syzarian
Joined 4 years ago
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Comment · 3 years ago
Comment · 3 years ago
Mathematics is a vast subject and I can’t keep track of all developments. In 2010 there was a paper on meadows. I’ve never heard the term before. In that paper it is written:
As usual in field theory, the convention to consider p / q as an abbreviation for p · q−1 was used in subsequent work on meadows (see e.g. [2,5]). This convention is no longer satisfactory if partial variants of meadows are con- sidered too, as is demonstrated in [3].
So, as I’ve stated many times, I talked about convention and indicated you can use whatever terms you want. In the paper quoted above they acknowledge what the convention is. That is that division is multiplication by the inverse. They are arguing that it is worthwhile in this new algebraic object to change the usual notion a bit. If people agree to a new usage of the word division then definitions will change accordingly. None of this is pertinent to the spirit of the original question given the context under which it was asked. All of this is highly technical.
Definitions and notions change as new mathematics is created (discovered?). This happens all the time. All you have to do is convince other mathematicians to go along with it.
https://arxiv.org/pdf/0909.2088.pdf
EDIT: Regarding what you wrote in your other comment: The analogy is not apt in my opinion. It’s hard to say zero can’t exist because the nonzero…. The moment you say nonzero means it does exist. I think a better way to look at the situation is:
I have an object that is a group under a binary operation f. There is another natural binary operation on that object that operates with f in a consistent way. That operation doesn’t form a group but if I add a symbol to my set and give these rules then both operations interact in a consistent, natural way. I get a group under the new symbol with the second operation while preserving the group under the first operation minus the new symbol.
With extended complex numbers you don’t quite preserve the structures or properties that one normally wants so I’d say it isn’t true division. It is division like.
Comment · 3 years ago
Are there any policies that Democrats could enact that would make them lose support? The answer is clearly yes. Is taxing vehicles enough to pay the true cost of road maintenance such an issue? I don’t know but it may be one that Democrats don’t want to tangle with. Such taxes are done at the state and national level and I think this is an issue they don’t want to give Republicans a win on.
Edit: The nation as a whole has shifted very much rightward the last 50 years. Instead of saying Republicans I should maybe say “right leaning politics”.
Comment · 3 years ago
It’s an issue that Republicans can attack Democrats on. Change is difficult for many people. Political calculations have been made to:
1. Not give an opponent an easy way to attack you that requires the masses to employ nuanced reasoning to understand why that attack is wrong.
2. Say things to your base to keep them voting for you even though you have no intention of implementing those policies.
Comment · 3 years ago
A 2000 pound vehicle driven an average of 15000 miles per year does $x of damage to the roads. X is a calculable figure. Tax the vehicle to at least pay $x. If it is in the pennies then tax that amount. That a heavier vehicle does more damage just means calculating the appropriate amount to tax it.
Comment · 3 years ago
The one point compactification of the complex plane is not a number system in the normal sense of what that means. Calling the use of the notational convenience 1/infinity a true division operation defies the common usage of the term in mathematics. You may call it whatever you want to though.
The answer given to the person who asked the original question was the correct one. You can’t do it because doing so would break consistency and that is of paramount importance when doing new things in mathematics. There are agreed upon usages of terms and symbols in mathematics. Why call something division in the true sense of the word when it breaks the conventional usage of what that term means? But, also, why invent a new symbol to denote what is analogous to division? So we abuse notation. This is done all the time. So on the one had we’ll say to calculus 1 students 1/infinity is 0 but also say infinity is not a number. Things are done for convenience but when asked, “Is this really division?” the answer is no.
Of course you can redefine all terms you desire and say things like: A circle can be squared, I just mean something different when I say circle than when you say it. But why do that? All of this is my opinion. You disagree and that is ok.
Comment · 3 years ago
I don’t think anywhere in the U.S. taxes for registering vehicles and gas are enough to pay for roads. These charges do not pay the “true” costs of vehicles’ damage to the roads.
Comment · 3 years ago
You are correct that the desire to subsidize commercial activity plays a role. Republicans may not have power in a state like CT but there is still a desire not to be beat up on an issue that will play well to the masses. For instance, Michelle Obama advocated for better nutrition for school lunch programs. No reasonable person could possibly have seen fault with this but after Fox News attacked her over this it became a dead issue.
EDIT: It is definitely not a dumb idea as you suggest. Many counties have sensible policies on this issue and yet they manage to trudge along without much trouble. As I said it is OK to advocate for pay-as-you-use government so long as people like you don’t pay for your negative externalities too often.
Comment · 3 years ago
In the U.S. the reason is the Republican party. If we pay an amount to register a vehicle based on weight due to the increased damage it does to the roads then it will be portrayed as a radical environmentalist plot to get rid of SUVs and trucks. Republicans as love the idea of pay-as-you-use government services except when it comes to infrastructure. For example, it’s OK for a $25 traffic fine to include $100 in court costs but don’t under any circumstance expect one to pay more for gasoline to pay for road maintenance.
Comment · 3 years ago
A child asked her mother, “Can I put my hand in the fire?”. The mother responded, “That’s a stupid question. Of course you can.”. The child put her hand in the fire and got severe burns on her hand. She learned then that instead asking “Can I…” she should have asked, “Is it advisable…”. Unfortunately for her she lived in a society in which people frequently say things like, “You have to file taxes on or before April 15.” when they mean, “You can file taxes after April 15 but you may incur fees and penalties if you do so.”. She later became a teacher and was very patient with her students.
Comment · 3 years ago
This is getting very far from where the original question came from. When talking to a layman one would say division is always multiplication by the inverse. There are nuances involved that a lay person simply can’t appreciate or understand. Had I known you knew about the extended complex numbers I would have answered differently. The extended complex numbers are not a ring, not a group, not an algebra, and so…is it really division then?
In math often times the answer we give depends on the knowledge of the person asking the question. For instance we tell calculus 1 students 1/x is not continuous as a function from R-{0} to R. Of course in the standard induced topology it is a continuous function but explaining this to calculus 1 students would be very difficult.
Comment · 3 years ago
I think I understand better where you are coming from. In computer science I don’t know what they typically mean when they say “division”. I’ll be more precise. In abstract algebra division means multiplying by the inverse. All of the notions of division mentioned in the Wikipedia page come from this idea. Computers can’t work with within the realm of the entire real number system. There they have notions of type. They like to extend common operators like “/“ to things that normally it doesn’t apply to. A computer language will sometimes return a value of int or some other type when the integer 5 is divided by 3. Depending on how the language designer wanted things to work. This isn’t division in a mathematical sense though.
Comment · 3 years ago
I'm having trouble following…
I know. Please don’t become a teacher.
Comment · 3 years ago
The word division means something in mathematics. There is general agreement in what that word ought to mean. You can define a binary operation in such a way that it doesn’t look like what we normally think of as division and label your operation division. In the same way you can define the symbol duck to refer to what most people call a chair. You won’t get anyone else agreeing with your new definition though.
Comment · 3 years ago
How does your pedantry contribute meaningfully?
If something doesn’t behave like 0 in a ring or other algebraic structure then using that label is confusing and simply not done. You are free to use any symbol you want but mathematics is a human endeavor and as such communication is important. Using the symbol 0 signifies something to those with mathematical training. Zero can’t have an multiplicative inverse because anything you call 0 that has an multiplicative inverse makes it behave like something other than zero. So no one would use 0 to describe such an element. In a ring, or abelian group, the symbol 0 is reserved for the additive identity element.
Similarly, I could say snkwoo is what most people call a chair. A grammarian would say there is no word snkwoo even though I just defined it.
Your original comment was wrong and bad. Instead of just admitting it or moving on you’ve decided to double down and make another bad comment.
[1] https://www.sfchronicle.com/crime/article/Is-the-San-Francis...
[2] https://www.finder.com/uk/police-statistics