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Re: 1/0 = 0

#561

Post by mlthoughts2018 »

drb91 wrote: post #560
mlthoughts2018 wrote: post #499

Earlier quoted context omitted.

Why is it conceptually wrong to say the average of an empty array is zero? My undergrad degree is in pure math and my grad degree is in mathematical statistics and I’ve never heard an idea like saying the mean of an empty array is zero is “conceptually wrong.” You bring up the history of CS, but even there you have debates about what convention to use for defining 1/0 for function totality and theorem provers. There’…

> Why is it conceptually wrong to say the average of an empty array is zero? It’s not conceptually wrong, it just means the “mean” you’re referring to calculates a different value than the “mean” we’re taught in school. So, underlying assumptions about the differences in “mean” should be communicated where it’s used.

Sure, I agree they should be communicated. Like, in the docs for “standard” mean functions, and not pushed into “specialized” mean functions, since needing this particular convention is not remotely special, and is rudimentary and expected in 99% of linear algebra and data analytics work, which are the largest drivers of these types of statistical functions.

Re: 1/0 = 0

#562

Post by milesvp »

ASalazarMX wrote: post #46
pmiller2 wrote: post #19

Earlier quoted context omitted.

It is inconsistent. If 1/0 = 0, then 1 = 0*0.

Wait, if 1 / 0 = infinity, then infinity * 0 = 1 This seems just as bizarre, since zero times anything shouldn't become 1, no matter how big or how many times you do it.

Look up the dirac delta function. It's a spike thats infinitely tall and infinitely narrow, with an area of 1. It's established now as a very useful tool in EE. But many people fought it tooth and nail because of this logic.

Re: 1/0 = 0

#563

Post by tobiasSoftware »

I'm writing a software language that has 1/0 = 0 as well. In my case, I am experimenting with several ideas, two of which are no runtime errors and no exceptions, so everything is either a logic error (aided by multiple return values) or a compiler error. It works fine in normal scenarios, but dividing by zero can't be caught at compile time, and operators can only return a single value, so I had to return something.

Re: 1/0 = 0

#564

Post by jibal »

semi-extrinsic wrote: post #519
jibal wrote: post #463

Earlier quoted context omitted.

"In this case, it's "obvious" that when e.g. a is zero, you want 1/a to be zero so you simply return b" b obviously isn't what you want from this harmonic mean when a is 0. And in any case x will be 2b if 1/a is 0, not b.

So, I did forget the correction such that you don't get 2b. The general case is described e.g. in the link below, and this is known as a "zero-corrected harmonic mean". It is actually "obvious" in some cases, but I'll concede that in other cases it's not. https://www.rdocumentation.org/packages/lmomco/versions/2.3....

Eh? That page says nothing about 1/0 being 0, or anything like that. That function takes the harmonic mean of the non-zero samples, and then adjusts the result by multiplying it by the fraction of the samples that are non-zero. It purposefully avoids dividing by 0. And the result of that function for a==0 and b!=0 is b/2, not b or 2b. So not only is what you claim to be obvious not obvious, it's flat-out wrong.

Re: 1/0 = 0

#565

Post by blub »

WalterBright wrote: post #375
whack wrote: post #72

My problem with "1/0 = 0" is that it's essentially masking what's almost always a bug in your program. If you have a program that's performing divide-by-zeroes, that's almost surely something you did not intend for. It's a corner case that you failed to anticipate and plan for. And because you didn't plan for it, whatever result you get for 1/0 is almost surely a result that you wouldn't want to have returned to the…

An issue with D that has repeatedly engendered heated debate is what should happen when a programming bug is detected at runtime. The two camps are: 1. The program should "soldier on" if it can. 2. The program should go immediately to jail, it must not pass Go, and must not collect $200. I'm solidly in the latter camp. If a program has entered a state unanticipated by the programmer, then there is no way to know how…

If the program is designed to "soldier on", then why not? The question is how is the program designed to react in face of failure: should it fail safe? And is it safe to crash on any error?

My point of view is that a program should abort if it encounters an irrecoverable error, such as imminent memory corruption. However, if it's designed to fail functional, then it could continue to work, either by reinitialising itself, or by aborting the current operation. However, it must be said that it's hard to design such programs, so the option of bailing out is very attractive.

Example: an out of bounds access is a typical programmer error. This should probably result in an abort during development, so that it can be fixed. However, it is possible to not want to abort just because something is not accessible when in production. The program could instead raise an exception and clean up the call stack up to its initialisation point, then notify the surrounding system that a problem was encountered and resume service. The surrounding system could then keep track of the encountered issues and try to perform a recovery after a threshold is reached: this is how Android recovery works for e.g. crashing system services, if I recall correctly, it first cleans some application settings, then system settings, then does an OS reinstall. Now in this example the trigger condition is a crash, but it doesn't necessarily have to be like that.

Re: 1/0 = 0

#566

Post by Veedrac »

pps43 wrote: post #505
Veedrac wrote: post #501

Earlier quoted context omitted.

It's a value type, so you can't really have NULL. I don't see the issue with those examples; if you have no samples, there's nothing to mislabel. An average blood pressure of 0 across 0 patients is only going to harm 0 patients.

> I don't see the issue with those examples Let me elaborate then. Average credit card balance predicts probability of default. Joe who maxed out his credit card is higher risk than Jane who pays back her entire credit card balance every month. Now we have Jack with no credit card. We predict that Jack is low risk because his average credit card balance is zero. Second example, defibrillator that monitors blood press…

I feel for the typical target market for Pony is going to be doing operations like that in floating point, or perhaps some explicit fixed-point non-integer type in financial cases. Whilst, yes, I can see this causing issues in some small subset of cases, the question really does come down to cost-benefit. Integer division by zero is not the only way for arithmetic to go wrong; would you expect every overflow to be checked too?

Re: 1/0 = 0

#567

Post by blablablerg »

pron wrote: post #536
blablablerg wrote: post #528

Earlier quoted context omitted.

In the definition of a field ∀ x . x/0 = undefined, not 0 or any other value you might prefer. A field is exactly defined by the field axioms: adding or removing any other axiom makes it no longer a field.

> A field is exactly defined by the field axioms: adding or removing any other axiom makes it no longer a field. This is clearly not true. Adding axioms (as long as they don't introduce consistency) can only reduce the set of models. Anything that satisfies the larger theory also satisfies the smaller theory. This is exactly why a group is a monoid is a semigroup. This is why fields are rings. In fact, this is why th…

>This is clearly not true. Adding axioms (as long as they >don't introduce consistency) can only reduce the set of >models. Anything that satisfies the larger theory also satisfies the smaller theory. This is exactly why a group is a monoid is a semigroup. This is why fields are rings. In fact, this is why the rationals or the reals are fields: they satisfy the field axioms, and more (for example, they are ordered).

Look at for example the set of real numbers. If you add the complex numbers to it, you can no longer speak of the real numbers. The real numbers form a subset of the complex numbers. The same thing happens if you add definitions to a field. You create something which has mayber has a field as a subset, but you can no longer speak of a field.

What is "undefined"? First let's look at the logic itself: What is true ∨ undefined ? What is true ∧ undefined? Now at the theory: What is undefined + x ? What is 0 * undefined ? What is 0 = undefined ? etc.[0]

Undefined is just undefined, no value, not a magic one, or just a random one, or one you prefer. Just void, the empty set. To understand it, look for example at the graph of y = sqrt(x), where x and y are real numbers. You see no points at negative x. Its undefined in the domain x >What is "undefined"? First let's look at the logic itself: >What is true ∨ undefined ? What is true ∧ undefined? Now at the theory: What is undefined + x ? What is 0 * undefined ? What is 0 = undefined ? etc.[0]

Now you apply boolean logic to the result of calculations, numbers and non-numbers, not on statements. I can also ask what is 5 AND true? Is that true? The question doesn't make sense, so the answer is: depends on the programming language.

Re: 1/0 = 0

#568

Post by pron »

blablablerg wrote: post #567
pron wrote: post #536

Earlier quoted context omitted.

> A field is exactly defined by the field axioms: adding or removing any other axiom makes it no longer a field. This is clearly not true. Adding axioms (as long as they don't introduce consistency) can only reduce the set of models. Anything that satisfies the larger theory also satisfies the smaller theory. This is exactly why a group is a monoid is a semigroup. This is why fields are rings. In fact, this is why th…

>This is clearly not true. Adding axioms (as long as they >don't introduce consistency) can only reduce the set of >models. Anything that satisfies the larger theory also satisfies the smaller theory. This is exactly why a group is a monoid is a semigroup. This is why fields are rings. In fact, this is why the rationals or the reals are fields: they satisfy the field axioms, and more (for example, they are ordered).…

> Look at for example the set of real numbers. If you add the complex numbers to it, you can no longer speak of the real numbers.

What does this have to do with adding axioms? You don't get the complex numbers from the real numbers by adding axioms. For example, the following is provable for the reals, but not for complex numbers:

    ∀x,y . x  y ∨ x = y
> You create something which has mayber has a field as a subset, but you can no longer speak of a field.

No. The class of all fields are all objects satisfying the field axioms. If you add axioms, you get get a subset of those fields. Each and every one of them is a field.

> Undefined is just undefined

I'm not looking for handwaving. Formal mathematics is math that can be done mechanically. Write down the axioms for undefined. Again -- I'm not saying it hasn't been done, but it's certainly not part of standard first-order logic, and it is unnecessary.

> To understand it, look for example at the graph

I understand what undefined means in informal mathematics. Try to see what it means in formal mathematics. But the important point is, try to understand why it is unnecessary in formal mathematics in most cases.

> Now you apply boolean logic to the result of calculations, numbers and non-numbers, not on statements.

No. You can write 1/x 20.

> The question doesn't make sense, so the answer is: depends on the programming language.

There is no such thing as "doesn't make sense" in formal math, even though there is such a thing in informal math. That's the whole point. Either the expression is ill-formed, i.e. not in the language or a "syntax error", or it must make some sense.

This is why it's useful and easy to say something is undefined in informal math, but not as useful and not as easy to do that in formal math.

Re: 1/0 = 0

#569

Post by blablablerg »

pron wrote: post #568
blablablerg wrote: post #567

Earlier quoted context omitted.

>This is clearly not true. Adding axioms (as long as they >don't introduce consistency) can only reduce the set of >models. Anything that satisfies the larger theory also satisfies the smaller theory. This is exactly why a group is a monoid is a semigroup. This is why fields are rings. In fact, this is why the rationals or the reals are fields: they satisfy the field axioms, and more (for example, they are ordered).…

> Look at for example the set of real numbers. If you add the complex numbers to it, you can no longer speak of the real numbers. What does this have to do with adding axioms? You don't get the complex numbers from the real numbers by adding axioms. For example, the following is provable for the reals, but not for complex numbers: ∀x,y . x y ∨ x = y > You create something which has mayber has a field as a subset, but…

>What does this have to do with adding axioms? You don't get the complex numbers from the real numbers by adding axioms. For example, the following is provable for the reals, but not for complex numbers:

> ∀x,y . x y ∨ x = y

Since R is a subset of C, you can write C as R with additional axioms. See for example:

http://www.math.mcgill.ca/gantumur/math249w15/numbers.pdf

>I'm not looking for handwaving. Formal mathematics is math that can be done mechanically. Write down the axioms for undefined. Again -- I'm not saying it hasn't been done, but it's certainly not part of standard first-order logic, and it is unnecessary.

Undefined is just undefined, that is no handwaving, it is a primitive notion. See https://en.wikipedia.org/wiki/Primitive_notion

You are trying to define the undefined in a formal system. Undefined is just the absence of a definition. Not 0, 3 , 2pi

> No. You can write 1/x 20. Yes, those are valid mathematical statements. 5 ∨ TRUE, or undefined V TRUE, I doubt it.

> There is no such thing as "doesn't make sense" in formal math, even though there is such a thing in informal math. >That's the whole point. Either the expression is ill-formed, i.e. not in the language or a "syntax error", or it must make some sense.

Well in language, in which I am corresponding with you, there is such a thing as 'makes no sense'.

You are somehow trying to capture everything in your logical system only to try to prove that 1/0 = 0 is part of a field, which it isn't. I've made my point here, it was nice talking to you.

Re: 1/0 = 0

#570

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