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Paul Lockhart said it better than I ever could: https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician's_Lament.pdf
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Education
Let me know what you think.
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Artistic expressions
This is Norman Wildberger's personal website where he expresses his unusual opinions: https://njwildberger.com/
There is a lot to unpack here, so I'll highlight a few important things:
- Debate with Daniel Rubin: He links to this on the homepage of his website. Rubin was incredibly respectful of Wildberger's ideas, ideas which I suspect most mathematicians would dismiss as nonsense pretty quickly. I appreciate that Rubin was willing to do this, because while I do not agree with Wildberger, it only grants more credibility to conspiratorial quacks when they are ignored by experts. One issue with this "debate", however, is that Rubin gave Wildberger most of the talking time and did not push back very much. He did, however, outline his objections more clearly in another video, which Wildberger neglected to link to or mention on his website. (At least that I could find. Feel free to correct me on this.) It can be found at this link: https://www.youtube.com/watch?v=GnepxZ-ZZOI
- Modern math compared to religion: Wildberger asserts that modern math is in some sense religious, believing in things on the grounds of faith alone. I might respect his objections a bit more if he didn't do this, since many of his other objections are at least understandable, but this assertion that mathematicians so desperately want their beliefs to be true, and that they don't have any real arguments is absurd, and potentially harmful. The assertion is baseless, and it paints mathematicians as complete fools, rather than the geniuses that many of them are.
- Conspiratorial wording: Wildberger uses a lot of conspiratorial wording such as "delusion" and "blindly accept." In this way, he appeals to people who are conspiratorially minded, and who want to feel like they are smarter than the experts. This idea of a widespread delusion is simply nonsense. Mathematicians do not blindly accept statements such as "...and then taking this to infinity..." and they frequently question the meaning of this sort of statement when applied to a context in which it has no formal definition or where its application cannot be justified. They don't simply search for evidence which agrees with their preconceived notions (as Wildberger would suggest) either. Take, for example this paper: https://vixra.org/pdf/1208.0009v4.pdf. A mathematician as described by Wildberger would blindly accept its conclusions, nodding their heads every time "as n goes to infinity" is mentioned. In the real world, however, any credible source will tell you that the problems that this article claims to solve remain unsolved. This is because a real mathematicians questions the use of limits in this paper, and recognizes it as invalid.
- The impact of Wildberger's conspiratorial wording: This is what really caused me to lose any remaining respect I had for Wildberger. Many of his followers hold the belief that modern math is a complete waste of time that does nothing for society. A trip to fantasy land that mathematicians get paid to take. This is problematic for two reasons. Firstly, it makes it seem as though mathematicians don't actually do anything, negating the sheer amount of work and effort that math takes. Secondly, it has lead many of his followers to believe that if mathematicians could only wake up, our technology would be drastically better, and millions of lives could be saved. (One look at the comments on one of his posts reveals just how many of them believe all of these things.) The irony in this is that without the concepts that Wildberger rejects, he wouldn't be making these blog posts on a computer, nor would we understand nearly anything of what we do today about the universe. Switching to Wildberger's ultrafinist math would kill, not save millions.
- "Are mathematicians scientists?": The short answer is no. They aren't supposed to be. Science uses inductive reasoning. Math uses deductive reasoning. Science can change with new evidence. Math is not evidence based, and proofs are set in stone. Science uses experimentation to draw conclusions. Math uses abstract deductive proofs. Science is observation based. Math is done in the abstract, and you can't observe abstract objects in the same way that you can observe physical ones. According to Wildberger, however, the approach of science is the only valid one. This completely misses the point of math, which brings us to my next point.
- Model vs. match: Mathematicians do not assert their axioms as objective truths. Math is not intended to be part of the physical world. Math, like any field of study, should be judged by its usefulness, regardless of how that usefulness arises. Math allows us to model things in reality, but it is not itself part of physical reality. It is a model, not a match, and that is the way it is supposed to be. This is because the physical world can be somewhat of an enigma. In theory, we shouldn't be able to make any predictions at all, because we don't know, for example, that just because F = ma this one time, that F will equal ma the next time we apply force to an object. The equation "F = ma" wasn't found in some deep dark cave signed "Creator of the Universe," we just observed that this equation is consistent with our observations. This is the beauty of math: Our mathematical models can make predictions about something without us actually needing to see it. That is why math is not observation based. Because that would defeat the purpose. Sure, the fact that math exists separately from the physical world means that it doesn't always match the physical world, but that is okay. No one is claiming that everything in math has a counterpart in reality.
- The law of (logical) honesty: Wildberger's law of honesty is a good one. The issue is that it is a moral principle, not a logical one. Not pretending to do something you can't is good life advice, but for the sake of logic, considering theoreticals is incredibly important and useful, and there is no problem with it. Wildberger says that this law of honesty invalidates a question such as "If you could jump to the moon, would it hurt?" I have no issue answering this question: Yes, it would. In fact, you would definitely die. You would accelerate incredibly quickly through the earth's atmosphere and into space, and if you weren't already dead, you would find yourself in the vacuum of space where your blood would boil. Wildberger would suggest that this wouldn't happen, because no one can jump to the moon anyway. My response to this is that it is possible to talk about what would happen if one were to jump to the moon, even if that won't happen. Why is it important to be able to use theoreticals though? No one really cares about what would happen if they could jump to the moon, but considering theoreticals can be very important. Wildberger agrees that it has been proven that there is no rational number equal to the square-root of 2. What exactly is this proof? Well, feel free to look it up if you want the details, but to summarize, it begins by assuming that the square-root of 2 can be written as a fraction, and demonstrates that this leads to a logical contradiction. That's right, we are not only imagining that we can do something that we can't, but we are using that assumption to prove that we can't by showing that it leads to a logical contradiction. It is undeniable that if the assumption that something can be done leads to a logical contradiction, it cannot be done, and yet under Wildberger's "law of honesty" (at least as he applies it) this sort of proof by contradiction is invalid.
Obviously, this does not address everything that Wildberger has to say, as he has said a lot, so if there is something specific that you want me to address, feel free to point it out.
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Miscellaneous
This "math overhaul" is ridiculous. Sure, if students are struggling, we should make math more intuitive for them, but that doesn't me that we should dumb it down or even eliminate it. Firstly, the concept that gifted students can just be shoved in with struggling students and then learn the same "big ideas" at different levels will never work, and advanced students will only be held back. Speaking from personal experience, schools already love to find excuses to keep students who are gifted at math in the same classes as everyone else, and this would only make that issue worse. A student at calculus level and a student at algebra level will never be learning the same "big ideas," and trying to force this on them will only drag students down. If parents ask for their child to be place in a higher class, even if they have real evidence to back up the fact that their child is significantly advanced, they may simply be told that their child was already being taught at their level, even when they are being lumped in with average and even struggling students. Secondly, this new curriculum will barely even teach math. The idea of keeping math in touch with reality has been taken too far. Students who end up actually pursuing a career in math will not be prepared for the abstract and theoretical. Moreover, this idea that there are "multiple roads" to calculus is ridiculous. If you want to learn calculus, you will need to understand algebra and trigonometry, not data science, computer science, or financial algebra. The topics of data science, computer science, and financial algebra aren't even topics in math! Data science and financial algebra are applications of math to business and economics, but they are by no means themselves areas of math, and working with computers will at best build some intuition for mathematical thinking in the form of coding, but computer science is not in and of itself math. (To be absolutely clear, theoretical computer science, which is the math behind the mechanisms of computation, is certainly an area of math, but this is at the very least college level, and between this and their focus on the real world, it is certainly not what is being referred to here.) Lastly, the example of a class activity given is especially concerning. Going on about how "everyone has their own strengths" and we are part of a "mathematical community" could possibly be somewhat inspiring, but is better suited for a team-building exercise, and not math class. This activity is a waste of time that will not do anything to improve mathematical intuition. That "mathematical community" won't help you if they're too busy drawing colorful lines to do actual math.
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Education
Many things are "debatable." Certainly when it comes to art many things are up to interpretation. Even in science, there may be reasonable controversy. A certain set of available data may seem to suggest multiple different underlying facts. Math, however, is unique. By those who know it well, it is sometimes referred to as a sort of logical game. It has certain rules, and when applied in the right sequence, those rules can lead to much more complex things. Nothing is left to interpretation. Everything is known for certain. Even so, controversy manages to sneak in, especially among those who are less experienced. When people run into certain counterintuitive concepts, they often assume that mathematicians must surely be mistaken. In this post, I will cover a few of these areas of controversy, and then discuss how they come about, and what is actually true. I will try to keep this at a level that everyone should be able to understand, but for those who want the more technical details, I'll be covering them in a second post.
0.99999... = 1:
This is undoubtedly the most notorious area of controversy in math. There are numerous debates on this on DART alone. Many people see this counterintuitive fact for the first time and assume it must be wrong. In reality, due to a flaw in our decimal system, not every number has a unique representation. Nothing magical is happening, these are just two different ways to represent the same number, not unlike 1 + 1 and 2. One could reasonably ask what defines a decimal representation of a number, and I will cover that in my second post. For the less mathematically inclined, if you are willing to accept that 0.99999... is a well-defined quantity, and that 10(0.99999...) = 9.99999..., the following argument should be enough to convince you that 0.99999... = 1 is a logical necessity:
x = 0.99999...
10x = 9.99999... = 9 + 0.99999... = 9 + x
9x = 9
x = 1
1 + 2 + 3 + 4 + ... = -1/12:
This sum clearly diverges to infinity. What is really meant by this statement is that if we extend the notion of summation to divergent series like this one, we will get -1/12. (I'll discuss how this "extended notion" works in my second post.) Unfortunately, many people take the clickbait thumbnails a little two literally, and start arguing against this, even though there is nothing to argue about.
Infinity:
A significant number of people are now saying that infinity must be purged from mathematics, because it is not truly a valid concept. Here are the main three points of contention that I have seen:
1. Infinity does not exist in the real world.
Infinity is a tool for math which is useful for the real world. No one is claiming that infinity "exists" in a physical sense. This would be like pointing out that in the real world, there are no perfect geometric figures, so we should not use geometry.
2. Infinity is not a well-defined concept.
I'm not really sure where people get the idea that infinity is not well-defined. Perhaps it is because there are multiple different definitions for depending on the context, but in every context that it is used, it is very well-defined.
3. Infinity leads to paradoxes.
Every paradox out there, even seemingly "sharp" paradoxes like the liar's paradox, are simply things which humans find it difficult to understand. Just because something is difficult for humans to understand does not make it invalid.
The Sleeping Beauty Paradox:
This simple paradox is widely debating even among (in fact, especially among) professionals. The paradox goes as follows: Sleeping Beauty is put under anesthesia on Sunday. A coin is then flipped. Regardless of its outcome, she is awoken on Monday and asked to guess the outcome of the coin toss. Then, if the coin landed tails, she is put back to sleep and looses all memory of being awoken on Monday. She is then awoken on Tuesday and asked to guess the outcome of the coin toss. You're Sleeping Beauty and you have just woken up. What is the probability that the coin landed heads?
The "halfer" position:
Clearly the answer is 1/2. A coin flip is always 50-50.
The "thirder" position:
Monday and heads, Monday and tails, and Tuesday and tails are all equally likely. If Sleeping Beauty is told it is Monday, the coin could have landed heads or tails with equal probability, and if she is told that the coin landed tails, it could be Monday or Tuesday with equal probability. Because of this, the answer is 1/3. This comes down to the fact that she is woken up twice when the coin lands tails, and only once when it lands heads. Given information can affect a probability, and in this case we are given that Sleeping Beauty has just woken up.
In reality, the controversy here is not because of the math, but because of ambiguous wording. What is meant by "You're Sleeping Beauty and you have just woken up?" Is it irrelevant, (as the "halfer" sees it) or is it intended to convey the given information "Sleeping Beauty just woke up?" (as the "thirder" sees it)
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Miscellaneous
I'll start things off with this simple little example:
1 is defined as s0. (sn means "the successor of n")
2 is defined as s1.
Addition is defined as follows:
a + 0 = a for any a by definition.
a + b for b > 0 is defined recursively by a + sb = s(a + b).
With definitions out of the way:
1 + 1 = 1 + s0 = s(1 + 0) = s1 = 2
Do you accept this proof as providing 100% certainty that 1 + 1 = 2? Why or why not?
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Miscellaneous
The title of this post is a question often posed by flat earthers to challenge the idea of a round earth. Because of its association with flat earthers, this question is often laughed at. I find this to be a shame, because its a quite interesting question. It also gives me the chance to do math and show off how smart I am!
I am going to start by answering a similar but related question. The earth is orbiting the sun at 66,660 mph [1]. Why doesn't the sheer speed of that orbit leave us all behind in the vacuum of space? The answer is inertia. Inertia is described in Newton's First Law of Motion: "An object at rest tends to stay at rest and an object in motion tends to stay in motion, unless acted upon by an outside force." How does that answer the question? Well, think about an airplane. Commercial airliners fly at about 600 mph. Do the passengers spend the entire flight pressed back into their seats at 600 mph? If one of them stepped out into the aisle, would he be propelled to the back of the airplane at 600 mph? No. Why? Because while the plane is moving at 600 mph, so are the passengers, the air in the cabin, and everything else in the plane. No additional force is required to keep them moving at that speed because of inertia. They will continue moving at 600 mph until acted upon by an outside force, i.e., the plane slowing down. In fact, if the plane were to vanish, they would continue moving at 600 mph until coming to an unpleasant stop due to the outside force of the ground. Military bombers have to account for this when they drop bombs. The bombs will keep moving at the speed of the airplane, so they have to calculate how far ahead of the target they need to be for their bombs to hit the target.
However, returning to the airliner, the passengers will be pressed back into their seats when the plane initially accelerates to 600 mph. This is because of Newton's Second Law of Motion, which states that Force = Mass * Acceleration. This tells us that force is only present when there is acceleration, and vice versa. So while the plane is accelerating, force is exerted on the passengers. While the plane is at a constant speed, no force is exerted on the passengers, and they are free to move about the cabin and eat disgusting airline food. Importantly, these effects are the same no matter what the speed is. A train traveling at 40 mph or a car traveling at 70 mph will have the same effects. This continues to be true no matter how large the number is. Inertia holds true whether you are traveling at one millimeter per millenia or 100,000 miles per second.
Returning to the question (finally!), we don't fly off the earth even though it is orbiting at 66,660 mph because it is traveling at a constant speed. If it were accelerating in some direction, we would feel a force. However, it is traveling at a constant speed, so we don't notice it.
Now, five paragraphs in, to the main topic. (Internet knights just love the sound of their own voice, don't they?) If the earth is spinning, that introduces a new consideration: rotation. Due to inertia, an object traveling in a straight line will continue traveling in a straight line, not a circle. So if the earth is spinning at 1000 mph, it should have left us behind long ago, right? Wrong. The big number of 1000 mph (1037.5646 if you want to be pedantic) looks impressive and scary. But when you do the math, it's anything but. Yes, the earth is spinning, but only once a day. That's 360 degrees per day, or 15 degrees per hour, or 0.25 degrees per minute, or 0.00417 degrees per second. So if we take your current velocity to be moving at an angle of 0 degrees, you are currently moving at 1037 mph at 0 degrees. Next second, you will be moving at 1037 mph at 0.00417 degrees, then at 0.00834 degrees, then at 0.01251 degrees, and so on. Notice what's happening? You're traveling at a constant speed, and your direction is barely changing. Going back to the plane example, if the pilot turned the plane 0.25 degrees every minute, would you notice? Not in the least.*
*Technically, you could if you put a level on the floor. Since planes bank in order to turn, you would notice the floor was slightly off level. However, my point is that you wouldn't feel your direction changing.
Intuitively, it's already clear why we don't fly off the earth. It's spinning so slowly that we can't even feel our direction changing, although we can measure it. But, for completeness sake, let's calculate how much acceleration we're all undergoing due to the earth's rotation. Let's also do the calculation in metric, because it's easier. The equation for centripetal force if acceleration = velocity squared / radius [2]. The earth has a radius of 6,378,000 meters, and it is rotating at 1,669.8 km/h or 463.83 meters / second. This gives us a centripetal acceleration of 0.0337 meters per second squared. That giant 1000 mph figure turned into a fraction of a fraction.
For the fun of it, let's find out how much force it would take to hold your average overweight American to the earth if gravity didn't exist. Suppose the overweight American weighs 100 kg. We can use Newton's Second Law of F=ma to find the answer of 3.37 Newtons, or 0.758 pounds. A quarter inch polyester ropecan hold 400 pounds, which is over 400 times stronger than needed [3]. So our conclusion is this: if gravity didn't exist, you could counter the acceleration of earth's rotation indefinitely using nothing more than string.
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Science and Nature
I come to this topic as a mathematician, not as a philosopher or a theologian. From my perspective, mathematical "sets" — specifically infinite sets — can provide a small insight into the nature of God.
By Robbin O'Leary, Professor of Mathematics
I won't get into the details of the mathematics of Infinite Set Theory now. (Check out https://www.quantamagazine.org/mathematicians-measure-infinities-find-theyre-equal-20170912/
if you want to know the math)
Stated simply, a set is a collection of objects. These objects might be words, numbers or anything we group into that set.
The things we group into sets are called "elements". We will notice right away that some sets are finite, and others are infinite. That is, the elements within some sets have no end, like the elements in the set of all rational numbers, for example.
Theoretical scientists can then treat sets as single objects and do higher form of math to learn things about our reality that couldn't be learned otherwise.
There are some startling findings about Infinite set theory that shed light on some of the concepts found in the bible.
Startling finding #1 - Subtracting elements from an infinite set does not make it smaller.
If you have the set of all numbers, and remove from it, the subset of all even numbers, BOTH the original set AND the subset remain infinite!
Infinity can be taken out of infinity with no reduction in infinity! The same goes for addition. Adding the infinite set of all even numbers to the set of all odd numbers give us a set that is EQUAL in size to the sets added!
Startling finding #2 - Adding infinite sets to infinite sets does not make them any bigger! Infinite sets cannot be reduced OR increased!
Startling finding #3 - Infinite set are all identical. Each element in one infinite set can be paired with elements in another infinite set such that all infinite sets have a 1 to 1 correlation. They are all the same size!
What do these 3 findings mean to Christianity?
Jesus claimed to be God. He said to Thomas, "How can you ask me to show you the Father if you have seen me?" Jesus claimed to have "come out from God". He was calling Himself a proper subset of God.
He shared a 1 to 1 correlation with God, which did not reduce God, or render Jesus Himself less than God. This truth also shows how Jesus could not know some things, and still be God.
God is the absolute infinite set, with The Father, the son, and the Holy Spirit being proper subsets of the set "God".
All three sets are equal. The same size. Each one is equal to God, exactly in a 1 to 1 correlation with God. God does not diminish by removing one subset, or increase when one subset joins the Godhead.
Infinite subsets of infinite sets have a quality called "reflection", which means they have the same qualities of the original set. So as a subset of God, Jesus carries all the qualities of God, and indeed the bible calls Jesus the express image of God, He is the incarnation of God.
Set theory offers an explanation for the inner workings of the trinity.
Mathematicians Measure Infinities and Find They’re Equal
By KEVIN HARTNETT
September 12, 2017
Two mathematicians have proved that two different infinities are equal in size, settling a long-standing question. Their proof rests on a surprising link between the sizes of infinities and the complexity of mathematical theories.
In a breakthrough that disproves decades of conventional wisdom, two mathematicians have shown that two different variants of infinity are actually the same size. The advance touches on one of the most famous and intractable problems in mathematics: whether there exist infinities between the infinite size of the natural numbers and the larger infinite size of the real numbers.
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Religion
1. Can you write a 6 digit number where...
2. Digits cannot all be in ascending order ex. (134579)
3. Digits cannot all be in descending order ex. (964321)
4. No digit can be zero.
5. No digit can be a multiple of a digit beside it. ex. (374852) 4 and 8 are side by side, and 4 is a multiple of 8.
6. No digit can be used more than once. ex. (375594) 5 is used twice.
7. No digit can be 1 more or 1 less than the digit before or after it. ex. (253894) 9 is one more than 8.
8. No digits beside each other can share a multiple. ex. (386947) 8 and 6 are beside each other, and 2 is a multiple of both. 6 and 9 are beside each other, and 3 is a multiple of both.
9. The six digit number cannot have more than 3 consecutive odd numbers.
ex. (495372) 953 and 7 are all consecutive and all odd numbers.
10. Finally, neither the first or last 3 digits of your 6 digit number should have a sum of more than 15
(352947) 9+4+7=20
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Can you do it?
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Gaming