Guest Essay by Kip Hansen
Spoiler Alert: This essay is not about the mathematical entity the imaginary number. I do think that an essay here about imaginary numbers of that sort would be interesting, but this isn’t going to be it. This essay, while not about the usual fare seen here – AGW; CAGW; Catastrophic Climate Change; Global Cooling; various oxides of carbon; the pH, level, or surface temperature of oceans; or the antics or ethics (or lack of ethics) of various international scientists and politicians — will hopefully be interesting to the majority of readers. It will ask more questions than it answers.
Last Saturday, 3 October 2015, WUWT’s indefatigable Willis Eschenbach published a guest essay regarding an NPR radio report by Ira Flatow that labelled “some recent pictures of flooding in Miami, Florida, as evidence that climate change is real and is already affecting Florida.” In response to a comment I made to that essay, Willis asked this very interesting question:
“…as you say, we can measure sea level with a “high degree of accuracy” … so are we measuring an imaginary thing? And if we average those highly accurate measurements, why would we not get a global average sea level? What am I missing here?”
In science, asking the Right Question is often, maybe always, more important than having the Right Answer. Let’s look at Willis’s questions and see what we can find out about the world and the world of science.
What are the questions here?
- Can we measure sea level with “a high degree of accuracy”?
- Are we measuring an imaginary thing (when we do so)?
- If we average those highly accurate measurements, why would we not get a global average sea level?
- What am I [we] missing here?
It is my idea here to ask a more generalized question — what are we measuring in Climate Science and are we measuring an imaginary thing when we do so? — but we can use “sea level” as the thought experiment example.
Let me address the first question first: Is it really possible to measure something like sea level (or surface air temperature 2 meters above the ground or sea surface temperature) with “a high degree of accuracy”?
When I stated in my original comment that we had been measuring sea level with a high degree of accuracy for years, I meant that we knew what sea level could be expected at various places at future times and had an idea what a more generalized “global sea level” might be and what changes had been seen over longer time periods like the last century or so. But for our thought experiment in this essay, let’s define “high degree of accuracy” as the commonly mentioned “annual anomaly” in the scientific literature. For “global average sea level” this is in single digit millimeters, usually 1.7/1.8 up to 3.4 mm per year, somewhere in that range. (For those thinking along on other paths, that might be tenths and hundredths of a degree Centigrade for global average surface air temperature and sea surface temperature, and even smaller, thousandths of a degree C for ocean water temperatures leading to a calculation of ocean heat content.)
Before we get very far, let’s ask “Why do we [they] want to measure global sea level?” The major reason seems to be, in our politicized world of global warming politics, that many want to measure global sea level to show that it is rising (which it has been for quite some time, at least the last 20,000 years) and that this continuing rise is 1) dangerous and 2) due to recent surface temperature rise over the last century, thus 3) due to Global Warming. The theme is to use sea level rise as a proof of increased thermal expansion of the water in the oceans and increased addition of water from melting land ice deposits, both asserted to be the result of Global Warming caused by increased atmospheric concentrations of greenhouse gases, primarily CO2, since the 1880s . We’ll see later in this essay that this is part of a larger modern scientific movement to produce “single numbers” to represent dynamic systems (some of which are properly known to be nonlinear dynamical systems).
Can we measure sea level to that (+/- 3 to 4 mm) degree of accuracy? Well, for sea level, even at a single precise location, the answer is “No, we can not.” Now, I am not trying to be provocative here, it is a simple matter of fact. If the sea would be so kind as to stand still, even for just a few moments, we could get in a very accurate measurement at a single spot, or even a lot of spots. Alas, the sea is never still, it is always moving up and/or down: tides, currents, wind chop, waves, wakes of passing vessels, rising and falling air pressure and, in most important locations, all of those at once. Thus, we cannot physically do it; the sea does not stand still long enough for us to make this measurement to that degree of accuracy. This gets only worse when we add in the information that both the dry land itself and the bottoms of the oceans, almost everywhere, are also in vertical motion and busy changing the volume of the ocean basins.
Many will protest: “Look here, Mr. Hansen. You can’t say that. There are scads of very scientific tables, charts, and journal articles very carefully telling us that now only can we make that measurement, we have been doing so for much of human history and [drum roll, please] since 1992 with [gasp!] satellites!”
It is my point here that what we are doing, where the doing is done, is not measurement, but derivation. Many measurements are taken, in many and diverse locations, at many and diverse times. In some cases, there are nearly continuous time series of measurements for particular locations. From these numerous individual measurements, for example, the tide station reports from the Battery in New York City, an interesting (but not to be detailed here) formula is applied to derive a figure, a single number, that represents the average difference between the sea surface and a geodetic bench mark (set in the bedrock of Manhattan Island years ago) over some period of time. We will skip the nearly infinite details as to whether the derived number represents a simple average between highs and lows, or is an average against time.
Let me point out that the NOAA CO-OPS system of tide stations has a very important and pragmatic purpose. Ships and boats need to know the depth of the water they will find in a particular spot – at a dock on the Hudson River or over the sand bar across the inlet – and at a particular time. Thus, tide tables are very important to sea going commerce and recreational boaters. It answers important questions such as: “Can I get there without hitting those nasty rocks (or going aground on that sticky mud) on the bottom? Can I stay here without being set down by the tide on those rocks or mud?” This system was never designed to measure “sea level rise” nevertheless it is used to compute changes in relative sea level trends in ports of American interest. Here are two Wiki articles on sea level: here and here. In the second article, this image is shown:
Notice please the difference between the trend calculated from tide gauges (orange line with grey error range) and the blue satellite measurements. Tide Gauge data (which measures Relative Sea Level at each tide gauge) accelerates while satellite data, which measures absolute sea level, keeps to its century long trend.
But what of those marvelous satellites? The official NOAA claim is: ”A series of satellite missions that started with TOPEX/Poseidon (T/P) in 1992 and continued with Jason-1 (2001–2013) and Jason-2 (2008–present) estimate global mean sea level every 10 days with an uncertainty of 3–4 mm.” Results can be seen on graphical form at NOAA’s Laboratory for Satellite Altimetry web site. It is interesting to see the difference in visual impact that results from the use of alternate coloring schemes and to observe the lumpiness of the oceans.
I know many of the readers here are familiar with the sea – Willis and I have each spent a hefty fraction of our lives living on the sea, and an ever greater fraction living at the edge of the seas. Three to four millimeters is between 0.12 and 0.16 of an inch – about the thickness of two American pennies stacked atop one another. Or, for our cousins in the United Kingdom, about as thick as a one pound coin. It is a rare and beautiful and awe inspiring sight to see the ocean smooth as glass to the horizon, or even just across the bay or harbor. In my one-third of a lifetime of living on the sea (totaling > 20 years), I have only occasionally seen the sea so smooth – the slightest breezes bring up wind ripples and chop that far exceeds 3-4 mm, and can build quickly to feet and meters. If a body of water is open to the ocean, undulating ocean swells march from one horizon to the other, swells also measured in multiple feet or meters, and not necessarily traveling in the same direction as the wind chop. This all adds up to a great deal of vertical motion of the sea’s surface – at times exhilarating and at times downright frightening.
Now if NOAA wants to claim that their satellites in their perfect orbits can somehow transmogrify the undulating, rising and falling, uneven surface of the Earth’s ocean to a resolution of +/- 3 to 4 mm, then very well. Who am I to say they can’t, even if I can’t imagine how they might even theoretically do so. Nonetheless, for our purpose here, let us make this distinction: they do not measure “global mean sea level every 10 days” – they don’t even claim to, their claim is that they estimate it. In every real pragmatic sense, they somehow derive a single number from a fabulously massive amount of data – data which in and of themselves are not direct measurements, but inferences of measurements made from other kinds of data.
Let’s quit fooling around. While it would be possible to measure sea level in individual locations, it is difficult and even when done it is not a true measurement, but a derivation from accumulated data and dependent on mathematical and statistical methods and definitions. If you ever find a particular section of sea at “sea level”, it will be totally momentary and accidental.
Sea Level, even “Sea Level at the Battery in New York”, is not properly represented by a single number, above and below some geodetic bench mark. What we call sea level is a derived, calculated number – an average of averages of an array of measurement time series. In this sense, as the calculated mid-point of a range over time, it is, in a practical sense, an imaginary number having no existence in the day-to-day life of the Port of New York.
There is, however, a pragmatic “sea level at the Battery in New York” – which itself is a predictable range above and below some depth of water at a certain point (a point referred to as Local Mean Sea Level) which, when modified by information of expected, predicted tides, can be extrapolated to other points in the harbor, which is useful for mariners despite its less-than-real aspect. It can be used in its gross form (fractions of feet or meters) to determine the depth of water over the bottom at a place and time important to a ship’s captain and crew. Here is the prediction of water levels, relative to MLLW, made for October 9th thru October 11th.
The bottom line is that sea level, anywhere and at any time, is not a direct measurement. Never. It is a calculated, derived number that represents a precisely defined, but actually quite complicated, idea.
In order to define global sea level, one must participate in an exercise of imagination along the lines of: Imagine that the planet has stopped spinning; that moon has never existed; that the planet is a perfect sphere (or perfectly regular ovoid or flattened sphere); that there is no wind; that the atmosphere is evenly distributed and air pressure is the same at all points; that the temperatures of the seas are all exactly even, everywhere, to all depths; that there are no currents;, that there are no ice caps; that the rivers have stopped flowing into the sea and that gravity is magically equal at all points on the Earth’s surface (it is not, btw): under those conditions, we could then say that global sea level would be precisely “there”, within 3 or 4 mm. My friends, this is what makes Global Average Sea Level, in this special sense, an imaginary number.
So, we have answered Question 2: Are we measuring an imaginary thing (when we do so)? Yes, we are “measuring”, in a sense, an imaginary thing when we say we are measuring sea level. The resulting calculated, derived number is a creature of our imaginations, an imaginary number.
Question 3 almost answers itself. “If we average those highly accurate measurements, why would we not get a global average sea level?” One can carry out a dizzying number of statistical and mathematical steps and arrive at some number – the more division steps involved the more precise looking the number will be. One can average any set of numbers. In this case, will one arrive at a number that is the “global average sea level”? Let’s look at Question 4 first and come back to this.
Question 4 is “What am we [originally “I”] missing here?”
This is a question of logic, and kind of follows on from an earlier essay I published here in February regarding Uncertainty Ranges. When one averages a series of numbers that are in reality themselves ranges, then the result must also be a range. In our case today, when averaging a series (or in this case, a computer-full) of imaginary numbers then the result must be another imaginary number, in the same sense as the numbers in the original data set. You can not average away original measurement error, you can not average away the fact that data given are themselves really ranges rather than single numbers, you can not average away the fact that original numbers themselves are, in the sense discussed here today, imaginary.
Before we too far afield here, let’s try to be clear on what the distinction is between a real number and what I have been calling here an imaginary number. This discussion takes place in the context of the measurement of characteristics of the physical world. For the result of a measurement to be a real number, the thing being measured must itself be measurable and the numerical result representing that measurement must represent something that exists in some meaningful and useful sense. However, the result of a measurement of a thing that itself is not physically measurable, but which can only be derived mathematically based on a definition that itself is an object of our imaginations (not something actually found in the real world), then that result should itself be considered, in this special sense, imaginary as well, despite its seeming precision.
There are innumerable averages of things that can be derived and calculated. Despite that, many of those averages are themselves imaginary, and their meaning and usefulness must always be thoughtfully considered. Such imaginary numbers may have some interesting meaning and some pragmatic usefulness but great care must be taken with their application, because, after all, they are imaginary and do not exist in reality.
Thus the average height of American citizens can be useful in determining the sizes of beds sold to Americans, at least indicating a range to be considered, it would be foolish to declare it the proper height of doorways for all new construction, even with an inch to spare tacked on, or to make exaggerated, scary, claims about public health threats based on the tiniest changes in such a number over some narrowly-selected time period.
Worse yet, and I hope there will be some comments in support of at least this idea, simple averages of averages of averages (all of which start with averaged, imaginary, derived numbers rather than actual measurements) are abominable absurdities. [ref: Simpson’s Paradox, etc.]
Here’s a ridiculous example: If we calculated the average altitude of the land in the state of South Carolina, first averaging the altitude of each county, then averaging the altitude of multi-county regions, and finally averaging regional altitudes, the result would be a number like (a totally pulled-out-of-the-air guess) 125 feet above sea level and when trended from the highest point in the Blue Ridge Mountains to the sea the state could be said to have a slope of XX feet per mile. It makes no difference in this sense if we weight the averages, krig the missing points, homogenize or smooth or smear. This procedure calculates and/or derives an imaginary number in the special sense of our working definition here. Thus, with our magic new imaginary number, it might be claimed that while some areas of South Carolina could be flooded by extreme high tides simultaneous with two feet of rain, on average the people there would not be prone to disaster as even the few expected flooded areas would quickly drain into the Atlantic. Applying such a totally mathematically correct yet imaginary number to the real world can result it wildly inappropriate conclusions. It was this type of logic powered by imaginary numbers that led a New York Times science journalist to erroneously claim that the global sea level rise caused by global warming (a real rise but an imaginary number) caused increased damages to New York City during Hurricane Sandy — the same error Ira Flatow made in the NPR segment about flooding in Miami, where the flooding referred to occurs at a spot that is below the long-term Mean High Tide, and was so when the street was constructed.
Now, coming back to Question 3: “If we average those highly accurate measurements, why would we not get a global average sea level?” If we average the very large data set of imaginary numbers for a specific moment in time, we will arrive at a new, even more imaginary, single number that could be called, if everyone were willing to allow it, “global average sea level”. Would it be pragmatically, practically, meaningful and useful? Maybe, but in a very limited sense…and we would have to be very careful as to what meaning we assigned to it.
Why? See my essay last year about Hurricane Sandy and damages to NY City. The purported sea level rise for the 50 year period 1960 – 2010 “caused by global warming driven sea level rise” should have been 4 inches (roughly half of the 8 inches over the last century). In actuality, only when we use the lowest estimate of subsidence for the Battery couple with the highest estimate of local relative sea level change do we see any positive contribution of absolute, global sea level change to the relative sea level at the Battery, the 0.59 inches in the upper right-hand corner:
What’s up here? The acknowledged century-long estimated global sea level rise did not show up at the Battery, not even over the most recent 50 year period. This should not surprise us – attempts to apply a single-number, “global sea level rise”, is ill-thought out – trying to apply an imaginary number to a specific real situation.
Today’s discussion is one way of looking at the current trend in Science in which attempts are made to reduce very complicated dynamic systems to a single number which can then be graphed against time, usually in attempts to do one or more of the following:
- to cast blame for the increasing or decreasing number on a substance or action or group, usually incorrectly
- using two such graphs of single numbers to correlate some single number with some other single number to sell a desired story, usually to cast blame or give credit, usually incorrectly
- to bring attention to [read this as: to cause public concern or worry about] some rising or falling single number in hopes of generating gain [in research funds, fame, public sympathy, public or political support], usually unwarranted
These single numbers, meant to somehow illuminate some feature of the real world, are often, maybe almost always, not real numbers representing real things, but imaginary numbers representing concepts that exist, on a pragmatic practical level, only in our imaginations, which may lack meaningfulness and usefulness, or both. In this special sense, we can rightly refer to them as imaginary numbers. And because they are almost never acknowledged as imaginary numbers which require special care in application, each of the three uses above is followed by “usually incorrectly” or “usually unwarranted”.
Now, even if you don’t agree with me, it should be interesting to discuss in comments some of the ongoing efforts to [mis-] use this special breed of derived number, the imaginary number, to sway public opinion in differing scientific fields around the world. I’d really like to hear your views and benefit from your experience.
# # # # #
Author’s Comment Policy: This essay is not really about global sea level, but I doubt we’ll be able to discuss it without also touching on the issues surrounding the issue of global mean sea level. I do know something about it and will try to answer questions.
I’d rather discuss the concept of “Are we chasing imaginary numbers?”
It’s just an idea…let’s talk about it.
# # # # #
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Thanks, Kip Hansen, for a very thought-provoking article.
Articles like yours make WUWT a place for learning.
Reply to Andres Valencia ==Thank you, sir.
Years ago when discussing global temp the official stance seemed to be the number was only good to show us longer term trends. Year to year trends meant little. After it became so political though we seem to pretend this product is accurate to tiny fractions of a degree rather then only longer term trends.
Kip, thanks for an interesting article. I had not expected such a detailed response to my question.
However, like many readers, I was put off by your redefinition of “imaginary numbers”, so much so that I didn’t read far into it at all when it was posted. You defended this, saying:
Kip Hansen October 10, 2015 at 11:37 am Edit
Now, suppose I wrote an article, and in the very first sentence I warned you that whenever I said “angry men”, I actually meant “nice women”. So then I started off saying something like “Angry men are the most dangerous kind of men. When you see angry men, you should get nervous” …
I assume you can see the difficulties with that.
And then suppose that when somebody busted me for that redefinition, I pointed out that they don’t own the English language, and that if they object to my use of the term “angry men” to mean “nice women”, that is just “hubristic” …
I’m sure you can see the problem. Yes, you are free to redefine “freedom” to mean “slavery” … but would that be conducive to or an impediment to communication?
And calling those like myself who think it would be an impediment to communication “hubristic”?
Well, your first paragraph was where I stopped reading. I jumped to the comments at that point, and stopped after your first comment. You are free to redefine anything. You are not free to diss people when they say that it’s not helping communication when you do that.
Finally, you may not realize it, but you’ve shot yourself in the foot by making it impossible to quote sections of your work. What would anyone who is mathematically literate make of this quote, for example?
Now that I’ve read your work, I have more substantial objections, but I wanted to get that out of the way first.
My best to you,
w.
Reply to Willis ==> Truthfully, the choice of the word “imaginary” was yours. There is no “redefinition” here, as many readers have already pointed out…simply using English words in a different usage, outside of the narrow higher mathematical nomenclature, and clearly stating so. There are a limited number of words in the English language, and different fields of knowledge are not allowed to “dibs” them and forbid their use outside their own field. A good example is the word “species“, which has several definitions in different nomenclatures: biology, monetary, ecclesiastical, logic, and then an everyday usage. In reality, even within the field of biology, the definition is in serious dispute.
Further, I have used the word quite correctly to refer to things, certain types of proffered numbers, as “imaginary numbers” using the common use of “imaginary” here: “1. a : existing only in imagination : lacking factual reality, b : formed or characterized imaginatively or arbitrarily” with some rather clearly stated caveats.
“What would anyone who is mathematically literate make of this quote, for example?” Personally, I would suggest thinking about it in context of the essay — but that’s just me…others may prefer to do something else.
Kip Hansen October 11, 2015 at 11:16 am Edit
Truthfully, YOU wrote the post, not me, so don’t try to blame me for your choices.
You have said that averages (a mathematical term) are imaginary numbers (another mathematical term). That’s called “redefinition”.
Sorry, but the nature of a quote is that it is offered OUTSIDE of the context of the essay …
w.
Reply to Willis ==> Respectfully, you need to re-read the essay more carefully before commenting further. Only one other person (apparently a High Priest of Mathematics) misunderstood my point so thoroughly.
Taking quotes out of context is not a virtue — it is what makes conversations impossible.
I’d love to hear from you again when you can restate the overall point of the essay in your own words, using 100 words or less. If you are even close, then we can discuss the finer points.
Are you back from your extended trip?
Kip Hansen October 11, 2015 at 3:21 pm
OK, so tell me what I misunderstood. I find no change in my understanding having re-read your post.
Unless you quote an entire document, all quotes are removed somewhat from the context. To make sense out of the lines I quoted, I’d also have to quote your entire introduction. Pass.
Sorry, pal. I’ve made a number of specific points about your claims below, here. How about you climb down off of your high horse and summarize my points in 100 words or less?
Or more to the point, forget about the summary—how about you just show where I’m wrong in my other comments?
Indeed I am. I was going to write up the rest, but it never came to pass. I had much more interesting scientific work to do.
All the best,
w.
Last Reply to Willis ==> From the git-go you have misunderstood even the area of science the essay is about. Quoting you : “I am saying that in an essay ABOUT MATHEMATICS…”
The essay is not about mathematics, which others have already pointed out to you. It is also not about the mathematicians “imaginary unit or numbers”, not about “AGW; CAGW; Catastrophic Climate Change; Global Cooling; various oxides of carbon; the pH, level, or surface temperature of oceans; or the antics or ethics (or lack of ethics) of various international scientists and politicians”, stated clearly “This essay is not really about global sea level, but I doubt we’ll be able to discuss it without also touching on the issues surrounding the issue of global mean sea level.” It certainly is not about simple averages of numbers.
I offer a good summary of what it IS about in the Post Script.
‘Are We Chasing Imagined Numbers’ would have done the job.
Reply to vukcevic ==> Probably right, that. Hard to please everyone with word choices — after all, look at what Mathematicians did when they chose “imaginary number” and “imaginary unit” as the nomenclature for “An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i 2 = −1. “
Thanks for the comment, I happen to be well familiar with both, i and j
Thanks Kip great post.
I appreciate your effort to define your terms.
Perhaps it also is a lesson in reading comprehension.
The persons claiming confusion and protesting that mathematics owns the term “Imaginary numbers”™ do provide examples of how the whole meme of Climatology has prospered in its impersonation of science, while refusing to define its terms, for far too long already.
Funny how what we are sure is so, tends to mislead our understanding of other peoples points of view.
john robertson October 11, 2015 at 12:01 pm
I am not saying that mathematics owns the term “imaginary numbers”.
I am saying that in an essay ABOUT MATHEMATICS, using the term “imaginary numbers” to mean something other than the normal mathematical meaning causes problems … as is proven by the number of people who turned away from the article or were unhappy with it for that very reason.
My aim is to write for comprehension, not for problems and dissension. Everyone here who is discussing the mis-use of the term is NOT discussing the essence of the article. Perhaps that’s OK with you and Kip.
Had I written the article, I’d be bummed to see people discussing my word choice rather than my ideas … but hey, that’s just me.
w.
Thankyou Willis, I think you reinforce my observation
“An essay ABOUT MATHEMATICS..”
Sorry to add to the confusion ,but I was lead to believe that Kip Hansen was discussing measurement of certain parameters of the real world and then what we claim to infer from these interpretations(measurements).
The 4 questions of Paragraph two or block quotation 2 is not the subject of this post?
Willis
“Had I written the article, I’d be bummed to see people discussing my word choice rather than my ideas … but hey, that’s just me.”
They do sometimes, because you often use the expression “warming” when you really mean “slows down the rate of cooling”.
When discussing physics, different processes are at work when something is “warming” than when the “rate of cooling is being slowed down”. It is unhelpful and confusing to use the expression “warming” and although this has been pointed out to you for many years, you often still use the expression out of its true and recognised scientific meaning and context. Just saying, perhaps you should reflect on that before too harshly criticizing Kip; stones and glass house etc. .
I am one of the people who noted that “imaginary” was an unfortunate use of words, but I did recognise that Kip had defined what he meant when using that expression. I suspect that it made good by line, which is a game frequently employed in the newspaper business to grab people’s attention and draw them in; it is an attention seeker, snazzy word and hence difficult to put down, and I suspect that explains why Kip decided to run with it. It was simply too irresistible to use a more appropriate expression for the principle that he was seeking to convey.
But the issue is the correctness and relevance of the point(s) being set out in this article and does it have real significance, or only passing significance. Whilst semantics are important debates should not really centre on that (which you often mention when someone picks you up for using the word “warming” out of its proper scientific meaning).
Like everyone else (well almost everyone), I enjoy reading your opinions and views, so welcome seeing your views on the substance of the point9s0 being made by Kip.
richard verney October 11, 2015 at 5:35 pm Edit
Naw, I gave that up years ago. People like you obviously think that if you put on a jacket, it doesn’t warm you up, and if you add an extra blanket at night, it doesn’t make you sleep warmer. OK, fine, got it.
So I stopped using it, although to me that’s just semantic silliness.
Anyhow, I doubt that you can find any recent example of me saying that.
Now, note that I made that change and stopped using that term just so folks would NOT discuss my word choices, but would instead discuss the ideas … exactly the course of action that I recommended for Kip, and that I have been repeatedly rubbished for recommending.
It’s not a question of whether I can understand it. I can. Most anyone who knows what an imaginary number is can still understand Kips claims.
But it doesn’t advance communication to assign a clearly defined mathematical name to something completely different, as is proven by the amount of electrons we’ve already wasted on the subject.
Which was my point.
Regards,
w.
Kip, here are my further objections to your work.
First, you’ve re-defined “imaginary number”, but it is not clear what you’ve redefined it to mean. For example, you say:
I’m sorry, but I don’t understand this distinction. Is every average an “imaginary number” in your world? It appears that you have defined an average as an “imaginary number” … but why?
And if an average is an “imaginary number”, then what about a trend? What about the error of an average? What about the first Principal Component? Are those “imaginary” as well?
In your world, it appears (but is far from clear) that everything but a measurement is “imaginary” … so that’s my first objection—no clear “bright-line” definition that would allow me to tell whether the result of calculation X is real or “imaginary”.
My second objection is that if averages in your world are “imaginary” … so what? Me, I call an average a “mathematical construct”, but that doesn’t make it wrong or useless as you seem to assume. We use averages in lots of ways every day. Businesses use monthly and yearly averages of their operations to make business decisions, and they don’t seem bothered by the claim that such averages are “imaginary”. The Social Security folks use the average of my best 35 years of earnings to determine my benefits, and they don’t seem to notice that the average is “imaginary”.
So that’s my second objection—I don’t understand what the problem is with averages on your planet?
My final objection is that often, the absolute value of the average is not meaningful … but the CHANGE in the average can be very meaningful. For example, nobody cares about the absolute average sea level height in New York. It’s generally not discussed in the slightest. We’re only interested in the the rate of change of the sea level—is it accelerating, slowing down, or staying the same.
And again, what difference does it make if the sea level in New York is “imaginary” or not? If it starts rising faster and faster, are you going to say “No problem, folks, it’s just an imaginary number, nothing to see here, move along”?
Me, I refer to such things as averages and the like as “mathematical constructs”. But this doesn’t mean that they are second-class mathematical citizens. To me this just means that averages do not exist in the real world, but are calculated values. Yes, there is no such thing as an “average temperature” or “average rainfall” … but California is in a drought, and the average annual rainfall is pathetic compared to its normal value.
Unless you think that’s just a coincidence, you’d have to admit that the changes in averages have meaning and value.
Best regards,
w.
Willis says:
…what about a trend? …the CHANGE in the average can be very meaningful. …We’re only interested in the the rate of change of the sea level—is it accelerating, slowing down, or staying the same.
That’t’s the critical observation. The trend is the most important metric, IMHO.
The alarming argument, made over and over, is that the SL rise is accelerating. They have to say that, because if the trend is not changing from the LIA (or worse for their side, if it’s starting to decline), then Occam’s Razor indicates that it’s natural.
That’s why I think we need a reliable, accurate method to measure the trend in sea levels. I know it’s not easy, and it probably can’t be measured accurately until we have several years (or decades) of measurements. But the claim has been made, and the only way to answer it is with verifiable measurements.
Finally, it seems that if the global SL was rising at an accelerating rate, there would be plenty of empirical evidence. But there’s not. In fact, a MSL marker carved into rock 174 years ago shows no change, and there are many other observations showing that the ‘acceleration’ scare is flat wrong.
I only partially accept that the rate of change is the important factor.
Let us assume that there is no rate of change but sea level rise at a uniform rate of 3 inches a year. We would have to adapt to that rising sea level in any event. This is so whether or not there is an increase in the rate of change.
Further, the rate of change does not necessarily confirm AGW. First, we do not know precisely how the change is made up; glacial melt, expansion of sea water, extracting water from below the water table and dumping it in the sea etc. Second, just because the rate of change increases, it does not mean that that is the consequences of manmade activity. It may merely be the result of natural processes (a rebound from the LIA, or even from the depths of the deepest lows of the current ice age), or it may be that the oceans are warming due to natural processes (less cloudiness, changes in solar insolation etc).
That said, I do accept that the absence in the increase of rate of change weakens the argument that sea level rise in of anthropogenic origins.
Richard V,
I agree with everything you wrote. It could well be all natural (meaning no human influence).
But the argument we’ve heard for many years is that AGW causes SL rise to accelerate. My point is that I don’t see any empirical evidence to confirm that. I don’t even see much real world evidence for a 3 mm rise.
Reply to Willis ==> (repeating what I said to your last comment)
Respectfully, you need to re-read the essay more carefully before commenting further. Only one other person (apparently a High Priest of Mathematics) misunderstood my point so thoroughly.
Taking quotes out of context is not a virtue — it is what makes conversations impossible.
I’d love to hear from you again when you can restate the overall point of the essay in your own words, using 100 words or less. If you are even close, then we can discuss the finer points.
Are you back from your extended trip?
Kip, to be fair, everyone takes quotes out of context. It’s how people converse; always referring to an entire article would be incredibly clumsy.
Willis has taught me one valuable technique: quote someone’s words. So now I start my replies by writing (for example), “Willis says…”, and then I paste in his words, verbatim.
That’s a really good way to avoid misunderstanding. IMHO, of course.
Reply to Willis and dbstealy re “Taking things out of context” ==> Looks like I’ll have to do a little teaching here:
“Out of context” and “taking out of context” is an idiom that has its own definition, I’ll give several from online dictionaries:
“to use only part of something that someone said, so that the original meaning is changed”
“without the surrounding words or circumstances and so not fully understandable” as in “comments that aides have long insisted were taken out of context”
“To report (something) without taking into account the context in which it occurred.”
and
Fallacy of quoting out of context
“The practice of quoting out of context (sometimes referred to as “contextomy” and quote mining), is an informal fallacy and a type of false attribution in which a passage is removed from its surrounding matter in such a way as to distort its intended meaning.[1] Contextomies are stereotypically intentional, but may also occur accidentally if someone misinterprets the meaning and omits something essential to clarifying it, thinking it non-essential.”
It is not a virtue, it is not a common practice (except in yellow-journalism). It is improper and creates a logical fallacy. Quotes taken from a larger piece of writing or speaking must be considered in light of the larger piece of writing and the general thrust or intent of it.
Honest.
Kip,
So what happens when someone wants to comment on something you wrote? They’re not allowed to quote what you said?
I’m not talking about distorting someone’s meaning. That is the fallacy, not simply responding to some particular comment.
dbstealey wrote;
“I start my replies by writing (for example)”
John Knight,
The quote was:
…I start my replies by writing (for example), “Willis says…”, and then I paste in his words, verbatim.
I was giving a hypothetical example of how I post a verbatim comment that I would like to reply to.
I’m not sure what you meant by your comment. Did you inadvertently post it too soon? Or maybe I’m just slow to understand it…
dbstealey,
I was just tying to demonstrate that just about anything can be misleading if taken out of context, and duplicating the exact words written by another is no exception.
John Knight,
OK, thanks for explaining.
That means the character of the person posting is very important. I’m sure Willis would never try to distort someone’s meaning by deliberately taking a quote out of context, and neither would I. Or you, or Kip. Most folks wouldn’t do that, because you can’t get away with it.
But I’m afraid that some would. After a few comments, we can decide if that’s what they’re doing, and disregard them. On the internet, it’s easy to see if someone is distorting a comment or quote. They don’t get very far.
But it’s not a logical fallacy to simply quote something that someone has written, and reply to it. If we couldn’t do that, conversations would become very weird and stilted; readers would have to try and understand what, exactly, anyone was replying to.
dbstealey,
Thanks for your thoughtful reply. I agree that quoting out of context is often fallacious on purpose when done, but there is the potential for someone to accidentally engage in such fallacy, particularly when the person quoted has indicated early on that they are not using terms in their customary manner. In this essay, Mr. Hansen opens himself up to much apparently valid criticism, if that initial proviso is not born in mind throughout . . and slips back into expectations based on the customary use.
dbstealey:
I think you know that I usually support your arguments and comments, but in this case I am writing to disagree with you.
You say
Sorry, but no. Most quotations are NOT ‘out of context’. As Kip Hansen says
An out of context quotation – either deliberately or inadvertently – misleads as to the message of the quoted article.
And you also say
Again, no. That is an example of the Appeal to Authority logical fallacy. The nature of a person presenting an argument does not – of itself – indicate the validity of the argument.
A person making or refuting a claim that a quotation is ‘out of context’ needs to explain why they think that. And ‘because X said it’ is not a valid explanation.
Taking your example as illustration, your opinion of Willis is not relevant to whether Willis has or has not quoted Kip out of context. You say your experience leads to you think Willis would not do that (at least, not deliberately). So what? My experience is that he has often done that to me and has refused to withdraw when his blatant error is pointed out (e.g. here). Neither your experience nor mine is relevant to whether Willis has or has not quoted Kip out of context. It is for Willis and Kip to explain their different understandings so others (including you and me) can assess whether or not Willis quoted Kip out of context.
Richard
JohnKnight and Richard Courtney,
Good points. I don’t want to argue about this because I save my arguments for the alarmist commenters.
I don’t disagree with anything, but I ask your opinion: do you think it is always wrong to quote a sentence or two out of an article like this, and reply to it?
dbstealey:
You ask me (and JohnKnight)
No it is not “always wrong” and I fail to understand why you consider I might think it is. I agreed with Kip’s explanation of ‘out of context’, and I quoted it, then I wrote
I stand by that.
Richard
Richard,
In that case, I think we are in agreement. As I wrote above:
“I’m not talking about distorting someone’s meaning. That is the fallacy, not simply responding to some particular comment.”
dbstealey
“…do you think it is always wrong to quote a sentence or two out of an article like this, and reply to it?”
Certainly not, but because of the provisional meaning the author is giving to the key phrase “imaginary numbers”, it is necessary to avoid presenting portions of the essay outside that provisional context he generated, to avoid quoting his words out of context.
The alternative to we readers/commenters being careful about that, would be for the author to include many repetitions/echoes of his initial distancing statement throughout the essay, such that it would be difficult to select a few sentences that did not make it clear he was not using the key phrase in it’s traditional math sense.
This sort of provisional space generation is very common in literature, and it would be quoting Mr.. Twain (for instance) out of context, if one quoted something he wrote as Huck Finn speaking, as though it was Mr. Twain writing his own thoughts outside the provisional context of a novel.
Mr. Hansen is doing something roughly akin to that, as I read it, in an attempt to convey a subtle truth about numbers that are often presented as though very definite and precise (imaginary), but are actually somewhat indefinite and vague when their origins are considered carefully (reality).
It’s an attempt at “counter-spin” to my eyes (not speaking in the classical mathematical or biological senses here ; )
Willis,
Just average the rainfall over CA with that of SC and problem solved. No drought and no flooding.
😉
Or if you live in a two roomed apartment, and one room is minus 100 degC and the other room is plus 140 degC, who would claim that you have a the perfect living conditions at an equitable 20degC.
Averages can often distort matters since you lose important information when averaging.
And what do the changes in averages tell you?
richard verney October 11, 2015 at 5:58 pm
Thanks, richard. Indeed they can. For example, the average human has one breast and one testicle.
Does that mean that we should eschew using averages?
No, it means that we should eschew using averages wrongly.
The changes in which averages? The changes in the average of my 35 most productive years tells me how much my Social Security benefits will be, and whether it’s worth working another year or taking retirement now.
So, you tell me the average you’re talking about, and I’ll tell you what changes in that average might mean … which may, of course, be nothing.
However, even the changes in apparently meaningless averages can tell us things. For example, what if after a while the average human has three-quarters of a testicle and one-quarter of a breast … what would that tell you?
Best regards,
w.
It is asking for trouble
Reply to vukcevic ==> Ah…if you review my total output here at WUWT, you’ll see that I sometimes like to stir the pot a bit, shake things up and poke the complacent with a stick to see what happens.
(Well, you better hope Apple doesn’t get wind of this, or you may be in for some serious trouble ; )
(Hmm . . i numbers. Judging by the reaction to imaginary numbers, you might have been summarily i shot, if you used i numbers instead ; )
Dang: I was going to comment about the HUGE UTILITY of imaginary numbers. With which we wouldn’t have PHASORS, for power systems analysis. Here’s a webpage and some pictures of “imaginary power” generators. http://www.tepcoegypt.com/Products-MV%20Reactive%20Power%20Compensation%20Banks.html
The key point here is THIS IMAGINARY NUMBER actually creates something USEFUL! And quantifiable. Unlike the “climate scientist’s” imaginary numbers which are…just that, made up and imaginary.
Reply to Max Hugoson ==> Alas, as I say in the first paragraph “This essay is not about the mathematical entity the imaginary number. I do think that an essay here about imaginary numbers of that sort would be interesting, but this isn’t going to be it.”
Maybe you’ll submit an essay here on that….it would be interesting.
You’ve got me interested, Max.
To understand how the graph I’ve shown above (it refers to magnetic fields and the AMO) is generated, go to the last animation in the link. There are two pointers in the graph yellow and orange. Assume that yellow is the solar magnetic cycle, and orange the earth core magnetic field. Further assume that the two pointers are rotating at different speed, ie. the phase marked as B is continuously changing, then by using slightly modified formula shown above the animation (exponents are changed to i*omega*t1) and (i*omega*t2) , and hey presto, the wave form is generated that so closely (wiggle) matches the AMO.
Apology. above was posted in wrong place, its proper place is further down, as re-posted
For those new to the ‘phasor’ terminology there is an easy to follow article Phasor Diagrams
Climate science shouldn’t shy away from this versatile tool. Static magnetic fields can be represented by ‘vectors’, but if they are harmonic (or idealised as such) oscillations then the ‘phasor’ is the next step. Two magnetic fields of such type are solar magnetic field and a component of the Earth’s field measured at surface, but assumed to be generated by thermal convection at the boundary of the earth’s mantle and the outer liquid core.
When the sum of two phasors is presented in Decartesian instead of the phasors’ polar system coordinates, result has an uncanny resemblance (correlation if you wish) with the N. Atlantic SST oscillation better known as the AMO
http://www.vukcevic.talktalk.net/AMOqi.gif
p.s.
It is hard to comprehend the power of imagination of Nikola Tesla, the inventor of alternative poly-phase electrical power systems (generator, transformers, motors etc) visualising all the physical processes involved, just with a few paper sketches, usually penned down after the whole idea was mentally worked out.
For beginners who got familiar with phasor diagrams in the previous link, this page is useful next step, with some nice animations.
http://resonanceswavesandfields.blogspot.co.uk/2007/08/complex-phasors.html#complex-rotor-animation
You will see references to the real and imaginary axis. In this case ‘imaginary’ doesn’t mean some is trying to fool the reader, it is a mathematical term to get over fact that there are positive and negative numbers (depending on your point of reference.
Here it is again:
To understand how the graph I’ve shown above (it refers to magnetic fields and the AMO) is generated, go to the last animation in the link. There are two pointers in the graph yellow and orange. Assume that yellow is the solar magnetic cycle, and orange the earth core magnetic field. Further assume that the two pointers are rotating at different speed, ie. the phase marked as B is continuously changing, then by using slightly modified formula shown above the animation (exponents are changed to i*omega*t1) and (i*omega*t2) , and hey presto, the wave form is generated that so closely (wiggle) matches the AMO.
vukcevic,
Thanks for these comments on real imaginary numbers ; )
I “think” I get what’s going on in this realm, but it’s hard for me to tell exactly what is the imaginary aspect(s) of what I’m trying to decipher/grasp. Would I be crudely correct to say something like; In “real” number-land, negative and positive values are not mirror images of each other, but in “imaginary” number-land, they can be?.
Mr Knight
My early experience of i-numbers is a bit too long for posting, you can find it here:
http://www.vukcevic.talktalk.net/i-numbers.htm
The requested URL /i-numbers.htm was not found on this server.
. . . Set phasors to chuckle ; )
Max says:
…THIS IMAGINARY NUMBER actually creates something USEFUL!
There are lots of mathematical discoveries that at the time appear to have nothing whatever to do with reality. They are considered oddities, which was the original perception of imaginary numbers when they were first discovered and proven.
But interestingly, just about every such mathematical discovery has real world applications. It seems mathematics really is the language of the universe. We may not understand a lot of what it is saying at first, but eventually it becomes clear. Even despite Godel…
☺
Mr. Hansen
Thank you for looking into and clarifying one of the important and controversial data issues. Despite the minor matter of the i numbers, your essay is appreciated judging by numerous comments. Thanks for the relaxed and polite way of dealing with the ‘nothing better to do’ detractors i myself included.
I agree.
It is important to expose the controversial accuracy issues of the “claimed” data.
Thank you.
Kip, I would still appreciate your answers to the issues I raised above. You’ve been more than happy to discuss my take on the “imaginary number” question in great detail, but unless I missed it, you haven’t said one word about the three specific and substantive issues that I raised in that comment … and now you are trying to declare that the discussion is over by adding a “Post Script” and walking away.
Well, perhaps it is over … but if so, you have not done your part. I asked specific questions which to date you have not been willing to answer.
w.
Kip Hansen October 12, 2015 at 7:30 am
One final comment about this part of your Post Script. Please stop blaming me for your word choice. I said “are we measuring an imaginary thing“. I said nothing about imaginary numbers, that’s all you, and your attempt to blame me for it is a joke. Man up and own your own choice of words, they’re not my words and have nothing to do with me.
w.
Last reply to Willis (really) ==> Good Grief! (h/t Charlie Brown).
Dang, that’s depressing. You didn’t strike me as a man who would turn and run rather than answer a few scientific questions.
Well, live and learn. I guess we’ll never find out what your bright-line definition of an “imaginary number” is that will let us determine what you were talking about, or what you have against averages or trends.
Before, I said that I’d asked questions that you were unwilling to answer. Now I see that in fact, I asked questions that you have categorically refused to answer.
Ah, well. I expect that kind of tactic from Phil Jones, not from skeptics. Like I said …
… depressing.
w.
Willis ==> I suppose it is depressing for you to discover that you don’t get to control every conversation, and that others, including myself, do not owe you answers. Bullying and blackmail-by-name calling make it even less likely that people will respond to you. You are often your own worst enemy.
Kip Hansen October 12, 2015 at 2:42 pm
Your idea, that I think I can control a conversation involving a herd of unruly netizens, is so far from reality that it’s not even wrong. NOBODY can control the conversation here at WUWT, that’s what makes it so interesting.
In any case, I said quite clearly what was depressing, and it had nothing to do with control. Let me quote it, since you didn’t do so and as a result you were just addressing your own straw man:
It was your moral cowardice that I found depressing, not anything about “control”. You had me fooled. I thought you were a man with some starch … foolish me.
In any case, I’m glad you’ve clarified the question of responsibility. According to you, it was my fault that you called what you were talking about “imaginary numbers”. And now it’s my fault that you refuse to answer my questions.
Is there anything that you take responsibility for, or is every bad thing in this discussion my fault?
Anyhow, I got it. You only answer scientific questions from people that you like and approve of. Or as Phil Jones famously said to Warwick Hughes, “Why should I give my data to you, when you’ll only try to find fault with it?”
Phil didn’t understand that that is exactly WHY he should have given his data to Warwick Hughes, because that’s how science works. It’s an adversarial game, and you are foolishly doing just what Phil did, refusing to act like a scientist because you don’t like the people who happen to be asking the questions. I don’t care in the slightest if you like me. It’s not a popularity contest, it is a discussion of scientific ideas. If you are a scientist, you’ll answer questions about your own work from anyone, PARTICULARLY those that you dislike.
Heck, my advice has always been, give your best ideas to your worst enemies, because if they can’t find anything wrong with them you’re likely home free.
Anyway, is that your final excuse for running away from the tough questions, blaming it on me because I’m such a big krool meany and I didn’t talk nice to you?
Or are you willing to put your personal feelings aside, act like a scientist, stop blaming me, and answer my questions?
Your choice … but don’t even think of blaming this choice on me. You’ve tried that twice already. Doesn’t work.
w.
Good Grief! More of the same — bullying and name-calling, name-calling and bullying.
So I take it you won’t be answering my scientific questions now, because you don’t like my style? Well, fair enough. I can be cantankerous.
But remind me again … what was your excuse for avoiding my questions back when I asked them, back when I was still a good guy, before you decided I was krool and heartless?
Why didn’t you answer my scientific questions back then? Click the link and read them. I was calm and polite in the asking. We could easily have avoided all of this. All I’ve done since then is ask you, again and again, to answer my questions.
You’ve decided that me asking you to answer simple scientific questions about your own claims, and then asking you again when you didn’t answer, and then again, and at the end, me calling you out on it when you finally flat-out refused to answer those questions, is somehow bad and wrong …
Let me suggest that you might want to re-examine your choices if you ever wish to gain any traction on this or any other skeptical website. That kind of evasion of scientific questions is the specialty of the climate alarmists, so you could probably make it fly over at say RealClimate or OpenMind.
But it doesn’t serve you well in scientific circles. Regardless of your reasons, even if your reasons are fully justified in your mind, even if the guy asking the questions is a total jerkwagon, you look simultaneously evasive, arrogant, and unsure of yourself when you refuse to answer scientific questions about your own claims. No bueno.
Best of luck, Kip. You seem like a decent man, and your refusal to answer simple scientific questions is a total mystery to me.
Regardless, and in all seriousness, I do wish you well and I sincerely hope you don’t get caught again in this kind of trap of your own making,
w.
Kip: (AKA “Pokey” ) (grin)
Another interesting essay and lively debate!
I’ve had no trouble distinguishing between the concept of imaginary numbers as you are using it, and the formal mathematical construct. But then, I tend to annoy both idiots and savants about equally.
Seriously? Your response is to ridicule what you call the “quality” my questions?
So you are now refusing to answer my questions because in your world they are “low-quality”?
Really?
Man, the last response I expected from my most recent comment was a new excuse for not answering them.
And that the excuse would be that I’m not going to answer the questions because I don’t like them?
It’s just a variation of the Phil Jones evasion, “Why should I give my data to you, when you’ll only try to find fault with it?”
Of course you don’t like my questions! Nobody likes to answer questions. We all think that we’ve explained things perfectly, and that if people would just open their eyes all would be obvious.
But that’s not the case. People have scientific questions, that’s the reality.
And I don’t get to go “Oh, Kip’s question is not up to my quality standards, no need to answer that one”. If I tried that BS on the folks who ask me questions of every level of quality, I’d get roasted for it, and rightly so. What do you think this is, one of those fake political interviews where the Russian President gets to approve of all the questions in advance?
THE NATURE OF SCIENCE IS THAT YOU DON’T GET TO JUST ANSWER THE QUESTIONS YOU LIKE!
w.
Willis ==> Saying “The quality of the “questions” speak for themselves.” is ridicule?
I’m sure you must feel your questions are exemplars of scientific enquiry.
I’m also sure other readers, if there are still any following this thread (which I doubt), will make their own evaluations of the scientific value or validity of your questions in their context here in the comments to my essay.
I just wanted others who may read your long series of comments here to know exactly what “scientific questions” you had asked and have been insisting that I answer.
Kip Hansen October 13, 2015 at 10:41 am
Kip, you still seem to think that you get to decide which scientific questions to answer. However, when someone does that, it’s no longer science.
Nonsense. I’d cited my entire comment containing the questions just above.
You, after giving us a very pompous lecture on what “out of context” means, merely selected from my nuanced and clearly posed comment, every sentence ending in a question … dude, that is the most out-of-context procedure I can imagine. Re-read your own definition above of out-of-context … then look at how you’ve mangled my words.
In any case, my point remains. You’ve given us a host of reasons for e.g. not giving us a bright-line definition of what an “imaginary number” is. I’ve gone through your post, and I can’t find any such definition. And without such a definition, I can’t answer the following kinds of questions, e.g.
Are all averages, no matter of what, “imaginary numbers”?
Or are just some special kinds of averages “imaginary numbers”?
Are trends “imaginary numbers”?
Does it matter if the variable being measured is intensive or extensive?
The average of 3 and 5 is 4. Does that make 4 an “imaginary number”?
The ocean height in point A is 3, and at point B it is 5. Is the average ocean height of 4 an “imaginary number”?
My income is 3 and my gorgeous ex-fiancee’s income is 4. Is our average income of 4 an “imaginary number”?
I continue to be mystified by your refusal to respond to a simple request to clearly define your own freakin’ term. You’ve redefined a common math term, “imaginary number”, to have some other meaning … but then you refuse to give a bright-line definition of or even discuss the new meaning.
I have constantly been offering you a chance to act like a scientist and, you know, discuss your own claims and define your own terms. I continue to be mystified that you don’t just take me up on my offer, instead choosing to dig your hole even deeper.
Kip, it doesn’t matter that you don’t like my questions or you don’t like me. If you refuse to answer scientific questions about your ideas, you’re not a scientist, you’re no better than Phil Jones, and you’ll never get traction on any scientific site.
With my best wishes,
w.
Willis ==> Maybe if you re-read the concluding summary:
and in reponse to you way above:
and
and
Please note that you are the ONLY reader that seems to have mistaken this essay to be about simple averages of everyday things.
You have worn out my patience, sir. Time for a little straight-talk:
Your attempts to bully others — in this case myself — into answering your mis-characterized “scientific questions”, which are based entirely on your own misunderstandings and, because of that fact, are the qualitative-equivalent of attack comments typically left by malicious clueless teenage-trolls, is typical of your behavior in the comment sections here at WUWT. Simply put, I do not like bullies and I do not feed trolls.
My apologies for being so blunt — maybe I should have just come out and said this at the beginning of your near-endless stream of nonsense.
[edited five minutes later to add an omitted emphasis to the first section — kh]
Mp. Hansen, I highly suspect flack is inevitable over this target.
Reply to JohnKnight ==> I fear you are right, sir.
Hi John
sorry about link not working.
Lets try again with this abbreviated version
On the x axis we have ‘real’ numbers with 0 (zero) in the middle , negative numbers to the left, positive numbers to the right.
Square of any number from either side of zero is to be found to the right hand side of zero, i.e. in the positive section, and consequently since none of them fall to the left of zero, i.e. in the negative section, we can not calculate sq. root of negative numbers.
This is a limitation of the one dimensional axis, implying that numbers in everyday use are simple and one-dimensional.
Since negative numbers also must have square roots we have to overcome this ‘minor’ difficulty. Let’s assume that numbers are more ‘complex’ than just one-dimensional, i.e. that they have second dimension, which we are not aware of up to now, so for a brief moment have to imagine it, thus so called ‘imaginary’ numbers.
It is much better and less misleading to just think of this new ‘complex’ idea as ‘complex’ numbers, encompassing both dimensions.
To represent this new two dimensional entity we need a two dimensional plane, where the old y axis is replaced with new i axis, which also has a positive and negative section. In this new system sq. roots of ‘real’ negative numbers now fall in the positive section of i axis.
In a two dimensional complex number, value of the first dimension is projected on the x axis and value of the second dimension projected on the i axis.
To make distinction between first and second dimension, we add an ito the value in the second dimension.
Hence a complex number is written as z = x+yi, if y=1 we just write i.
Fact is that all numbers are complex, but if the value in the second dimension is zero, i.e. y=0, then number is simple and called ‘real’ number, and if value in the first dimension is zero i.e. x=0, we have so called an ‘imaginary’ number
It would be far more appropriate for terms ‘imaginary’ and ‘real’ to be dropped all together, and just think of all numbers as the ‘complex’.
Note that to understand complex numbers we do not need to start with notion that i = sq. root (-1)
In complex numbers whereby value in either dimension is not equal to zero, are found all over complex plane. Using polar coordinates, instead Cartesian to describe complex plane it is a much neater and mathematically more efficient way of applying complex number calculations, I would compare it to changing from the Roman to Arabic numerals annotation. By doing so we automatically encounter periodic functions (sin & cos) in which case complex numbers become an essential tool for studying harmonic oscillations as found in mechanical, electrical & electronic engineering, radio communications, electro-acoustics etc.
Anyone familiar with vectors should have no difficulty in comprehending the notion of two dimensional numbers.
The above including number of graphics, was roughly content of an essay I had to write in the last term of my grammar school, subsequently I went to University, and had no problems (as encountered by many students) with the understanding applications using complex numbers
Thank you much, vukcevic, your patience is admirable.
It seems to me my impression was correct, though my question was perhaps a bit clumsy. I do see this “disregard” for the often apparent reality of “real numbers” as wise and quite useful, now. I always suspected as much, but it’s good to have a better idea of how this manifestation of freedom of the mind has born good fruit.
John, tnx.
It just emphases futility of the squabble further up the thread. English language has disadvantage since expressions as imaginary, differential, integral etc, are used in informal conversations as well as strictly precise mathematical formulations. In other minor languages, like mine own, colloquial terms are very different while those used in science are identical to the above ‘internationalised’ terms.
To paraphrase ‘most of maths is fiction, people who think that things that happen in fiction are not real, they are wrong.’
Mod, I apologize … but I’m entering this test reply to see if my “chemistry set and bullets” preprocessor is packaging the right codes to become the beauteous neo-HTML that this site accepts as input. Pardon me, please.
• bullet 1 with bold stuff
• bullet 2 with italics text
• bullet 3 with underlined points
CO + H₂O + heat → CH₄OH + stuff.
H₂SO₄ + 2 Fe → Fe₂SO₄
²³⁸U is fertile; ²³⁵U is fissile. A fact that could ‘blow you away’. LOL
And this¹ is a footnote²
— end of test —
GoatGuy
_______
¹ first footnote
² second footnote.
[Reply: Please use the ‘Test’ page for this. Thanks. ~mod.]
Kip Hansen October 14, 2015 at 7:14 am
Willis ==> Maybe if you re-read the concluding summary:
Thanks for the response, Kip. I’ve read that several times. Unfortunately, it doesn’t allow me to distinguish between real numbers and “imaginary” numbers.
Under the first part of your definition, if I understand it, if someone calculates an average (or some other “single number”) representing some factor of a complex system (say the average depth of the ocean), that does NOT necessarily make it an imaginary number.
However, under the first part of your definition, if they use that single number to cast blame, or if they compare that single number to some other single number, or if they use that number to generate funds or public sympathy, then it IS necessarily an “imaginary number”.
Unfortunately, that definition is completely contradicted by the final quoted paragraph of your summary. It says that “imaginary numbers” are not “real numbers representing real things”. Instead, “imaginary numbers” are numbers that represent “concepts that exist, on a pragmatic practical level, only in our imaginations”.
Now, in my understanding, the following numbers are NOT “real numbers representing real things”, but numbers that represent concepts that exist only in our mathematical imaginations:
• averages
• medians
• variances
• standard deviations
• skewness
• kurtosis
• first differences
• principal components
etc.
So by the definition in your final paragraph quoted above, all of those are “imaginary numbers” no matter what they are averages or variances of. They exist only in our mathematical imaginations—you can’t directly measure say variance in the real world.
So I fear that I still don’t have enough information from your definition to tell if a number is a real number or an “imaginary number”. For example, which of these are real numbers and which are imaginary numbers?
1. The average expenditures of my wife and myself.
2. The standard deviation of the rainfall amounts in Sheboygan, Wisconsin.
3. The average shirt size of the American male.
4. The median lifespan of people in Buenos Aires.
5. The average of one day’s worth of minute-by-minute measurements of the sea level measured in a stilling well in Galveston, Texas.
6. The average per-capita energy use graphed against average per-capita GDP (PPP values), categorized by country.
7. The average of the highest 35 years of your earnings, which is used to calculate your Social Security benefits.
8. The median human dietary requirement for Vitamin D.
According to the your definition, these are all, every one of them, not “real numbers representing real things”, but “imaginary numbers” that represent “concepts that exist, on a pragmatic practical level, only in our imaginations”.
But are all of them “imaginary numbers” in your world, or are some of them real numbers, and why?
Finally … so what if they exist only in our imagination? Why does this make them second-class numerizens, “imaginary numbers” in your terms? All of those are numbers which are actually used in practical, beneficial ways in the world. It is valuable to know how much Vitamin D people need. It is useful for the city planners in Sheboygan to know how much rainfall there has been in the past.
So I’m still very unclear about just what it is that distinguishes an “imaginary number” from a real number, and I don’t understand why you think it makes a difference.
My best to you, and thanks for pointing out the paragraphs you think define an “imaginary number”.
w.
I still don’t feed trolls … go beg attention elsewhere.
Kip Hansen October 14, 2015 at 12:17 pm
Oooh, you’ve invented a brand new excuse to avoid answering scientific questions.
Well, folks, there you see it. He refuses to give us a clear “bright-line” definition of what he is talking about. He refuses to discuss my scientific questions about his vague definition. And now we get yet another excuse for his not answering questions. This time I’m a “troll”. A peculiar kind of “troll”, one who is doing nothing but asking clearly stated scientific questions …
As to “begging attention”, what bizarre fantasy is that? My work attracts about a million page views per year … I have much, much more attention on me and my work and my claimed failings than most men would ever desire. I don’t need more.
Kip, I ask you questions because you’ve advanced a vague theory that somehow some numbers are good and some numbers are bad. It’s an interesting theory, but you haven’t told us how to tell the difference.
Me, I’m just trying to figure out which numbers are which on your planet. What’s mystifying to me is, why don’t you just answer the questions? Seems like if you truly believe in your theory, you’d want to explain it to everyone’s satisfaction.
I gotta confess, though … near as I can remember, that’s the first time ever that anyone has seriously called me a troll. Makes me think you’re not entirely clear on the meaning of the word. I’m asking real scientific questions for a real reason—you have not given us a clear definition of the central idea of your claims, your concept of the “imaginary number”.
Best regards,
w.
Mr. Eschenbach,
It’s kinda hard for me to imagine what the problem is for you, in understanding what I and it seems to me dozens of other people understood Mr. Hansen was trying to convey about the “rubberyness” of some numbers presented on the mass media as hard and precise.
If you hear the talking heads on the news saying; “Good news on the economy, employment rose a quarter of a percent again last month, beating expectations for the twelfth month in a row now.” . . Are you convinced anything particularly good has really happened . in reality-land? Don’t you treat such “numbers” as sort of illusory and highly suspect in terms of reflecting a significant truth beyond the speaker having those words in front of them to read?
I don’t see any reason to have a problem with the general idea Mr. Hansen seems to me to be presenting . . it seems almost self evident to me.
This may come as a shock to you. but I don’t trust Siants, the big lumbering institutionalized quasi Entity, any further than I trust the TV talking heads. That Moron decided CO2 in a pollutant, I’m not bowing to that Idiot. He’s not my Idol.
Mp. Hanson . . illusory? . chasing illusory numbers? My inner poet cried out in it’s tiny little voice; *Yeah, it’s the correct term, AND it fills the metaphor perfectly* . . What would you suggest I tell the little imp? ; )
Reply to JohnKnight ==> Thank you — I can not really believe that W.E. has so thoroughly failed to understand what you and the (probably) a couple of hundred other readers here managed to comprehend, most at first reading — albeit a few only after a little discussion in the Comments. He is a normally bright guy.
His dogged insistence that I answer his questions, each based on his own mis- or non-understanding of the topic, and the labeling them as “scientific questions”, has me utterly baffled. When combined with his bullying commenting style it does make for comments that are the qualitative-equivalent of those nasty attack comments normally left by malicious clueless teenage-trolls (although W.E. does have a better vocabulary than most of that ilk). Given this, I have finally decided to quit responding to him, in effect, applying the same policy I do with other trolling : Don’t Feed the Trolls.
Overall, his behavior in the comment sections here at WUWT, which often follows the pattern he has exhibited here with this essay, has always been a subject of some interest to me in a sociological study sort of way. Can’t say I’ve come to any conclusions yet though.
I do agree that “illusory” might be a better choice if I were to write about this idea again — but as I’m sure you’ve already realized, the word originally comes from responding to a question from Mr. E and I stuck with it in this tangential, answering essay — I tried to head off the expected objections — stepping on a few nomenclaturists toes — in the first sentence with almost zero success!
I have an upcoming essay about “The Single-Number Fallacy” (logical fallacy) following on to this general idea — don’t worry — I shall not call the Single-Numbers “imaginary numbers” (once burned…)
Mr. Hansen
I would advise not to use word ‘precision’ in the same sentence, else you are for lot of bother again.
Reply to vukcevic ==> You are probably right….but part of the Fallacy of the Single-Number is that these sorts of numbers are almost always proffered in a form that implies “great precision”…..which is a hefty percentage of what is wrong with them. For example, this from NOAA for August 2015:
Here the “precision” is claimed at 1/100ths of a degree C.
Now I understand why you are cautioning me, but I need to address this issue somehow, and would appreciate your advice.
I have seen references make this differentiation of defintion:
Is this what you are talking about?
How would you suggest I speak about this aspect of my posited Single-Numbers? Perhaps I should refer to them as being “proffered with claims of great accuracy” ?
Appreciate you willingness to help….
JohnKnight October 15, 2015 at 1:13 am
Thanks, John. I understood that part very well The problem lies in your phrase, “some numbers” … my question, the question that Kip has repeatedly refused to answer, was and still is, which numbers? Because if we can’t tell the good numbers from the bad, what use is his theory?
And his examples don’t help. For example, as to precision, he says:
This simply reveals his lack of knowledge of his subject. The SEAFRAME sea level measuring systems measure sea level routinely to that accuracy, by using a combination of an acoustic measuring system and a “stilling well”. This is a simple vertical tube placed in the ocean, with a tiny hole drilled in it.
http://newsimg.bbc.co.uk/media/images/44371000/gif/_44371966_sea_level_station_416.gif
As you can imagine, the level inside the stilling tube rises and falls with the tides … but not with the waves of passing vessels and the like.
Do I treat them as illusory? It depends on the particular numbers and the particular claim. Again I point you to the problem—merely saying that some numbers can’t be trusted doesn’t help us unless we have some method to distinguish between the real numbers and the “imaginary numbers” … and Kip has refused to answer questions about how he is making that distinction.
I don’t either, as a general principle … but as always, the devil is in the details. Merely saying some numbers can’t be trusted doesn’t help us in the slightest, unless we know which ones.
While I appreciate the example, this may come as a shock to you, but I have no clue who either Siants or “that Moron” are …
All the best,
w.
Kip here is an example that illustrates the “accuracy” issue you raise.
..
Suppose you had a measuring stick with marks at the 1-foot, the 2-foot, 3-foot, 4-foot, 5-foot, 6-foot, and 7-foot lengths.
…
Now using this stick, you measure 10,000 adult American males, recording each measurement to the nearest foot.
…
If you take the average of your measurement results, you’ll find that it comes very close to 5.8333 feet. (which is 5 foot, 10 inches)
…
In fact if you used 20,000 observations, your results would be even closer to 69.7 inches.
…
The standard error is inversely proportional to the square root of the number of observations. So your 1-foot measuring stick will get closer to the population average with a higher number of observations.
..
http://neurobiography.info/images/stats/se.gif
“Is this what you are talking about?”
Nope.
Term ‘Single precision numbers’ (not to be confused with precision of single numbers) refers to the method of digital coding of numbers in 32 bit byte
http://www.h-schmidt.net/FloatConverter/IEEE754.html
it is also referred to as Single-precision floating-point format
https://en.wikipedia.org/wiki/Single-precision_floating-point_format
Sorry, but I wouldn’t be able to help with the rest.
Reply to vukcevic ==> Yes, I get the “single precision number” angle — I’d be stepping into another cow-pie of nomenclaturists objections. I will avoid combining the words “single” and “precision” (along with “imaginary”).
Thank you.
Reply to Steve Jones ==> Thanks, Steve. Appreciate the example.
Mr. Eschenbach,
“Thanks, John. I understood that part very well. The problem lies in your phrase, “some numbers” … my question, the question that Kip has repeatedly refused to answer, was and still is, which numbers?”
You’re welcome . . And if I were someone who didn’t yet understand what you do about the “example” I conjured, how would you answer if they asked you what you are asking Mr. Hansen? …which numbers, etc?
Is not the question itself asking about something rather “illusory” and difficult to pin down with any single “equation”?
JohnKnight October 15, 2015 at 3:15 pm Edit
John, Kip refused to just give us a clear definition of what is and isn’t an “imaginary number”. He’s given us a number of excuses for his actions, but none of them included the excuse that the answer was too “difficult to pin down” … so please, don’t give him ideas for new evasions.
However, following your general thought, since he had refused to give us a general definition, I decided to see if we could put some bounds on it by asking him about specific examples. These examples would not be either “illusory” or “difficult to pin down”, to use your words. I asked:
Since he also refused to answer those questions, I fear that your objection that the concept is vague or hard to define lacks merit.
w.
Mr. Eschenbach,
“John, Kip refused to just give us a clear definition of what is and isn’t an “imaginary number”. ”
Sure, and I didn’t give you one either, but you understood me just fine it seems. And it seems to me Mr. Hansen is trying to sort of brainstorm here, with us, on how to best approach what he sees as a difficult to define problem, certainly in terms of CAWG propaganda weaponry, so to speak.
Why not give it a try, and see if you can’t come up with something more definitive, using that newscast example you seem to understand in terms of the numbers being illusory and suspect insofar as reflecting realistically on much of anything, despite sounding hard and precise etc?
As we start hearing monthly updates on the current temperature of the earth, in hundredths of a degree updates, what shall we say? How shall we respond in a way that helps more people know that’s just an imaginary thermometer in mommy Earth’s mouth ; ) or anything truly like that?
JohnKnight October 15, 2015 at 11:51 pm Edit
Sorry, I have no idea what you’re referring to here.
DEAR HEAVENS, WHY DO YOU THINK I PUT FORWARD THOSE EIGHT QUESTIONS???
Mr. Hansen is REFUSING to brainstorm. I’ve asked those questions to try to brainstorm about the issue, to try to narrow down what he’s talking about. Unfortunately, he will not even say which of those are real numbers and which are “imaginary numbers”.
You truly don’t seem to understand what’s happening. I am and I have been attempting to get to something more definitive. Unfortunately, Mr. Hansen flatly refuses to engage with me in any shape or form. He won’t define what he’s talking about. He won’t say whether my examples are real numbers or imaginary numbers. He just says he doesn’t like the questions … sorry, that cuts no ice with me.
Now, if you think you understand what Mr. Hansen’s “imaginary number” is, I’m happy to discuss it with you … so how about you answer the questions, and we can start from there. Here they are again:
Not a hard question. Give me your answers and your reasons, and you can do what Kip refuses to do, what did you call it … ah, yes, to brainstorm here, with us, on how to best approach what he sees as a difficult to define problem
Finally, you say:
Mmmm … given the response to this post, I doubt talking about an “imaginary thermometer” will get you much traction anywhere. To me, claiming that some measurement is “imaginary” is a horrible claim. The problem with the current global temperature records include but are not limited to:
• it is an average of an intensive property, which
• makes the uncertainty of the results larger than folks think, plus
• the temperature records have a high Hurst Exponent, which
• makes the uncertainty of the results larger than folks think, plus
• the records are short, spotty, irregular, and
• the number of records changes over time, and
• they contain confounding factors (UHI, land use changes, instrument changes, location changes) which
• have either been removed or not, and the removal may be accurate or not, which
• makes the uncertainty of the results larger than folks think
Now, any or all of those are perfectly valid messages to help people understand the meaning and importance of the thermometer records.
But telling folks “Don’t worry, it’s just an imaginary thermometer”? Man, I’d never try that on on. Saying it’s an “imaginary thermometer” has no real meaning. It doesn’t point to the real problems. It doesn’t further understanding.
And more to the point, it’s NOT an imaginary thermometer. It’s a real thermometer, just one with real and large problems.
My best to all,
w.
Mr. Eschenbach ,
“You truly don’t seem to understand what’s happening. I am and I have been attempting to get to something more definitive.’
Go for then, batter up, sport.
“DEAR HEAVENS, WHY DO YOU THINK I PUT FORWARD THOSE EIGHT QUESTIONS???”
To sabotage rational dialog, I think you’re “cointelpo”.
JohnKnight October 16, 2015 at 2:04 am
Thanks, John, but you have the metaphor backwards. I am pitching the questions, and I’ve invited Kip to bat. Since he refused, I invited you to bat … is this your form of a refusal?
Asking rational questions and trying to get a definition of mathematical terms is “sabotaging rational dialog”????
On what planet is that true?
w.
Reply to JohnKnight ==> I tried to warn you. The minute you give him any attention at all, even negative attention, he just laps it up and then goes on and on and on…more-of-the-same…bully and name-call … demanding answers to his inane self-proclaimed, mis-characterized, pseudo-“scientific-questions”.
He’ll never stop until you quit feeding him attention.
Kip Hansen October 16, 2015 at 9:30 am
Thanks, Kip, I’m glad to see that you are now coming up with a new excuse for not answering simple questions about your own claims. Now it’s because I’m a “bully” … ooooh, is poor little Kip getting bullied?
Kip, to “bully” someone you have to be able to threaten them. The classic example is from school, “Give me your lunch money or I’ll beat you up”.
Bullying is less than effective if the threat is “Give me your lunch money or I’ll … I’ll … I’ll do nothing to you.”
So if you are feeling bullied, perhaps you could let us know just what you think the threat is? Because as far as I know, I have absolutely no way to threaten you, and more than that, I have no desire to threaten you.
In other words, claiming oh poor me, now I’m the victim of krool bullying by the eeevil Willis is just one more in your endless list of excuses.
As to me going on and on, there’s a very simple way to stop me—ANSWER THE QUESTIONS ABOUT YOUR OWN CLAIMS, so that we can understand what it is you are talking about.
But noooo, you’d rather whine about being “bullied” than answer simple questions about your work.
Pathetic.
w.
Mr. Hansen,
Sorry I ruined your comment thread ; )
I heard ya, I was just doing a bit of “research” of my own.
I say he’s an imaginary troll ; )
Reply to JohnKnight ==> Weird, huh?
You can email me at my first name at the domain i4 decimal net if you want to share your research findings.
Cheers.
Yeah, very weird . . too weird for me to swallow. Like I said earlier; I highly suspect flack is inevitable over this target
Hi John
re chasing illusory: I vaguely remember reading somewhere:
‘Don’t believe everything you think. Thoughts are just inclusions of the mind’
Thanks for the suggestion, vukcevic . . I thought I heard; *Garbage in, garbage in . .* and lilting laughter . .
Very interesting set of comments.
Thanks Kip.
I’ll say….”very interesting” indeed.
You’re welcome.
Excuse the lateness of this reply. I initially thought that I would merely hint at this in my comment above but decided to elaborate.
The mean is the expectation of the distribution ie the temperature anomaly if there were no local factors creating a divergence from the climate mandated temperature. This equals the mode and the median for a normal distribution so irrelevant which you choose.
Do you really want the mean when the distribution is more skewed? If you think that the temperature is proportional to the amount of energy in the atmosphere around the station, then yes but air that’s falling warms without more energy input and cooling as it rises up a mountain. The minimum temperature is also highly dependent on the dew point so what is the justification to use the mean? The min and maximum are highly dependent on when cloud cover arrives over the station.
More importantly is the missing data. I’ve noticed that in the Aus records that the data that’s missing the most is mostly the very hot days. Such missing data can make a large difference to the annual mean but little to the median.
Why this fascination with the mean as an indicator when it physically has little meaning and will be affected more by dodgy numbers? Surely the median is a better indicator of what the climate does.
Median filtering in image process is used for speckle noise rather than blurred noise and surely that is the bigger problem in using minimum and maximum station temperatures (dew point, cloud cover, wind shifts).
Reply to Robert B. ==> In support of your view:
This follows the general idea behind my essay here — “average temperature” in the sense Marcia is using is an example of my “imaginary numbers” — a single-number proffered to us as the essence of a huge and vast complex dynamic physical system, an essence that does not exist in the pragmatic real world.
Thank you for stopping back and elaborating your earlier comment.
The CET is one of the more accurately calculated regional temperatures with the UHI corrections.
Here you can see daily maximum, minimum and 20 year averages all compared to the insolation.
http://www.vukcevic.talktalk.net/CET-dMm.gif
From the graphs I could make number of i/relevant observations, but there is no need to do so since most are self-evident.
Kip, thanks for posting Marcia Wyatt’s comment.
This illustrates a problem with averages of intensive variables like temperature. To remind folks, an “extensive” variable of a given substance like a litre of water is something like mass, where if you double the amount (extent) of the substance, you get twice the mass. But temperature is an intensive variable. if you double the amount of water, you don’t get twice the temperature.
Now, if you have a block of some unknown substance, you can get an accurate measure of the mass with one measurement. But you can’t do the same with temperature, because it might be hot at one end and cold at the other, and frozen in the middle where you can’t measure it.
This causes problems … but not insuperable problems. What the measurement of intensive quantities involves is:
• It means that you need more than one measurement to give you an answer.
• It means that the uncertainty of your result is greater than if you were measuring an extensive variable.
• It means that the more measurements you take, in both time and space, the more accurate your answer is.
However, it doesn’t mean that the answer is “imaginary” or useless. Statisticians long ago recognized the problem you and Marcia are referring to, and people have developed statistical tools to get value out of those types of measurements.
FOR EXAMPLE: In mining exploration, you are looking for an “ore body”, an area where there is a concentration of the desired item. Now, the average density of the ore in the ore body is an intensive quantity. As both you and Marcia point out, measuring an intensive quantity is difficult, and in addition, the location of the measurements influences the answer.
SO … in response, did people:
a) Throw up their hands and say it’s an “imaginary number”, or,
b) Develop methods for getting the best estimate of the true value for the intensive variable given the limited information available?
The answer, of course, is b). Statisticians developed things like kriging, and Kalman filters, and first-difference analysis, and optimum interpolation, to deal with exactly the problems that you and Marcia seem to think are insuperable.
In fact, although Marcia seems unaware of it. the value of the surface air temperature at any time is pretty tightly constrained by the amount of ground stations that are currently in place. The problem is not the number of stations, although more stations would help. But the main problem is that we can’t really trust the individual station time series because of undocumented moves, instrumentation changes, and the like.
These problems are somewhat obviated by the use of satellite measurements of the temperature. Of course it brings up its own problems, but the coverage is much better and the trends are not affected by UHI, station moves, or thermometer changes.
Now, here is the important point. All that any of these techniques can do is VARY THE UNCERTAINTY of the calculated result. In general, they can’t change an intensive variable into an extensive variable.
So suppose I want to measure the temperature of my swimming pool. I toss in one floating thermometer and tie the string to the ladder. I record the temperature every three hours for ten days and take an average. The average is 20°C. Is this an imaginary number?
Absolutely not, no way. It’s just a number with a large amount of uncertainty. We don’t know the temperature of the other end of the pool, or the bottom of the pool. Maybe the other end is always in shade. And the bottom is not warmed as much as the surface by the sun, but then the surface loses more heat through radiation and evaporation than the bottom, that kind of thing. So maybe the temperature is really 20° ± 5°C.
But I’m not satisfied by that, so I get 8 thermistors and I put them at the four corners of the top and the four corners of the bottom of the pool. In this case, I record the eight temperatures every minute and average them and I get 23.3° ± 0.5°C … is that an imaginary number?
Again, absolutely not. Again, it’s just a number with an associated uncertainty. In this case, as you’d expect, since we have more measurements in both time and space, the answer that we get is more certain. But that’s just the nature of intensive measurements—all we can ever get are estimates with associated (and usually largish) uncertainties.
So let me suggest that if you want to distinguish between real numbers and what might be called “illusory numbers”, here is a bright-line definition— illusory numbers are those without an associated uncertainty. They give the illusion that they are scientific, when in fact they are not.
A corollary, of course, relates to the importance of accurate estimates of total uncertainty. This is an entire branch of statistics, one that in climate science is often given short shrift, or wrongly calculated, or ignored entirely … but that’s another post entirely. My experience in general is that in climate science, someone who overestimates the uncertainty of their results is a rara avis indeed …
w.
Mr. Eschenbach.
I give you credit for trying to actually give a definition, and I hope you won’t react too negatively, but it seems to me you’re not yet quite getting the problem Mr. Hansen is attempting to find a way to deal with.
The problem, as I see it, is not about one such “illusory” number as you defined it there, but dozens, crammed together with “illusory” number glue, to arrive at a sort of “hyper-illusory” single number, presented as though the reading on a giant thermometer.
Sort of like what Mr. Mann did when he used different data sets to construct his infamous hockey stick graph, on steroids, so to speak. A whole slew of “Illusory” numbers as you defined them, blended together in “illusory” ways, and presented as though a single measurement.
Just my take . . .
An intensive property is independent of the size of the sample and the same in any part of it.
Willis, you’re imagining that the mean of max and min thermometer readings of atmospheric readings are intensive properties like density (quotient of two extensive properties – mass/volume) or concentration (eg. amount of solute/volume of solution). Your pool example is more like the mining example where carefully analysing many samples will give you a good estimate of total thermal energy divided by the heat capacity but thats a liquid, and a small quantity.
Strange considering what you wrote about SST being limited by evaporative cooling that’s exponential with temperature that you can’t appreciate that its not the same as concentration.
The average of subsurface ocean-temperatures is more like your idea of illusionary numbers.
Robert B October 18, 2015 at 5:14 pm
Thanks, Robert, but that’s only half true. An intensive property is independent of the size of the sample, but it is NOT the same in any part of it.
w.
The average of concentration is not an intensive property unless the sample is homogeneous. I know that we tend to call the total amount of one component divided the total amount of all components in the system an average concentration but its not the average of the densities of the samples except where the whole sample is analysed evenly, in which case you might as well add up the amounts and then divide.
Sorry the above was written in a rush (and so is this). Σmi/Vi /n ≠Σmtotal/Σvtotal. Think about what you need to do to get that average to be meaningful and can you do it with SST and/or min/max measurements 2m above the ground. You need to consider more than how much thermal energy in the mass of air is represented by the measurement and if the uncertainty for the 1km3 around the station is most likely ±1°C, whats the point?
(fingers crossed that the html come out OK).
≠m(total)/V(total). I’m going to stop and let someone else convince you.
JohnKnight October 17, 2015 at 9:24 pm
Thanks, John. If you could give an example of such a “hyper-illusory” single number, that would be useful.
All the best,
w.
Mr. Eschenbach,
Not with certainty . . I’m trying to give you my impressions, as a layman, observing the discussion. Are there any numbers at all that you would consider beyond plausible, like monthly global temps in hundredths of a degree?
JohnKnight October 18, 2015 at 3:43 am Edit
Thanks, John, but without examples it’s very hard to respond to your claim that there are “hyper-illusory” single numbers.
The problem is not the number of decimals in the answer. It’s the lack of accurate estimates of uncertainty in the answer that makes it illusory and problematic. Climate science in general has forgotten that science without error bars is not science at all …
All the best,
w.
Willis,
Well, isn’t the entire IPCC ~CAWG is settled science~ claim, an assertion about such a hyper-illusory number; 100% chance it will happen (if we don’t cough up trillions of bucks ; ) with lots of error bars involved along the way to reaching that certainty conclusion? The error bars don’t seem to have played much of a limiting/reality check roll in coming up with that (to my mind) hyper-illusory number.
Isn’t that 100 a good example to your mind, of what I tried to “define”,?
JohnKnight October 18, 2015 at 2:10 pm
John, how would I know? The idea of a “hyper-illusory number” is your own, so I don’t have a clue what the definition might be. That’s why I asked for examples …
w.
Willis,
” The idea of a “hyper-illusory number” is your own, so I don’t have a clue what the definition might be.”
?? . . I wrote what my idea is, such as it is;
+Sort of like what Mr. Mann did when he used different data sets to construct his infamous hockey stick graph, on steroids, so to speak. A whole slew of “Illusory” numbers as you defined them, blended together in “illusory” ways, and presented as though a single measurement.+
“That’s why I asked for examples …”
I gave you three examples that capture the idea, I feel;
Employment numbers
Current global temperature
The certainty of catastrophic anthropogenic global warming
I’m really not sure what you want . . I mean, I’m just trying to give you my impression, not a testable scientific hypothesis.
JohnKnight October 18, 2015 at 11:43 pm
John, as you may note I was asking for examples of your “hyper-illusory single number”. The examples you list above were all given by you as examples of Kip’s “imaginary numbers”.
I give up, John. If you want to wave your hands and offer your “impressions”, there’s no meat there for me to respond to.
Having said my piece, I suspect I’ll just leave you to discuss these matters with Kip unless you do offer something with substance.
All the best,
w.
Willis,
“John, as you may note I was asking for examples of your “hyper-illusory single number”. The examples you list above were all given by you as examples of Kip’s “imaginary numbers”.”
The notion that you don’t realize I used your definition of illusory number to generate a different way to say imaginary number, is not believable to me.
If you’d like a meatier way for me to say that, just ask.
“Are there any numbers at all that you would consider beyond plausible, like monthly global temps in hundredths of a degree?”
Hi John
I think that the global temps are hopelessly inaccurate, but we can assume that intra-decadal trends may just about resemble to reality. I say that, because there is another global variable (geomagnetic poles–dipole) having similar trends, but it leads the GT by about a decade
http://www.vukcevic.talktalk.net/GT-GMF1.gif
Do I have an explanation? Not for a time being, that I would be able to fully justify.
Yo, Vuk (hopefully not a derisive term in Euronese ; )
That is interesting, and I find the “electric universe” (as I’ve heard it called) explanation(s) more and more plausible as I learn more about this conceptual framework . . all these cycles pulsing and swirling everywhere . . one wonders why such things wouldn’t effect our climate(s).
(assuming we have climate . . some apparently deny it ; )
Dr. Svalgaard from Stanford university, made an attempt to invalidate it, but inadvertently confirmed the hypothesis
http://www.vukcevic.talktalk.net/MV-DrS.gif
Svalgaard – yellow and brown lines, directly superimposed by Svalgaard on the existing
Vukcevic graph – blue, green and red lines.
To be fair correlation fails before 1850, but then, we do not have much of reliable instrumental data of magnetic intensity before 1850, Gauss invented magnetometer in mid 1830’s, and was not in wide use until decade or two later. There is no need to comment on accuracy of global temps before 1850,
p.s jo – pronounced ‘yo’ means ‘good’ in Hungarian
Impressive stuff, Vuk, thanks for elaborating. Very interesting.