2. Thus we have addition as an explanation of physical events. Next we can examine geometry: how come the distance across this round stone seems to be proportional to the distance around it? I measure this stone and I can find this number that relates the two, which I call pi. Again I can see this physical phenomena in nature and create mathematics to try to understand it. Astronomy and the rest of the sciences all logically fall in suit of this idea, with people wanting explanations to why certain "stars" were at certain locations during the year, etc. In fact, in the case of sciences, the mathematics and the explanation of the phenomena are heavily interwoven to the point where they are interchangeable and complementary.
Since our conclusion is that mathematics does not exist without those studying it and therefore man made, we must lastly address the reason we do not "discover" mathematics. Let us bring up another question: does the mathematics exist before it is known? Since we've concluded that mathematics is an explanation, like science, let us draw another comparison between the two. Quantum mechanics, the funky physics of the small, was obviously not always known in the past and was formulated by scientists in the early 20th century. However, the reality of how these particles behaved was always true in nature, regardless of our understanding of it. Essentially what this means is that nature is right regardless if science is. With the case of mathematics, there is no equivalent world where mathematics exists regardless of humans. Any math we "see" is superimposed on the world; it is our metric to quantify reality. We yearn to explain that which is around us, the goal of academia. Meanwhile, reality simply is. Mathematics is mental; it is a framework we use to understand, and like science, it is not reality itself.
Edit:
I have another discussion on this topic that responds to a few of the criticisms of this argument, many of which were my own (I am not personally convicted either way but I want to facilitate conversation on this topic) I take a slightly modified position, of which I posted here.
Examining a brief history of mathematics we can see that mathematics has always come about as an explanation of the world we see. Imagine the earliest forms of algebra: I have 3 apples, and after picking 2 more, I have 5 now. The question would be: why do I have 5 now? Why not 4 or 6 or something else? The answer we know now is that this a physical manifestation of addition, by adding 2 I must increase my original number by 2. Thus we have addition as an explanation of physical events. Next we can examine geometry: how come the distance across this round stone seems to be proportional to the distance around it? I measure this stone and I can find this number that relates the two, which I call pi. Again I can see this physical phenomena in nature and create mathematics to try to understand it. Astronomy and the rest of the sciences all logically fall in suit of this idea, with people wanting explanations to why certain "stars" were at certain locations during the year, etc. In fact, in the case of sciences, the mathematics and the explanation of the phenomena are heavily interwoven to the point where they are interchangeable and complementary.
Since our conclusion is that mathematics does not exist without those studying it and therefore man made, we must lastly address the reason we do not "discover" mathematics. Let us bring up another question: does the mathematics exist before it is known? Since we've concluded that mathematics is an explanation, like science, let us draw another comparison between the two. Quantum mechanics, the funky physics of the small, was obviously not always known in the past and was formulated by scientists in the early 20th century. However, the reality of how these particles behaved was always true in nature, regardless of our understanding of it. Essentially what this means is that nature is right regardless if science is. With the case of mathematics, there is no equivalent world where mathematics exists regardless of humans. Any math we "see" is superimposed on the world; it is our metric to quantify reality. We yearn to explain that which is around us, the goal of academia. Meanwhile, reality simply is. Mathematics is mental; it is a framework we use to understand, and like science, it is not reality itself.
Edit:
I have another discussion on this topic that responds to a few of the criticisms of this argument, many of which were my own (I am not personally convicted either way but I want to facilitate conversation on this topic) I take a slightly modified position, of which I posted here.
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