I understand they are MIPS embedded-type deals with specially designed firmware (TCP fingerprinting indicates that at least the network stack isn't derived from any public RT OS sources... so I'm guessing it's an HP original)
I assure you that the energy expended in getting fission products into the sun exceeds the extracted energy from said spent fission products by a large margin.
So no, sending shit into the Sun is NEVER A SOLUTION. Thanks.
It's more likely that Matt was having a little online fling with a few "MySpace hos", kids from school, xanga, amihotornot... whatever. It is quite prevalent. I think Mom, Dad, and the androgynous horde that is Slashdot needs to wake up a bit.
I remember doing it like 8 years ago, when webcameras were new and expensive. Getting that digital cameras for christmas a GODSEND because you didn't have to chance developing the film after documenting your escapades.
Poor kid. Needed a BIT more computer savvy. And never burn the stuff to CDs without obfuscating or mixing in with more mundane stuff (multi-session audio CDs with no autorun in the data section were good choices... and they doubled as mood music mix CDs)
That WMF flaw is older than the commercial internet. It was an artifact of supporting OLE in WMF and how thread control (hah) in Windows 3.1 worked... kept backwards compatible to this day. It was a shitty design from the getgo, malice or "terrarist fightin' tool" have nothing to do with it. Also, Steve Gibson is a tool. Seriously, get your security news from ANYWHERE else.
I think maybe the thing that people forget is that there is a difference between science (as in, the study of whatever), and SCIENCE (guys with clipboards in labcoats and $100,000+ equipment).
You don't need to consult Bill Nye to know how to make informed, every day decisions.
But if you're going to listen to someone's opinion on immigration reform and how it affects your decision on whether to send your kids to private school or not, don't take Joe Blow on the radio's insight to heart, look to the opinion of a sociologist on the issue.
There are many people studying different areas of our increasingly complicated and specialized lives, and they will provide valuable knowledge the helps humanity. You just have to look for it... and beware the appeal to authority (i.e. celebrities: the point of the article).
My ol' mother went to Catholic school for 9 years and she knew more about sex and birth control and all that than any of her other peers once she got to public high school.
I've been trying to put something like this together for ages. It has been a perverse dream of mine to do system management from javascript for the longest time. This totally rocks. And with dynamic binding? GLEE
I problem I never figured out (and which always kept me from getting anywhere) was how to reconcile threading/forking with the spidermonkey engine in a way that could be manipulated cleanly from the scripts themselves.
function MyFunc() {
var i=0;
{
let i=i+1;
alert(i);
}
alert(i); }
You would see "1", then "0"
lexical scoping in ecmascript is very simple, to support the construction of closures, as it is primarily an event handling language. The let keyword would provide a built-in way to introduce a blocking scope (it is possible now, but it involves defining an anonymous object and using a method scope, IIRC)
That being said, the cases where you need such a construction are fewer than you would imagine, as it can make code intent unclear.
HTML, from it's inception, was designed to be layout indepedent. The size and placement of objects is determined from inside out, expanding to fit.
CSS follows this model, to keep things from horribly breaking when a browser decides to turn off some CSS feature or substitute a stylesheet for print preview or what have you.
Saying that some container should expand to an arbitrary size that hasn't been determined yet breaks the model. That becomes especially problematic when you nest a series of parent-relative-sized elements within each other. You'd have to use a two pass model... where you establish minimum sizes for elements, and then potentially increase the size of some of them working back outwards for vertical alignment (without introducing reflow; if you do then you have to go back and recalculate the widths of everything).
I think it's a fundamentally hard problem unless you introduce a layout language that is properly seperated from the content so it can be more clearly expressed. Sort of like "framesets", for positional layout, and then CSS+HTML for document-flow content within each piece.
And we "fixed" this with x86_64. The extended instruction set allows for more orthogonal expression of what you want to do with your ops w/r/t regs and memory (although not all of them are equivalent length, the more common ones are shorter, so what does it matter?)
So uh, this memory-mapped IO that I'm using instead of emulated PIO, and these programmable DMA controllers, and the cascading interrupt muliplexer, and this hybercube bus with cache coherency... that all is just a figment of my imagination.
Meanwhile my Sun has OH LOOK, a crossbar, and MY GOD! this newfangled PCI bus. WHAT HATH SCIENCE DONE?
He's got a pretty good grasp of theory from the harmonics and progression of the piano portion, and he assembled a dynamite drum arrangement.
I think it's likely he can actually play both instruments; it's hard to imagine an original piece for keyboard unless you've a passing acquaintance with playing it; if not you might create something "impossible" to perform but still attempt it in this spliced-together video for the entertainment value. Not a good case for needing to know how to play the drums from listening to the arrangement... but he had a set to record from... he has to be able to play them a little, otherwise why have them at all?
Using QEMU with the closed-source kernel driver mode, or when using Xen in paravirt. w/32-bit Windows, you can make a PCI or PCI-express card available exclusively to a guest, making it appear in their PCI configuration space. Of course, the VGA for the system will still be emulated by the VM. It would be a good idea to do the initial install without the device available, and the make it available on a subsequent boot of the Windows image so it discovers it and prompts for drivers. IIRC, this does not work for AGP cards. So PCI add-in cards or PCI-e are the only options.
Attach a seperate monitor to this card, and the guest will be able to install drivers for it and generally use it as if it were running dedicated. Don't attempt to initialize or otherwise use that card from your host/privledged guest, because I can't imagine what the consequences of that would be.
I haven't tried this myself, but we're going to soon when we get some VT-capable hardware. Everything I've read about these capabilities suggests this is possible, and I can't wait to try it. I think the reason you don't hear much about it is because most people still have a single video card, or they are using PCI forwarding for cards that attach to SANs or network cards and stuff for speed.
Pi is definitely a well-defined real, and so is e!
Real are constructed by taking bracketed infinite sets of increasing and decreasing fractions (Q) that converge on the real.
So pi is defined as the the pair of: [largest in the set of fractions definitely less than pi, smallest in the set of fractions definitely greater than pi] Think of the inner and outer circumscribed polygons closing in on a circle. (The formulas with the delta/epsilon bits evade me, but they definitely exist)
And a classic example, the real number 1 is [0.99999999... (sum(9/10^n)), 1.0000000... (1+1/10^n)]
But infinity can't be bracketed. (What fraction is trivially larger than a fraction trivially smaller than "infinity", when you can't even use infinity in the definition for the fraction expansion?)
You have to prove some expansion, that by mathematical induction, as n increases, the different between the least upper and most lower bounds decreases monotonically. You'll find you can't create a suitable expression defined in terms of Q to place in the left and right sides. (There is a proof of this, but I'm not familiar with it).
Which leads to R not having infinity as a set member.
In terms of analysis (not in terms of computer science), we've been to this brink many times, pondering expressions near division by zero and such. It's why we have the Calculus and Complex Analysis in the first place (thank Leibeniz, Newton, Gauss, Cantor...).
Introducing these terms is like throwing away Calculus and saying: we don't limits and sequences and that notational nonsense, we need absolute, point-in-time answers for every formulation, damn the conseqences.
Of course, this real projective number line lacks the information content carried by similar techniques (Newton's infinitesmals, or other types of transfinite quantites). It's useless. It really is.
He was just pissed that fields are defined by a set (with no infinities or anything), and two operators (addition and multiplication), and a bunch of axioms. Damn it, he wanted complete closure and wanted all six common arithmetic operators to be onto because it "felt right". So instead we've got an object which is barely a commutative ring (with operators with tons of funky corner cases), and we haven't gained anything in terms of new theorems or strong relation statements from the extra axioms he has to tack on.
I have a feeling that calculus, and measure theory, and all that good stuff has a better handle on infinities than this theory... don't you think?
Thanks for clarifying my earlier statement... one-sided limit and limit with direction (in the complex plane) are congruent, while the two-sided limit is like the limit where the magintude of delta -> 0 in the complex plane.
And I had the differentiable at two-sided limit existing at backwards, durr.
He's trying to combine the real projective line and the extended real line... whhhyyy...
Let's not confuse parameterized expressions and limits. Infinity is not a quantity you can substitute into an expression.
If you define h(t) = f(t)/g(t), where t is a member of the reals, well guess what? h(t) at t = infinity is not defined because T IS IN THE REALS. Infinity is NOT A REAL.
Forgive the bold but this is the most important part that nobody in this thread seems to grasp.
Now, you are allowed to say limit(h(t)) where t->inf, but that doesn't mean the same thing as t = inf. It means "t grows without bound". Which is the same thing as defining a parameter u = (1/t) and saying u->0, which is something that you could intuitively understand. At there still, you do not define t=0, because then u=1/0, and division by zero is not allowed.
But you could study the resulting composite expression and show where terms cancel out if they did become trivially small, and you are left with the constant 3/2 which is what the expression would tend to. But again, you can not actually evaluate it at infinity, that isn't possible in the realm of standard analysis.
I admit, not being mathematician myself, I never before truly taught about the peculiar nature and deep significance of our ubiquitous zero. Because of its "triviality" it masks all levels of structure it may have.
Well, neither am I. But it doesn't seem so strange that the constant in between the negative and positive integers has no sign. The signedness is only significant in that it encodes the "direction" away from the origin (0).
Of course, the zero vector has no direction. If one tries to compute it (using trigonemtric functions), you will find it's value is undefined. As such, it is not in the unordered set of directions. Fancy that! (This is like asking what's the hue in the HSV model for black in RGB (0,0,0)) A nullity vector is meaningless, just as nullity is meaningless, so let's not go there.
In physics, a parameter takes on the units, not the value. The unit doesn't "cancel out" because it's being multiplied by zero. It's like a type on a variable in a computer program. Just because an integer is zero doesn't make that type of variable (void) or anything.
And I completely agree with your last paragraph. Except for the last part. That zero remains should not be troubling. In any model, you must restrict your input parameters so that the output parameters make sense. If you divide, if you take a square root, the special cases where the values can be out of range in that operation must be checked for and not allowed in their domains back out to the input parameters. This goes for dividing by zero or ANY OTHER CONSTRAINT. It is one of many. But certainly when addinging, or multiplying, or using many other operators, a zero is a useful and meaningful quantity.
You don't need provably correct code to avoid a divide by zero. Whenever you divide, it's not because you're manipulating data structures or hash tables. It's because you've been given a formula or heuristic where division was suggested or required. Only somebody forgot to make sure the divisor can't be zero (surely a sign the original EQUATION or METHOD is wrong, or that input in some other part of the system is being checked for a different valid range).
How many times have you written code where you divided by a non-constant factor determined at runtime? I mean, really? I doubt it's very frequent. All the more reason to be cautious about how you implement said code.
He has introduced an algebraic object that is not a ring or field, that is only different notationally compared to IEEE Inf/Nan semantics (aside from NaN != NaN).
What problem is trying to solve with these constructions? It certainly won't make arithmetic any more or less sound... dividing by zero is a problem of phrasing the question, rather than the semantics of the representation of the answer.
An example:
Suppose we have, through some process, yielded an expression f(t) = g(t)/h(t). The end result of f(t'), let's say, is 5. And let's say we have an additional piece of information, g(t) = 10. Then we can deduce h(t) = 2, and perhaps determine either g^-1 or h^-1 with additional information at data points (t_1, t_2, etc.)
Now let's say f(t') = nullity. So, h(t) is 0. Or maybe not, because g(t') might be nullity at t'. If we knew h(t') is neither (0, nullity) then we know g(t') is nullity. And if h(t') is 0 or nullity, then we can't determine what g(t') is... it could be anything. The data point where the answer is nullity gives us no usable information, other data points (t_1, t_2) must be considered.
The same would happen in the first example if h(t') = 0; that datapoint t' would not be possible, a t' not in the domain of f(t).
This example might seem silly, but my point is that the nullity doesn't really tell you very much, other than that you propogated forward a division by zero and it collapsed your quantity to the number out of the number line. It's a unexceptional exception. It's like NULL in an outer join or normalized representation... not a good sign. But in any other system, you would either have a contradiction, or in the case of software, an exception, which is just as valid in determining how to handle the situation, or to work around it.
Division by zero SHOULD be an exception: Either... 1) your model is wrong mathematically 2) you translated your model into code incorrectly 3) you have an off-by-one bug 4) your model is ill-conditioned and underflows 5) you're not handling precision correctly 6) you need arbitrary precision
It's one (or more) of those things when you get that DIVIDE BY ZERO exception and your program crashses. THINGS THAT SHOULD BE FIXED. Fix the code, don't wrap it in a class and hope it doesn't happen again!
In his case, he didn't introduce a "catch-all" symbol, he introduced classes of infinities (ordinal and cardinal) with constructions and everything... it was actually useful for stuff.
Dividing by zero is never defined. x/0 is NOT EQUAL to +/- Inf, no matter what your calculator or computer says.
The limit exists, sure enough, but the limit takes directionality in account in the expression, which is where the sign (or complex directional infinity) comes from.
Please don't misuse limit expressions and perpetuate falsehoods about our precious additive identity!
And zero isn't signed. By definition. Now, the construction [-0.00000000, +0.00000000], the bounds which define the real number zero might look like +/-, i.e. signed zeroes, but they two are really just limits with direction, and the value of the limit (i.e. 0) has no sign. At all.
Slashdot Mirror
I understand they are MIPS embedded-type deals with specially designed firmware (TCP fingerprinting indicates that at least the network stack isn't derived from any public RT OS sources... so I'm guessing it's an HP original)
I assure you that the energy expended in getting fission products into the sun exceeds the extracted energy from said spent fission products by a large margin.
So no, sending shit into the Sun is NEVER A SOLUTION.
Thanks.
It's more likely that Matt was having a little online fling with a few "MySpace hos", kids from school, xanga, amihotornot... whatever. It is quite prevalent. I think Mom, Dad, and the androgynous horde that is Slashdot needs to wake up a bit.
I remember doing it like 8 years ago, when webcameras were new and expensive. Getting that digital cameras for christmas a GODSEND because you didn't have to chance developing the film after documenting your escapades.
Poor kid. Needed a BIT more computer savvy. And never burn the stuff to CDs without obfuscating or mixing in with more mundane stuff (multi-session audio CDs with no autorun in the data section were good choices... and they doubled as mood music mix CDs)
Nice flamebait.
Of course, this FBI study was completed before the results of the congressional elections were finalized, so...
That WMF flaw is older than the commercial internet.
It was an artifact of supporting OLE in WMF and how thread control (hah) in Windows 3.1 worked... kept backwards compatible to this day.
It was a shitty design from the getgo, malice or "terrarist fightin' tool" have nothing to do with it. Also, Steve Gibson is a tool. Seriously, get your security news from ANYWHERE else.
I think maybe the thing that people forget is that there is a difference between science (as in, the study of whatever), and SCIENCE (guys with clipboards in labcoats and $100,000+ equipment).
You don't need to consult Bill Nye to know how to make informed, every day decisions.
But if you're going to listen to someone's opinion on immigration reform and how it affects your decision on whether to send your kids to private school or not, don't take Joe Blow on the radio's insight to heart, look to the opinion of a sociologist on the issue.
There are many people studying different areas of our increasingly complicated and specialized lives, and they will provide valuable knowledge the helps humanity. You just have to look for it... and beware the appeal to authority (i.e. celebrities: the point of the article).
My ol' mother went to Catholic school for 9 years and she knew more about sex and birth control and all that than any of her other peers once she got to public high school.
I've been trying to put something like this together for ages. It has been a perverse dream of mine to do system management from javascript for the longest time. This totally rocks. And with dynamic binding? GLEE
I problem I never figured out (and which always kept me from getting anywhere) was how to reconcile threading/forking with the spidermonkey engine in a way that could be manipulated cleanly from the scripts themselves.
Any thoughts on this?
function MyFunc()
{
var i=0;
{
let i=i+1;
alert(i);
}
alert(i);
}
You would see "1", then "0"
lexical scoping in ecmascript is very simple, to support the construction of closures, as it is primarily an event handling language. The let keyword would provide a built-in way to introduce a blocking scope (it is possible now, but it involves defining an anonymous object and using a method scope, IIRC)
That being said, the cases where you need such a construction are fewer than you would imagine, as it can make code intent unclear.
HTML, from it's inception, was designed to be layout indepedent. The size and placement of objects is determined from inside out, expanding to fit.
CSS follows this model, to keep things from horribly breaking when a browser decides to turn off some CSS feature or substitute a stylesheet for print preview or what have you.
Saying that some container should expand to an arbitrary size that hasn't been determined yet breaks the model. That becomes especially problematic when you nest a series of parent-relative-sized elements within each other. You'd have to use a two pass model... where you establish minimum sizes for elements, and then potentially increase the size of some of them working back outwards for vertical alignment (without introducing reflow; if you do then you have to go back and recalculate the widths of everything).
I think it's a fundamentally hard problem unless you introduce a layout language that is properly seperated from the content so it can be more clearly expressed. Sort of like "framesets", for positional layout, and then CSS+HTML for document-flow content within each piece.
No, I take it.
Then why do you care?
And we "fixed" this with x86_64. The extended instruction set allows for more orthogonal expression of what you want to do with your ops w/r/t regs and memory (although not all of them are equivalent length, the more common ones are shorter, so what does it matter?)
So uh, this memory-mapped IO that I'm using instead of emulated PIO, and these programmable DMA controllers, and the cascading interrupt muliplexer, and this hybercube bus with cache coherency... that all is just a figment of my imagination.
Meanwhile my Sun has OH LOOK, a crossbar, and MY GOD! this newfangled PCI bus. WHAT HATH SCIENCE DONE?
He's got a pretty good grasp of theory from the harmonics and progression of the piano portion, and he assembled a dynamite drum arrangement.
I think it's likely he can actually play both instruments; it's hard to imagine an original piece for keyboard unless you've a passing acquaintance with playing it; if not you might create something "impossible" to perform but still attempt it in this spliced-together video for the entertainment value.
Not a good case for needing to know how to play the drums from listening to the arrangement... but he had a set to record from... he has to be able to play them a little, otherwise why have them at all?
Using QEMU with the closed-source kernel driver mode, or when using Xen in paravirt. w/32-bit Windows, you can make a PCI or PCI-express card available exclusively to a guest, making it appear in their PCI configuration space. Of course, the VGA for the system will still be emulated by the VM. It would be a good idea to do the initial install without the device available, and the make it available on a subsequent boot of the Windows image so it discovers it and prompts for drivers.
IIRC, this does not work for AGP cards. So PCI add-in cards or PCI-e are the only options.
Attach a seperate monitor to this card, and the guest will be able to install drivers for it and generally use it as if it were running dedicated.
Don't attempt to initialize or otherwise use that card from your host/privledged guest, because I can't imagine what the consequences of that would be.
I haven't tried this myself, but we're going to soon when we get some VT-capable hardware. Everything I've read about these capabilities suggests this is possible, and I can't wait to try it. I think the reason you don't hear much about it is because most people still have a single video card, or they are using PCI forwarding for cards that attach to SANs or network cards and stuff for speed.
Pi is definitely a well-defined real, and so is e!
Real are constructed by taking bracketed infinite sets of increasing and decreasing fractions (Q) that converge on the real.
So pi is defined as the the pair of:
[largest in the set of fractions definitely less than pi, smallest in the set of fractions definitely greater than pi]
Think of the inner and outer circumscribed polygons closing in on a circle. (The formulas with the delta/epsilon bits evade me, but they definitely exist)
And a classic example, the real number 1 is [0.99999999... (sum(9/10^n)), 1.0000000... (1+1/10^n)]
But infinity can't be bracketed. (What fraction is trivially larger than a fraction trivially smaller than "infinity", when you can't even use infinity in the definition for the fraction expansion?)
You have to prove some expansion, that by mathematical induction, as n increases, the different between the least upper and most lower bounds decreases monotonically. You'll find you can't create a suitable expression defined in terms of Q to place in the left and right sides. (There is a proof of this, but I'm not familiar with it).
Which leads to R not having infinity as a set member.
In terms of analysis (not in terms of computer science), we've been to this brink many times, pondering expressions near division by zero and such. It's why we have the Calculus and Complex Analysis in the first place (thank Leibeniz, Newton, Gauss, Cantor...).
Introducing these terms is like throwing away Calculus and saying: we don't limits and sequences and that notational nonsense, we need absolute, point-in-time answers for every formulation, damn the conseqences.
Of course, this real projective number line lacks the information content carried by similar techniques (Newton's infinitesmals, or other types of transfinite quantites).
It's useless. It really is.
He was just pissed that fields are defined by a set (with no infinities or anything), and two operators (addition and multiplication), and a bunch of axioms. Damn it, he wanted complete closure and wanted all six common arithmetic operators to be onto because it "felt right". So instead we've got an object which is barely a commutative ring (with operators with tons of funky corner cases), and we haven't gained anything in terms of new theorems or strong relation statements from the extra axioms he has to tack on.
I have a feeling that calculus, and measure theory, and all that good stuff has a better handle on infinities than this theory... don't you think?
Thanks for clarifying my earlier statement... one-sided limit and limit with direction (in the complex plane) are congruent, while the two-sided limit is like the limit where the magintude of delta -> 0 in the complex plane.
And I had the differentiable at two-sided limit existing at backwards, durr.
He's trying to combine the real projective line and the extended real line... whhhyyy...
Let's not confuse parameterized expressions and limits.
Infinity is not a quantity you can substitute into an expression.
If you define h(t) = f(t)/g(t), where t is a member of the reals, well guess what?
h(t) at t = infinity is not defined because T IS IN THE REALS. Infinity is NOT A REAL.
Forgive the bold but this is the most important part that nobody in this thread seems to grasp.
Now, you are allowed to say limit(h(t)) where t->inf, but that doesn't mean the same thing as t = inf. It means "t grows without bound". Which is the same thing as defining a parameter u = (1/t) and saying u->0, which is something that you could intuitively understand. At there still, you do not define t=0, because then u=1/0, and division by zero is not allowed.
But you could study the resulting composite expression and show where terms cancel out if they did become trivially small, and you are left with the constant 3/2 which is what the expression would tend to. But again, you can not actually evaluate it at infinity, that isn't possible in the realm of standard analysis.
I admit, not being mathematician myself, I never before truly taught about the peculiar nature and deep significance of our ubiquitous zero. Because of its "triviality" it masks all levels of structure it may have.
Well, neither am I. But it doesn't seem so strange that the constant in between the negative and positive integers has no sign. The signedness is only significant in that it encodes the "direction" away from the origin (0).
Of course, the zero vector has no direction. If one tries to compute it (using trigonemtric functions), you will find it's value is undefined. As such, it is not in the unordered set of directions. Fancy that! (This is like asking what's the hue in the HSV model for black in RGB (0,0,0))
A nullity vector is meaningless, just as nullity is meaningless, so let's not go there.
In physics, a parameter takes on the units, not the value. The unit doesn't "cancel out" because it's being multiplied by zero. It's like a type on a variable in a computer program. Just because an integer is zero doesn't make that type of variable (void) or anything.
And I completely agree with your last paragraph. Except for the last part. That zero remains should not be troubling. In any model, you must restrict your input parameters so that the output parameters make sense. If you divide, if you take a square root, the special cases where the values can be out of range in that operation must be checked for and not allowed in their domains back out to the input parameters. This goes for dividing by zero or ANY OTHER CONSTRAINT. It is one of many.
But certainly when addinging, or multiplying, or using many other operators, a zero is a useful and meaningful quantity.
You don't need provably correct code to avoid a divide by zero.
Whenever you divide, it's not because you're manipulating data structures or hash tables. It's because you've been given a formula or heuristic where division was suggested or required. Only somebody forgot to make sure the divisor can't be zero (surely a sign the original EQUATION or METHOD is wrong, or that input in some other part of the system is being checked for a different valid range).
How many times have you written code where you divided by a non-constant factor determined at runtime? I mean, really? I doubt it's very frequent. All the more reason to be cautious about how you implement said code.
This entire fucking thread is offtopic.
Mod this whole meta-judeochristian-philisophical-wtfbbqry down. Including this post.
BOMBS AWAY. Make sure to use all five of your points. Thanks.
He has introduced an algebraic object that is not a ring or field, that is only different notationally compared to IEEE Inf/Nan semantics (aside from NaN != NaN).
What problem is trying to solve with these constructions? It certainly won't make arithmetic any more or less sound... dividing by zero is a problem of phrasing the question, rather than the semantics of the representation of the answer.
An example:
Suppose we have, through some process, yielded an expression f(t) = g(t)/h(t). The end result of f(t'), let's say, is 5.
And let's say we have an additional piece of information, g(t) = 10.
Then we can deduce h(t) = 2, and perhaps determine either g^-1 or h^-1 with additional information at data points (t_1, t_2, etc.)
Now let's say f(t') = nullity.
So, h(t) is 0. Or maybe not, because g(t') might be nullity at t'. If we knew h(t') is neither (0, nullity) then we know g(t') is nullity.
And if h(t') is 0 or nullity, then we can't determine what g(t') is... it could be anything.
The data point where the answer is nullity gives us no usable information, other data points (t_1, t_2) must be considered.
The same would happen in the first example if h(t') = 0; that datapoint t' would not be possible, a t' not in the domain of f(t).
This example might seem silly, but my point is that the nullity doesn't really tell you very much, other than that you propogated forward a division by zero and it collapsed your quantity to the number out of the number line. It's a unexceptional exception. It's like NULL in an outer join or normalized representation... not a good sign.
But in any other system, you would either have a contradiction, or in the case of software, an exception, which is just as valid in determining how to handle the situation, or to work around it.
Division by zero SHOULD be an exception:
Either...
1) your model is wrong mathematically
2) you translated your model into code incorrectly
3) you have an off-by-one bug
4) your model is ill-conditioned and underflows
5) you're not handling precision correctly
6) you need arbitrary precision
It's one (or more) of those things when you get that DIVIDE BY ZERO exception and your program crashses. THINGS THAT SHOULD BE FIXED. Fix the code, don't wrap it in a class and hope it doesn't happen again!
In his case, he didn't introduce a "catch-all" symbol, he introduced classes of infinities (ordinal and cardinal) with constructions and everything... it was actually useful for stuff.
This nullity is... well.
Dividing by zero is never defined. x/0 is NOT EQUAL to +/- Inf, no matter what your calculator or computer says.
The limit exists, sure enough, but the limit takes directionality in account in the expression, which is where the sign (or complex directional infinity) comes from.
Please don't misuse limit expressions and perpetuate falsehoods about our precious additive identity!
And zero isn't signed. By definition. Now, the construction [-0.00000000, +0.00000000], the bounds which define the real number zero might look like +/-, i.e. signed zeroes, but they two are really just limits with direction, and the value of the limit (i.e. 0) has no sign. At all.