Of course, what you suggest could be done without having to introduce special transaction elements between the customer and credit card vendor. You can simply use encryption here.
encrypt(Kvendor, "Authorize $20 to [vendor] from account 000001 at bank foo") encrypt(Kvendor, encrypt(Kbankfoo, sign(Kself, "Authorize $20 to [vendor] from account 000001 at bank foo")))
SSL can easily serve all of these purposes. Of course, you would not send exactly the same message to both (this makes a known-plaintext attack possible).
Obviously, it's not. The article is also not "new" information by any account (everything said was in B.S.'s original blog entry).
I also want to stress that while this is a blow to the faith we placed in SHA-1 (why were we trusting a hash invented by the NSA, anyway?) it doesn't do much for most uses of SHA-1. For example, it could allow you to perform a successful man-in-the-middle attack against an SSL-encrypted HTTP session, but the user might notice something was up when response times were measured in days;-)
It also does not impact the strength of encryption used for such sessions.
Personally, I still think people should push for his show. It's not that it's likely they'll do it (so it doesn't matter than he's unavailable), it's about showing them how hungry fans are for "good" Trek. If they shelve it for 2 years and come back and give it to Berman again... you'll get the same thing all over again.
So what should a large proportion of the world's climatologists do if they seriously believe that mankind is in danger of causing seriously climatic damage?
Work all the harder to disprove their thesis... The strength of a theory rests on a) the reproducability of experimental results and b) the resilience to disproof.
If you have a theory, then as a scientist it is your job to attempt to disprove it. If you would prefer to engage in politics, then fine, but don't call it science.
So let me see if I understand: it was a bad idea to host projects on SourceForge because they're giving people free bandwidth as long as they conform to the definition of open source, as interpreted by the people who coined the term? Or are we forgetting who the OSI is these days (I may have missed a Slashdot groupthink memo)?
And no one has said that the OSI is going to start revoking certifications because you use your own license. Theyr'e simply pushing to reduce the number. In practice that will mean a long series of negotiations, and that's going to benefit a lot of projects and people.
"what would [...] be wrong (in an ethical sense) with ignoring or even causing pain in others?
And if it were not wrong, where does that lead us? I'm not claiming to have any answers--just questions."
You're getting into what objective right and wrong are. There are plenty of books on the topic, and I suggest you read them. Slashdot is not the right place to get into something so complex.
The simple answer is that there is nothing about being a chemical soup that prevents you from structuring the kind of society in which you would want to live. The concept of fairness is very much a survival trait, in ensuring that unfairness to us is viewed with scorn.
To put it in other terms: you don't need a god or a soul to make the world a better place to live.
It's sad when biographies matter more than the science
No, just stop there.
That's totally unfair. Someone said that quoting Crichton was out of line because he was an author of trashing airport novels. I simply responded by presenting his credentials in order to rebut the obvious ad-hominem attack.
As for the ad-hominem attacks on RealClimate.org:
"the egregious Crichton" "Will-full ignorance" (refering to George Will, who I feel no love for, but it bothers me that a site claiming to be about science drags itself down to this level)
Just to cite two. They also tend to do things like explain what the current consensus in the scientific community is... which it turns out exactly maps to the views held by the site. This is quite the boost to their position, but I just thought I'd take a look at the data... sadly, their data is a report which didn't actually perform a survey of researchers so much as cherry-pick the literature that seemed to best support their point of view.
Seems they did not choose to include Mt Wilson's results even though they explain the current warming trend in terms of solar radiation... hmmm that's odd. So what's this "consensus" thing? Is it a vote? A survey? Or just a political football that is used to defend a theory which doesn't stand on its own merits?
Sigh. First off, consensus has nothing to do with the Inetnet. In fact, the Internet isn't particularly about science. The Internet is the product of engineering, and a well engineered tool it is.
However, even if the Inernet were a theory (the product of science), it would have nothing to do with consensus.
Science doesn't work on consensus. Science has to do with proof, theory and disproof. Anything else is social and other than being an interesting topic for research in itself, has nothing to do with science.
When you hear someone speaking of scientific consensus, they are talking about their PERCEPTION of the process from the outside.
He goes on to explain that many important scientific discoveries have been in direct conflict with the consensus.
And have been accepted instantaneously if they were backed by evidence, at least after from Newton onwards. See Newton, Maxwell, Einstein and Bohr for details.
Oops! Didn't read the article, did you?
Feel free. He gives some excellent examples that I was unaware of as well as some that I was. It certainly demonstrates his educational background in biochem (what is it about biochem folks who go on to be popular writers? Is there a writing prereq or something?), but other fields have many similar examples.
Computer science, for example, moves at a faster face than most, and new ideas are accepted often by fiat when their implementations become widespread, however I recall the decades old notion that B-tree filesystems were not efficient being challenged by one Hans Reiser. His work was soundly ridiculed and even when he produced a working implementation, the establishment in filesystem research spent quite a bit of time telling him why it didn't work... until it became obvious that there were an awful lot of people using it to boot from....
What is it that we're arguing here? Are we trying to establish that the right way to interact with science is to gather consensus? If so, please wake me when the fires stop burning. That's when I'll calmly explain that you need to take all probabilities into account, and when research is young (say, less than 10 years old), the chances that the consensus is wrong is HIGH.
"Without the concept of a soul, how can one feel pity/sadness for any organisms"
Let me answer your question with another quote from your post:
"pain, anguish, cognition and the like are merely electrochemical processes"
Pity would fall under "and the like".
We can regard our capacity for empathy, remorse and compasion as noble if we like, but that does not change the fact that they are simply responses to a stimulus as evidenced by recent research which has begun to track down where exactly in the brain visual empathy takes place.
One wonders if aliens were to come to earth if they would not say, "poor things. Their lack of a soul means they have no capacity to [insert cryptic emotion we can't comprehend here]."
Second, you could use this argument to justify absolutely anything. If all you have to do to disprove a scientific finding was to simply say "all these scientists might be wrong"
You're taking an absolutist stance in which the goal is to "justify" and "disprove".
That is not the goal. We (who feel that the scientific method has more merit than politics) simply wish to let the process of theory and trial work itself out without turning every morsel of conclusion into a talking point. Let the Manns and McKitricks debate the evidence and let others in the field reproduce data. This is how it works. Heck, the most conclusive piece of data yet was published in 1998. Anything less than 10 years ago is rarely something that has had time to be assailed by reasonable scrutiny from the scientific community.
That we're simply too impatient to wait for the conclusion is not a matter that is of interest to the scientific method.
One further thing: your response included 4 links. Two were to the same article (different sites publishing it) and one was to a Wikipedia entry that basically said: Crichton is controversial. All of the links refered to his FICTIONAL story, "State of Fear".
What really shocks me about this is that you pointed me at RealClimate.org previously, and they have a long piece about the very speech that I referenced. Why did you point me to articles about his fiction?
I'm sorry Aaron, I stopped reading when you started quoting Michael Crichton
That's too bad because what I quoted above is quite true. Science is not about consensus, it's about fact. Politics is about consensus.
He's an author, and a bad author of trashy airport thrillers at that.
To quote from his biographical blurb:
Educated at Harvard University, Cambridge, Massachusetts, A.B. (summa cum laude) 1964 (Phi Beta Kappa). Visiting Lecturer in Anthropology at Cambridge University, England, 1965. Henry Russell Shaw Travelling Fellow, 1964-65. Entered Harvard Medical School, M.D. 1969; spent one year as a post-doctoral fellow at the Salk Institute for Biological Sciences, La Jolla, California 1969-1970. Visiting Writer, Massachusetts Institute of Technology, 1988.
He may not be in the thick of this particular field, but with that list of credentials, I'll take his opinion on the matter of what the line between science and politics is over yours or mine or GWB's for that matter any day.
I guess you didn't find the time to read much on RealClimate.org as you said you'd try to do?
Actually, I did. I stopped after the 10th or so ad-hominem attack on people who held ideas that they did not like. Science is also not about discrediting those who hold conflicting theories.
Michael Crichton gave an excellent speech on the topic of global warming (with the amusing title of, "Aliens Cause Global Warming") in which he pointed out the poison in the word "consensus". To quote:
"I regard consensus science as an extremely pernicious development that ought to be stopped cold in its tracks. Historically, the claim of consensus has been the first refuge of scoundrels; it is a way to avoid debate by claiming that the matter is already settled. Whenever you hear the consensus of scientists agrees on something or other, reach for your wallet, because you're being had.
"Let's be clear: the work of science has nothing whatever to do with consensus. Consensus is the business of politics. Science, on the contrary, requires only one investigator who happens to be right, which means that he or she has results that are verifiable by reference to the real world. In science consensus is irrelevant. What is relevant is reproducible results."
Discuss the facts all you like. Discuss findings. Discuss measurements that might hold important meaning. But if you are going to discuss consensus, then you're talking politics, not science.
Anthropic warming is a hard topic. It's hard because it's wrapped up in our understanding of climate, and much as people would like to think that our understanding of climate is stable, it's anything but. A spike in temperatures (relative to what seemed to be normal warming previous) between the mid-1900s and today is an anomoly. Human influence is one way to explain that anomoly, and I grant that it's not a too bad as theories go. It is, however, only a theory. Other theories include the idea that warming has various step-functions related to solar influence, and that we may be observing such an event common to this sort of period between ice-ages.
To make warming our number-one hot political topic for the environment distracts us from some HUGELY important topics, so I (and some others like me) would just like to be a bit more certain before we take pulic focus off of the things that kill millions of people every year.
Thank you for making the distinction between public and scientific discourse and what that means for the status of an "accepted" conclusion. That's a point I find myself bringing up all to often.
"I regard consensus science as an extremely pernicious development that ought to be stopped cold in its tracks. Historically, the claim of consensus has been the first refuge of scoundrels; it is a way to avoid debate by claiming that the matter is already settled. Whenever you hear the consensus of scientists agrees on something or other, reach for your wallet, because you're being had."
He goes on to explain that many important scientific discoveries have been in direct conflict with the consensus. So please, let's not use the word "consensus" in this context. Discuss warming trends (there's some hard data you can point to), anthropic influence and other real topics, but science is not a popularity game. If you talk about consensus, you're only talking about politics, not science.
For my part, I have no fundamental problem with the idea behind Kyoto, though a) I think there are better places to spend time and money that would save more lives (e.g. reducing chemical toxic waste dumping) and b) the details of the treaty are almost certainly a mass of political potatoes that are getting lobbed around for individual gain, so one should not be too quick to judge its detractors.
I've still not been sold on anthropic warming, but I'd welcome more debate in the US on emissions. At the very least it's not a bad idea to keep our emissions under control with an eye toward air quality (though keep in mind that air quality isn't necissarily served by a focus on CO2 levels).
I think it's still worth pushing. Here are the reasons:
JMS could pick someone he trusts -- and yes, I think there are a few -- to execute his plan for him the way Abrams has done with Lost
Paramount may have decided to let the series lay fallow for a year or two today, but I doubt that that would hold in the face of a successful media blitz in favor of JMS's show
Like I said before, it's not likely that even if the effort is successful in stirring up Paramount that they would ever select JMS's treatment. Instead you want to push them to re-evaluate the creative team and Berman in particular
If they're holding off for 1-2 years, now's a great time to start directing their attention at the fans... Paramount can be a bit... "slow".
You need to improve Trek regardless of the fact that in a sane world, we'd go off and do something else now, because they're never going to let it die. The merchandising alone makes a new series (even if it flops) worth putting on the air. So the question is: do you make crap or something good?
Probably neither. It's probably so unlikely that it's not worth categorizing. The reasons are many, but it comes down to this: Straczynski is pig-headed, but not stupid. He would never agree to do it without complete creative control, and there's practically no way Paramount would allow that to happen. It would be along the lines of Bill Gates giving over control of the Windows platform to Linus Torvalds without even retaining the right to veto checkins.
Without complete creative control, you know what would happen: JMS would put forward his bible saying, "we tear down half of Starfleet and kill off a few notables to shake things up," and Paramount would reply with notes like the following:
Sounds good, but I don't think we want to kill anyone who has an action figure
"Tear down" might be too strong. Let's just say that there's a night-club bombing somewhere on Ferengi-prime
Great ideas, but you need to sex it up a bit
The first scene needs to really pull the viewers in: make it a fist fight
My kid loves tribbles, can you work one in as a major character?
He's been through this before with TNT and Crusade (the mess that you saw on TV was the result). He won't do it again.
All that said, YES, petition Paramount to do it. I think that at the very least it shows a massive lack of confidence in current show-runners and might upset the apple-cart enough to get someone creative in there.
Re:May be a big deal...
on
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SHA512 will be suceptible to similar attacks
Well, agreed, but if SHA-512 is attacked in the same way, the reduction of effort on the same order doesn't cripple it. If SHA-512 only gets you back up to 2**80 after using a variant of this attack, then you're back to the acceptable range that SHA-1 was believed to be at previously, no?
It's not as if SHA-1 has been "broken", so much as a short-cut was found in generating collisions. That short-cut only makes it so much easier. You then have to find your threshold for "too easy". After all, you can always generate collisions brute-force, so no hash is "secure" given enough computational resources. It's all a matter of acceptable difficulty.
Re:May be a big deal...
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the chances of a collision are the same as they always were and always will be [...this shows...] fewer attempts than 2^80 to generate collisions
You could be right, but two points:
The article doesn't make that distinction clear and how could you be certain of the chances of collisions up front? If it's proven at some point that there's some vast space within the set of all possible bit-string results that SHA-1 would never have generated (which is what I thought this paper demonstrated, though obviously it's not in general circulation, so I don't know that), why would that be difficult to accept?
Re:May be a big deal...
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SHA-1 Broken
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I know some folks like to type out long decimal numbers to impress the magnitude of the numbers on the reader
Well, the point was that the numbers (2**69 vs. 2**80) were in the article, my purpose in showing numbers at all was to demonstrate the size of the numbers involved to those who didn't think in powers of two.
Slashdot Mirror
carrier pigeon
Obviously, it's not. The article is also not "new" information by any account (everything said was in B.S.'s original blog entry).
;-)
I also want to stress that while this is a blow to the faith we placed in SHA-1 (why were we trusting a hash invented by the NSA, anyway?) it doesn't do much for most uses of SHA-1. For example, it could allow you to perform a successful man-in-the-middle attack against an SSL-encrypted HTTP session, but the user might notice something was up when response times were measured in days
It also does not impact the strength of encryption used for such sessions.
Try this out for size: http://solar-center.stanford.edu/sun-on-earth/vars un.html
Once someone can take that graph and explain how humans are responsible for it, I'll be first in line to demand changes to the way we do things.
So what should a large proportion of the world's climatologists do if they seriously believe that mankind is in danger of causing seriously climatic damage?
Work all the harder to disprove their thesis... The strength of a theory rests on a) the reproducability of experimental results and b) the resilience to disproof.
If you have a theory, then as a scientist it is your job to attempt to disprove it. If you would prefer to engage in politics, then fine, but don't call it science.
So let me see if I understand: it was a bad idea to host projects on SourceForge because they're giving people free bandwidth as long as they conform to the definition of open source, as interpreted by the people who coined the term? Or are we forgetting who the OSI is these days (I may have missed a Slashdot groupthink memo)?
And no one has said that the OSI is going to start revoking certifications because you use your own license. Theyr'e simply pushing to reduce the number. In practice that will mean a long series of negotiations, and that's going to benefit a lot of projects and people.
"what would [...] be wrong (in an ethical sense) with ignoring or even causing pain in others?
And if it were not wrong, where does that lead us? I'm not claiming to have any answers--just questions."
You're getting into what objective right and wrong are. There are plenty of books on the topic, and I suggest you read them. Slashdot is not the right place to get into something so complex.
The simple answer is that there is nothing about being a chemical soup that prevents you from structuring the kind of society in which you would want to live. The concept of fairness is very much a survival trait, in ensuring that unfairness to us is viewed with scorn.
To put it in other terms: you don't need a god or a soul to make the world a better place to live.
It's sad when biographies matter more than the science
No, just stop there.
That's totally unfair. Someone said that quoting Crichton was out of line because he was an author of trashing airport novels. I simply responded by presenting his credentials in order to rebut the obvious ad-hominem attack.
As for the ad-hominem attacks on RealClimate.org:
"the egregious Crichton"
"Will-full ignorance" (refering to George Will, who I feel no love for, but it bothers me that a site claiming to be about science drags itself down to this level)
Just to cite two. They also tend to do things like explain what the current consensus in the scientific community is... which it turns out exactly maps to the views held by the site. This is quite the boost to their position, but I just thought I'd take a look at the data... sadly, their data is a report which didn't actually perform a survey of researchers so much as cherry-pick the literature that seemed to best support their point of view.
Seems they did not choose to include Mt Wilson's results even though they explain the current warming trend in terms of solar radiation... hmmm that's odd. So what's this "consensus" thing? Is it a vote? A survey? Or just a political football that is used to defend a theory which doesn't stand on its own merits?
Sigh. First off, consensus has nothing to do with the Inetnet. In fact, the Internet isn't particularly about science. The Internet is the product of engineering, and a well engineered tool it is.
However, even if the Inernet were a theory (the product of science), it would have nothing to do with consensus.
Science doesn't work on consensus. Science has to do with proof, theory and disproof. Anything else is social and other than being an interesting topic for research in itself, has nothing to do with science.
When you hear someone speaking of scientific consensus, they are talking about their PERCEPTION of the process from the outside.
He goes on to explain that many important scientific discoveries have been in direct conflict with the consensus.
And have been accepted instantaneously if they were backed by evidence, at least after from Newton onwards. See Newton, Maxwell, Einstein and Bohr for details.
Oops! Didn't read the article, did you?
Feel free. He gives some excellent examples that I was unaware of as well as some that I was. It certainly demonstrates his educational background in biochem (what is it about biochem folks who go on to be popular writers? Is there a writing prereq or something?), but other fields have many similar examples.
Computer science, for example, moves at a faster face than most, and new ideas are accepted often by fiat when their implementations become widespread, however I recall the decades old notion that B-tree filesystems were not efficient being challenged by one Hans Reiser. His work was soundly ridiculed and even when he produced a working implementation, the establishment in filesystem research spent quite a bit of time telling him why it didn't work... until it became obvious that there were an awful lot of people using it to boot from....
What is it that we're arguing here? Are we trying to establish that the right way to interact with science is to gather consensus? If so, please wake me when the fires stop burning. That's when I'll calmly explain that you need to take all probabilities into account, and when research is young (say, less than 10 years old), the chances that the consensus is wrong is HIGH.
"Without the concept of a soul, how can one feel pity/sadness for any organisms"
Let me answer your question with another quote from your post:
"pain, anguish, cognition and the like are merely electrochemical processes"
Pity would fall under "and the like".
We can regard our capacity for empathy, remorse and compasion as noble if we like, but that does not change the fact that they are simply responses to a stimulus as evidenced by recent research which has begun to track down where exactly in the brain visual empathy takes place.
One wonders if aliens were to come to earth if they would not say, "poor things. Their lack of a soul means they have no capacity to [insert cryptic emotion we can't comprehend here]."
Second, you could use this argument to justify absolutely anything. If all you have to do to disprove a scientific finding was to simply say "all these scientists might be wrong"
You're taking an absolutist stance in which the goal is to "justify" and "disprove".
That is not the goal. We (who feel that the scientific method has more merit than politics) simply wish to let the process of theory and trial work itself out without turning every morsel of conclusion into a talking point. Let the Manns and McKitricks debate the evidence and let others in the field reproduce data. This is how it works. Heck, the most conclusive piece of data yet was published in 1998. Anything less than 10 years ago is rarely something that has had time to be assailed by reasonable scrutiny from the scientific community.
That we're simply too impatient to wait for the conclusion is not a matter that is of interest to the scientific method.
One further thing: your response included 4 links. Two were to the same article (different sites publishing it) and one was to a Wikipedia entry that basically said: Crichton is controversial. All of the links refered to his FICTIONAL story, "State of Fear".
What really shocks me about this is that you pointed me at RealClimate.org previously, and they have a long piece about the very speech that I referenced. Why did you point me to articles about his fiction?
That's too bad because what I quoted above is quite true. Science is not about consensus, it's about fact. Politics is about consensus.
He's an author, and a bad author of trashy airport thrillers at that.
To quote from his biographical blurb:He may not be in the thick of this particular field, but with that list of credentials, I'll take his opinion on the matter of what the line between science and politics is over yours or mine or GWB's for that matter any day.
I guess you didn't find the time to read much on RealClimate.org as you said you'd try to do?
Actually, I did. I stopped after the 10th or so ad-hominem attack on people who held ideas that they did not like. Science is also not about discrediting those who hold conflicting theories.
Actually, I posted very different things, with the same quote. But, of course, you didn't bother to read either post, did you?
Anthropic warming is a hard topic. It's hard because it's wrapped up in our understanding of climate, and much as people would like to think that our understanding of climate is stable, it's anything but. A spike in temperatures (relative to what seemed to be normal warming previous) between the mid-1900s and today is an anomoly. Human influence is one way to explain that anomoly, and I grant that it's not a too bad as theories go. It is, however, only a theory. Other theories include the idea that warming has various step-functions related to solar influence, and that we may be observing such an event common to this sort of period between ice-ages.
To make warming our number-one hot political topic for the environment distracts us from some HUGELY important topics, so I (and some others like me) would just like to be a bit more certain before we take pulic focus off of the things that kill millions of people every year.
Thank you for making the distinction between public and scientific discourse and what that means for the status of an "accepted" conclusion. That's a point I find myself bringing up all to often.
As for the article itself, let's please stop talking about "consensus".
Seriously, if you want a nice review of the topic of scientific consensus, here's a bit from a speech given by Michael Crichton He goes on to explain that many important scientific discoveries have been in direct conflict with the consensus. So please, let's not use the word "consensus" in this context. Discuss warming trends (there's some hard data you can point to), anthropic influence and other real topics, but science is not a popularity game. If you talk about consensus, you're only talking about politics, not science.
For my part, I have no fundamental problem with the idea behind Kyoto, though a) I think there are better places to spend time and money that would save more lives (e.g. reducing chemical toxic waste dumping) and b) the details of the treaty are almost certainly a mass of political potatoes that are getting lobbed around for individual gain, so one should not be too quick to judge its detractors.
I've still not been sold on anthropic warming, but I'd welcome more debate in the US on emissions. At the very least it's not a bad idea to keep our emissions under control with an eye toward air quality (though keep in mind that air quality isn't necissarily served by a focus on CO2 levels).
You need to improve Trek regardless of the fact that in a sane world, we'd go off and do something else now, because they're never going to let it die. The merchandising alone makes a new series (even if it flops) worth putting on the air. So the question is: do you make crap or something good?
Probably neither. It's probably so unlikely that it's not worth categorizing. The reasons are many, but it comes down to this: Straczynski is pig-headed, but not stupid. He would never agree to do it without complete creative control, and there's practically no way Paramount would allow that to happen. It would be along the lines of Bill Gates giving over control of the Windows platform to Linus Torvalds without even retaining the right to veto checkins.
Without complete creative control, you know what would happen: JMS would put forward his bible saying, "we tear down half of Starfleet and kill off a few notables to shake things up," and Paramount would reply with notes like the following:
- Sounds good, but I don't think we want to kill anyone who has an action figure
- "Tear down" might be too strong. Let's just say that there's a night-club bombing somewhere on Ferengi-prime
- Great ideas, but you need to sex it up a bit
- The first scene needs to really pull the viewers in: make it a fist fight
- My kid loves tribbles, can you work one in as a major character?
He's been through this before with TNT and Crusade (the mess that you saw on TV was the result). He won't do it again.All that said, YES, petition Paramount to do it. I think that at the very least it shows a massive lack of confidence in current show-runners and might upset the apple-cart enough to get someone creative in there.
SHA512 will be suceptible to similar attacks
Well, agreed, but if SHA-512 is attacked in the same way, the reduction of effort on the same order doesn't cripple it. If SHA-512 only gets you back up to 2**80 after using a variant of this attack, then you're back to the acceptable range that SHA-1 was believed to be at previously, no?
It's not as if SHA-1 has been "broken", so much as a short-cut was found in generating collisions. That short-cut only makes it so much easier. You then have to find your threshold for "too easy". After all, you can always generate collisions brute-force, so no hash is "secure" given enough computational resources. It's all a matter of acceptable difficulty.
the chances of a collision are the same as they always were and always will be [...this shows...] fewer attempts than 2^80 to generate collisions
You could be right, but two points:
The article doesn't make that distinction clear and how could you be certain of the chances of collisions up front? If it's proven at some point that there's some vast space within the set of all possible bit-string results that SHA-1 would never have generated (which is what I thought this paper demonstrated, though obviously it's not in general circulation, so I don't know that), why would that be difficult to accept?
I know some folks like to type out long decimal numbers to impress the magnitude of the numbers on the reader
Well, the point was that the numbers (2**69 vs. 2**80) were in the article, my purpose in showing numbers at all was to demonstrate the size of the numbers involved to those who didn't think in powers of two.