NOT Off-topic: Mathematics vs. Science

By Ctein

For the holidays I usually give you all a column that delves into some far-flung aspect of science. This time I'm waxing a little more philosophical, but it is most decidedly not off-topic—it sprang out of, and directly relates to, photographic discussions here and abroad.

Here's an important difference between mathematics and science. Mathematics is fundamentally an intellectual exercise. You start with a set of postulates or axioms and you see how far you can run with them, creating an extensive and self-consistent (more or less) system of theorems and proofs. It's all in the logic and it's all internal to that system. The system you create doesn't necessarily relate to the real world nor to any other mathematical system you create. It's a beautiful entity unto itself.

Now, there are some wonderful mysteries about this. Many mathematical creations that were created purely as an intellectual exercise later turn out to be of real practical value. Sometimes it takes decades for the practical value of a particular intellectual edifice to be revealed. It is as if you arranged gears and cogs and levers on a table purely on aesthetic and compositional principles and 20 years later someone realized that would be a great working machine for their task.

Another is that deep connections sometimes turn up between an entirely different and apparently independent mathematical creations. Both of these things create wonderful arguments for mathematical philosophers having to do with whether mathematics really is just an intellectual creation that happens to be useful or if it's a genuine discovery about a physical reality. We're not going there today. The main thing to realize is that in mathematics connection to the real world is not a necessity, just sometimes useful.

Science is all about the real world. You can construct the most elaborate and apparently ironclad logical analysis, based on indisputable first principles, and if a laboratory experiment (properly done, of course, and that's a lot harder than most photographers realize) disagrees with that analysis, you have to toss the analysis out. It doesn't matter that you can't figure out what's wrong with it, the real world has told you something is wrong with it.

Here's how this ties into photography: lots and lots of would-be photographic experts treat photography as if it were mathematics. You see this most especially in discussions of the "quantitative" aspects of the craft like sharpness, resolution, diffraction, depth of field, exposure ranges, etc. I call them "would-be experts" because they never subject to their analyses to testing. They then issue pronouncements about how digital cameras cannot match the resolution of film, or how you simply can't stop down a lens beyond a certain point without obvious and visible degradation, or that depth of field does this that or the other, or...you get the idea.

These are the kind of pronouncements that folks with real expertise and experience debunk regularly. Said debunking is regularly disbelieved by an unfortunate fraction of the populace. Why? Because they are entranced by the theory and the analysis.

If someone has produced an extensive and learned discourse on the effect of diffraction on image resolution, but real-world experience and experiments show that the photographs don't behave that way, many people still feel that it needs to be explained where and why the logic is wrong. They believe that the argument really isn't settled until that is accomplished.

Sorry; that's not the way science works. That's the way mathematics works. If you create a theorem and a proof in mathematics and I say it's wrong, I have to show where the logic fails. If you do the same thing in science, all I have to do is show that real world results do not correspond to the intellectual edifice. It's not my job or problem to figure out why the edifice falls down. It's the responsibility, solely, of its creator.

Those are the words of wisdom I want to leave you with this holiday season. Photo-tech is science, not mathematics. Data (emphasis: good data) always trumps logic and theory.

•     •     •

Note: Further breaking my vow that I was going to stay off-topic until I got caught up through the new year, next week I'll be tossing your way a peculiar photographic puzzle I've run into. Then maybe I'll get back to being off-topic. Or not. I did warn you about "shinies," didn't I.

Ctein

Trained in physics, experimentalist Ctein (it's his only name, and it's pronounced "kuh-TINE") has been writing for photography publications for four decades. His weekly column for TOP appears on Wednesdays.

Original contents copyright 2012 by Michael C. Johnston and/or the bylined author. All Rights Reserved. Links in this post may be to our affiliates; sales through affiliate links may benefit this site.
A book of interest today:

(To see all the comments, click on the "Comments" link below.)
Featured Comments from:

Miserere: "My father-in-law is a mathematician at a prestigious American university; his field of research is differential geometry, and his particular work has no practical applications. He is fond of saying, 'reality is overrated.' As an astrophysicist, I usually respond, well, crap, reality is all I have to work with!

"I see parallels here with arguments in photography that I see happening every day on photo forums. My favourite is how big a print one can make from an x-megapixel image. I have wonderful 30x20-inch prints made from 10MP images, whereas the Internet Sages tell me (and continue to misinform others) that those files can only be printed up to 13x8.7".

"I hope they read your column, Ctein, and put away their damn calculators and just make a print already."

Edd Fuller: "Seems like a lot of people believe the translation of Descartes' 'Cogito ergo sum' is 'I think, therefore it is.'"

Doug: "Even after years of it, I'm still amazed when someone posts a question on a photography forum that they could've answered for themselves with a few trial snaps. Digital is virtually free, folks. Do the experiments.

"See also this xkcd comic."

Zalman Stern: "In my experience, there is little disagreement between good mathematical analysis of photographic systems and their performance in practice. Doing a correct analysis without measurement is generally impossible unless one has the full set of design parameters for the lenses, sensors or film, camera mechanicals, etc., but the state of the art in modeling is up to the task of predicting reality reliably. E.g. lens correction profiles can be generated from design data without measurement or computed from calibration test shots without design data and both methods produce good results. (And yes, in both methods there is often some hand tweaking involved to fix problems from the correction model not being accurate enough.)

"Diffraction is very well understood and can be modelled mathematically to high precision.  Such analyses generally show a very gradual falloff in resolution, which matches experience. There are plenty of reasons one might choose to shoot at slower apertures even if they are less optimal in resolution at the plane of focus.

"A certain contingent of pundits around digital photography have made diffraction an issue, largely on the basis of overestimating the value of attempting to exactly optimize resolution. I suppose if you've spent $50k+ on a medium format digital back, any photo that doesn't exhibit exquisite sharpness comes as an affront to one's sensibilities and absolutely requires theoretical explanation. After all, every pixel is sacred. This is not about the disconnect between mathematical theory and photographic reality, it is about promotion of half-baked analysis in service of equipment bias.

"All that said, when it comes to advice on how to actually practice photography I have a pretty simple approach: I look at the person's photos. If they have work that I can learn from, I solicit their advice. Otherwise not. Neither mathematical analysis nor experimental technique is all that relevant."

Mahn England: "I'm an Architect by training. At University we were introduced to the applied sciences of structural engineering, acoustics, and lighting. In 30 years of practice since graduation I have never had to calculate a bending moment, reverberation time, or the spectral output of a luminaire. But as I have found also with photography, others are trained specifically to bother with these important things so that I can get on with what I do knowing that I will get somewhere near the mark...designing a building or taking a photo. To paraphrase the Roman architect/engineer Vitruvius: Firmness, Commodity and Delight...in equal measure."

John Woods: "Sir Arthur Eddington, who provided experimental tests of Einstein's theory of gravity, was interested in the philosophy of scientific method. He famously said: 'Do not trust a theory until it has been supported by experiment; and do not trust experimental results until they have been supported by theory.'"

Comments

[Richard Tugwell]

Dear Mr Ctein

You say

"lots and lots of would-be photographic experts treat photography as if it were mathematics. You see this most especially in discussions of the "quantitative" aspects of the craft like sharpness, resolution, diffraction, depth of field, exposure ranges, etc. "

Woud you include yourself in this group? Seems to me that most of your discussions are about just this very thing - "the quantitave aspects"

[Craig]

Nice essay. It is irritating how many people online go around repeating things they've read but clearly have never actually tested for themselves. The diffraction issue is a good example. One sees charts in reviews showing that sharpness drops off after f/8 even on full-frame cameras, however when I tested it myself using a Canon 5D Mark II and a very sharp macro lens, I found that even in a 100% view, the difference in sharpness between f/8 and f/16 was minimal; it was only at f/22 that the softening effect of diffraction was really noticeable, and in a "fit image to screen" view, even that was doubtful. With the higher resolution of a Nikon D800, it's possible that the results would be different (if sensor resolution is really the limiting factor), but even then the question remains whether a 100% view on a 100 ppi monitor is really the most appropriate test case. How many of us ever really intend to create monstrously huge prints that will be viewed up close with a magnifying glass?

[Bill Pierce]

The nice thing about digital is you can often conduct an experiment more quickly than you could with film. The bad thing is that given a new photo toy, in my enthusiasm I often confuse quick and incomplete.

[Marc Rochkind]

This is an excellent and useful distinction between mathematics and science. It might be extended to included another related discipline, engineering. ("Mathematics vs. Science vs. Engineering")

You say the "Science is all about the real world," which is true enough, but, more specifically, it's about gaining an understanding of the real world.

Engineering is about designing and building things that operate in the real world. Much of it is based on science, but some of it is also based on empirically-gained knowledge that might not yet be explained by science. (For example, information on the strength of fasteners based on failure experiments rather on materials science.)

And, as long as I've added engineering to the mix, why not add another discipline: Art?

I think of photography as a combination of engineering (in turn based in part on science) and art. For instance, there are some engineering rules that can be used to arrange basic portrait lighting. The science behind these rules is perhaps interesting, but the rules themselves can be practiced effectively without knowing the science, if any, behind them.

The actual portrait is taken with lighting that might follow the engineering rules closely, or not at all, or something in between. That's art.

[Andrew]

I'm afraid that the analogy doesn't do justice to either mathematics or science. Science cannot be separated from mathematics. An experiment can do nothing but support or discredit a mathematical model.

Forum-style speculation may not be tied to reality, but it has none of the rigor of mathematics.

[Peter]

As Winston Churchill once said, "No matter how good the strategy, it may be useful from time to time, to check what is actually happening on the battlefield."

[bill]

LOL, a favorite topic of mine, thanks.

A postulate; maybe your logic and math is sound but you are measuring and evaluating the wrong data.

b

[Salvador Moreno]

Brilliant post. Sometimes the trees (the measures) don't leave us see the forest (the art).

[Damien Martin]

I work in a research field in physics, and I very strongly disagree that you have to toss the analysis out, or not care where the error is, when doing science. I also disagree that the responsibility to find the error lies solely with the creator.

If a theory is logically consistent, and a properly done experiment is performed (and analyzed correctly) disagrees with the the theoretical result, then you have discovered that one of your initial postulates is incorrect. This is useful information about the real world! You can try and design future experiments to determine which of your postulates was wrong.

If, on the other hand, you made a sloppy error then there is little to be learned except that the theory cannot even stand mathematically.

It is not even true that we say only the creator of this theory is responsible. If we found a result that conflicted with Einstein's general relativity, people would try and construct new gravitational theories that took different postulates from Einstein. In fact, this has already been done in the form on MOND and TeVeS theories (whether the data supports these alternatives or even requires different versions of gravity is a different and long conversation). The scientific community would not take the view "that's Einstein's problem, not ours" even if Einstein was still alive.

To take an example that was in the news: the Higgs boson is predicted as part of the standard model of particle physics. It is clear from our theory that "something" needs to play its role, even if Peter Higgs suggested the very simplest thing it could be. There are thousands of papers describing possible types of Higgses, or even "Higgs-less" models that have other consequences such as "non-perturbative interactions". If we had found nothing new -- a possibility that our current understanding of field theory claims is not an option -- we would have to carefully analyze our mathematics (axiomatic field theory is not as mature as we would like) and analyze our postulates for field theory.

I will agree with your claim that the primary difference between science and mathematics is that science is beholden to the real world. The rest of the commentary about what scientists do when they find disagreements does not, however, match with the reality of how the scientific community operates.

[JimF]

Hi Ctein;
Mathematics is a science. You may not think so but many mathematicians do unless of course you an ardent admirer of early Karl Popper or you are a holdover to the idea that mathematics is one of the seven original liberal arts (but then grammar was one of those original seven). Indeed in many areas mathematics does make use of the scientific method.
However I do follow your point in your article. But I think the distinction is best described not by science or mathematics but by observation that photography, like painting or sculpture may use mathematics, it is fundamentally an art ( the expression or application of human creative skill and imagination, typically in a visual form such as painting or sculpture, producing works to be appreciated primarily for their beauty or emotional power - Oxford dictionary). Like said sculpture or painting. And to carry it further, painters or sculptors don't routinely discuss the math they used in the creation of their art (although it is at times interesting)they discuss the finer points of the art. Which I think is your point. The technical aspects of the camera are not the art anymore than the fact the painters brush uses a particular hair or glue to hold the hair to the brush. The photograph is the art like the painting is the art. The camera is to the photograph as the brush is to the painting.

[Terry Crooker]

Thank you!

[John Krumm]

Sure, just when I was about to publish my grand photographic unification theory, you have to cast doubt.

I enjoy a good clean test. Currently Roger of Lens Rentals makes my favorites with his multi-lens comparison charts. I also enjoy a good narrative mixed in, the classic field testing notes. But I really don't like long threads at DPR arguing that we should all be shooting with "uni white balance" if we want to maximize expose-to-the-right benefits.

[Nik]

Great article, Ctein. I enjoyed reading this.

[Peter Morgan]

If "real-world experience and experiments show that the photographs don't behave" the way that a theory expects, surely there's room for a discussion (or even, if the mismatch is large or is considered significant, an argument) about how to change the theory (or the acknowledged scope of the theory) or about how to recalibrate our interpretation of the experimental results (in extreme cases by expending much effort to discover why a neutrino experiment was "bad") so that the data and the theory again behave enough the same for the purposes of engineering? It may be that it's more often that a theory is bad than that a dataset is bad, but, as you say with emphasis, there is "good data", which implicitly acknowledges the existence of bad data, and to a lesser extent of causes of bad data, bad photographs and causes of bad photographs, etc.. The development of better theories, which to some extent determine how we characterize data as good data, is ongoing. The development of very good theories, however, takes decades and is very difficult ---there are few people as good at it as was Maxwell--- no less than creating very good experiments is very difficult. I think it takes some talking to get the work done and to make it accessible to people who might find it useful.

[HomoSapiensWannaBe]

Interesting post, and you could go one further with Math -> Science -> Technology.

One reason I quit working in the high-end audio equipment business was the tiresome psychological tricks that many people play that affect their objectivity in evaluating sound and making rational equipment purchases.

The "more bits are better" argument comes to mind. No matter how many times objective blind testing demonstrates that practically no one can consistently tell the difference between 16-bit/44.1khz and higher bit rates such as SACD and 24-bit/96khz PCM, TRUE BELIEVERS insist there is a difference. Not just a little difference, but a huge difference. These audio geeks often lace their praise for the perceived superiority of higher than CD bit rates in lofty terms that would make a jaded wine critic reach for the thesaurus.

I continue to be surprised at just how good my Canon S100 images can look at low ISOs when compared to a Nikon D600 with 24-85VR lens. My mind tells me that all that extra resolution, dynamic range and pixel size should make a giant difference in any situation -- and sometimes it is clearly better -- but reality the S100 looks as good for many shots and situations. So much, that I'm not reaching for the D600 as much as I thought I would. Not exactly buyers remorse, but a lesson learned... I hope!

[SeanG]

This response may be off topic but...

A few years ago I had the pleasure of listening to Roger Penrose's lecture at the Institute for Mathematics and its Applications — http://www.ima.umn.edu/public-lecture/2004-05/penrose — entitled "Does Mathematics Rule the World?"

The subject was the possibly extricable link between science and mathematics.

[Hugh Crawford]

To quote Yogi Berra - "In theory, there is no difference between theory and practice. But in practice, there is."

Part of the problem is that most photo theory is about a particular part of photography but does not take into account the entire process or "stack" to borrow an I.T. term.

A student with a camera lens that has barrel distortion and vignettes also has an enlarger lens that has barrel distortion and vignettes. Student replaces crummy enlarger lens with Componon-S and wonders why prints look so much worse.

So one may know all about how a lens "should" behave at all magnifications, but not much about what happens when you use it to take pictures at distances between one meter and 500 meters, in air, with one sort of light or another, of objects that may or may not be flat, on film or a sensor that might be flat with a sensitive layer of no depth or maybe not at all flat with a deep emulsion that scatters light internally.
Then film gets processed in chemicals that might enhance the micro-contrast and apparent sharpness with lower overall contrast or do just the opposite. Depending on how the tank is agitated and how tall you are (I'm not kidding)

The effects of development technique confusingly look a little like the effect of over or under corrected spherical aberration. Or how clean the lens is, or the coating. And all of these factors scale differently by size of the print and size of the camera,

And with film you then run it through an enlarger with it's own set of artifacts , and condenser enlargers have so many things that can change image quality that it makes my head hurt.

Did you ever hear the one about the darkroom worker whose print quality went to hell when he stopped smoking? (I'm still not kidding)

All this stuff has analogous factors in the digital world except for ones about the size of the photographers and their smoking habits.

Given all this whack-a-mole photographic process interaction it's easy to come up with rules of thumb that work for one photographer in the context of their total working environment that are wildly at odds with theory.

[Drew Medlin]

"If it disagrees with experiment, it's wrong."
-Feynman.

[Mark]

While I can't disagree with anything said here, I reluctantly admit I am a dolt when it comes to math. And although photography may be the eye's ability to see and then capture what's seen, my complete lack of math skills (yes, I can add, subtract, multiply and divide, but that's it, can't do fractions at all) makes photography hard for me as I don't really understand stops and how many clicks of the dial equals how many stops, etc. Wish I did.

[Mark Alan Miller]

Some sciences are so tightly bound to mathematics at a theoretical level that those doing research in the field can't just say, "look, this result shows everyone is wrong" without being prepared to suggest why.

I used to work for a medical research institution, where many of the researchers needed nothing more than arithmetic, for very routine stats. Advanced mathematics was irrelevant to their work. The guys working at a molecular level needed fancier math, of course, but they were in the minority.

[David Aiken]

If I had to try to sum up where I think mathematics and science differ in their relationship to the real world in a single generalisation, I'd say that mathematics describes and science explains. Neither makes the world behave the way it does but both can be useful, in different ways, in helping us do things in the world.

And, of course, it is possible to describe something that doesn't exist, novelists do that all of the time, but it's not possible to explain something that doesn't exist even if you can explain why it doesn't exist which is a different explanation entirely.

Still, it is possible to do a damn good scientific experiment without relying on anything other than observation, Galileo did that with a couple of balls of different mass and proved an important point in the process. Getting as much mileage as possible from that point does require a bit of mathematics and it is worth remembering that t's a lot easier to describe something without giving an explanation at some point in the process than it is to explain something without giving a description at some point in the process.

And all of the above is a generalisation so it's governed by the "it's the exception which proves the rule" clause.

[Dennis]

JimF - I'd like to comment on a couple points you made. Photography, by & large, is far removed from art. Most of it is utilitarian and much of it lacks any more thought than one applies to heating a cup of water in a microwave oven. To the extent someone pursues photography in a more thoughtful manner, there's a balance of art and craft; more or less of one or the other depending on the photographer. But I also don't see much behind the old camera == paintbrush analogy. A paintbrush can't record more or less dynamic range. It doesn't restrict you to recording daylit scenes or limit the amount of detail you can record if you opt to produce a really big print (well, I suppose a very coarse paintbrush might). It doesn't affect your ability to focus on fast moving athletes.
But I still like the point Ctein makes, which is that real life results are far more useful to photographers than theory.

[Tregix]

"Physics is to mathematics as sex is masturbation."

R. Feynman

Joke apart, you are clearly an experimental physicist and not a theoretical physicist. What you describe as "would-be experts" are closer to theoretical physicists than to mathematicians. I would rather call them “theoretical photographers”. There are many of them on this planet. Much more than theoretical physists ;-)

Tregix.

[ctein]

My Dear Mr. Tugwell,

I am most impressed. You managed to read the entire essay without coming close to catching its message. Perhaps a case of too long, did not actually read? Well then, let me try it in more concise form: the problem lies not in discussing quantitative aspects of photography, but in treating them as mathematics rather than experimental science. Armchair theorists do the former. I do the latter.

And, not so by the way, I am widely recognized as a photographic expert and have been for decades, not a “would be-”… even if not in your ever-so-exalted judgment.

Let's stop the not-so veiled jousting and snide remarks, okay? They don't serve TOP well.

pax \ Ctein
[ Please excuse any word-salad. MacSpeech in training! ]
======================================
-- Ctein's Online Gallery http://ctein.com 
-- Digital Restorations http://photo-repair.com 
======================================

[The Grumpster]

Mathematics is the only pure truth that exists. Science may be pretty good at explaining the world around us, but the world around us may be merely a fictitious creation for all we know.

Even after the universe has expanded infinitely into nothingness and there is no space nor time left, mathematics will be still be just as true as it has always been.

[Jan]

Dear Ctein,
after thinking about it I am not really sure what you are trying to communicate in this article.

Whether or not mathematics is science or not is simply a matter of definition.
As far as I know, your distinction between science and mathematics is not commonly accepted. Some people associate science with natural science, i.e. physics, chemistry and so on with the exclusion of mathematics, but others do not, i.e. include mathematics.
Historically, I doubt that the exclusion of mathematics was ever commonly accepted, especially among the antic greeks, but I am no historian.

Furthermore, I doubt that (natural) science in the way you refer to it is 'about the real world'. It is only about the real world accessible to scientific methods.
Now, even if one resticts oneself to the latter (e.g. excluding a good novel), it is an inherently mathematical problem to define what this could possibly mean (in terms of mathematical logics).
Usually scientists solve that problem by restricting themselves to tiny parts of whatever the accessible world appears to be and simply ignore the mathematical implications.
Ironically, at times of Newtonian mechanics, the experimentally accessible world seemed much more universal than today.

Finally, even in mathematics, one does not have to disprove the n-th attempt of the quadrature of the circle. This is what it is all about.

Regards, Jan

[ctein]

Dear Edd,,

Oh, I like that!

And it has me thinking, it is a correct statement if you're talking about subjective rather than objective data. If you're talking about likes and dislikes, for example. And perhaps there's the makings of another essay in there, as some people tend to confuse the two. E.g., statements to the effect that such and such is [objectively] lousy, as opposed to such and such didn't work for them. One runs across it quite a bit in comments about whether a particular photograph, body of work, or camera is a “good” one.

~~~~~~

Dear Andrew,

In 1000 words or less, do I really do justice to anything? [grin] No deep end of the pool, I just dabble in the water and make pretty splashes.

~~~~~~

Dear bill,

Oh yes, the “you're not measuring what you think you're measuring” problem. It's a good one!

I have at times pondered the idea of some columns about how to run good photographic experiments, but I really don't think I'm up to the task. I don't know how to condense hundreds of hours of high end college training into a few thousand words.

~~~~~~

Dear Damien and Peter,

I don't disagree with you at all. I think we have a bit of pronoun trouble, that's all.

If MY experiment contradicts YOUR theory, it is not MY job as the experimentalist to show why YOUR theory is wrong. It's sufficient for ME to simply show that someone, YOU, the theory creator, or some other theorists, need to go back to the drawing board and figure out why it failed. This was the point I was making.

The problem you see in many photographic debates is that the “theorist” asserts in some form or another that is the "experimentalist's" job to figure out why the theory fails. Now, mind you, there's nothing wrong with the experimentalist wanting to engage in such games, and I frequently enjoy doing so. I like problem-solving. But it's not their obligation to; they don't own the problem in the conflict between data and theory, the theorists do.

Few people are good theorists and experimentalists, both. They are to be treasured. I tend to be much better at the latter than the former. If any of my articles give the impression that I'm a good theorist, it's only because when I'm really good at is doing the experiments to test the theory.

Incidentally, photographically speaking, Maxwell was a much better theoretician that experimentalist; read this:

http://theonlinephotographer.blogspot.com/2007/05/james-clerk-maxwells-big-mistake.html

pax \ Ctein
[ Please excuse any word-salad. MacSpeech in training! ]
======================================
-- Ctein's Online Gallery http://ctein.com 
-- Digital Restorations http://photo-repair.com 
======================================

[Marc Garrett]

Great article Ctein; thank you for sharing.

Reminds me of a Peter Norvig post from a couple years back "On Chomsky and the Two Cultures of Statistical Learning." http://norvig.com/chomsky.html. Here's the Chomsky quote that set it off: "It's true there's been a lot of work on trying to apply statistical models to various linguistic problems. I think there have been some successes, but a lot of failures. There is a notion of success ... which I think is novel in the history of science. It interprets success as approximating unanalyzed data."

I take it Chomsky would not agree with your definition of science. :-)

[Rod Graham]

I'm a cardiac nurse. If I call a doc about an abnormal rhythm I've seen on the telemetry monitor, the first thing he will ask me is: "What does the patient look like?"

Make those prints and SEE the actual results.

[Jakub]

The only actually useful (ie non-oneoff) result of an experiment is a testable hypothesis, which can be further tested and refined. It's like you've read part of Popper but ignored what makes science valuable and, you know, interesting, or more than "what I found out a little bit for myself."

That said, in my limited experience the most theoretical, physics-oriented photographers seem (almost uniformly) to lack artistic talent as defined by interesting photographs.

[Lenya R]

This is an interesting argument but, with all my enormous respect for Ctein I think that its representation of how a mathematical result is accepted or rejected is a misconception. Mathematical proofs are often very complex and it is nearly impossible for someone but the author to find an error (sometimes for the author as well). In reality, mathematical results are subjected to the same tests as in science - they have to make sense, and there has to be a reasonably clear reason why they are true, in order for them to be accepted. Of course, a single counterexample makes the result clearly wrong and nobody has to find the mistake in the author's argument in such case. But even if there is no known counterexample, to an extent, the main burden on the author is to explain why the result is correct, the proof is in some sense secondary. Sometimes an explanation comes much later than the formal proof but in that case the explanation is often much more interesting than the proof itself. It is said that any idiot can prove a theorem, it is much harder to discover a theorem.

[Will Frostmill]

Ctein,
Thanks, this is a great article. I'm inspired now to test the handful of things* I care about with my camera, rather than spend another hour poking at old forum threads trying to find an answer. (Not that there's anything wrong with that, - it does help to find out what questions have been asked.)

I began to think this way when I started setting up a slide copying rig and realized that I didn't have a theoretically perfect sensor and lens combination, but that probably didn't matter for the quality of the scan I needed. A working definition of "good enough" let me put together something that was in fact testably good enough.

Maybe you'd like to give some examples of how to construct testable photographic hypotheses that actually test what they appear to? I know how difficult it is to put together a decent experiment.

Will
*there's a lot I would like to know, but if I restrict myself to the cameras and lenses I actually own, there's only a few nagging questions left.

[Robert Roaldi]

Here's what I recently found out. I own an Oly E-620 that's not supposed to be too good at 1600 ISO, according to the interweb. I had never used it at 1600 ISO since buying it 4-5 months ago. So I took a walk outside one evening in the dark after a snowstorm and shot a bunch of pics at 1600 ISO and shutter speeds of 1/10 to 1/40 in really peculiar outdoor street lighting. I see noise on the screen but the 8x10s out of my inkjet look just fine to me.

[Tom Hill]

Very interesting post. Also, very interesting comments. I commend everyone.

Just an observation. For many, the merit of a theory or "stance" comes from the ability to conduct rhetoric, to be able to have successful discourse on the subject. Of course, someone who's schooled in language and communication might be able to "prove' things beyond the merits of their emperical evidence. The language becomes the basis of the proof. It's easy to see this throughout history.

I defintely don't want to say discourse or rhetoric is bad. I do want to say there is more to "truth" than simple rhetoric. Just because someone speaks well does not mean they have all the answers.

Cheers

[Al Patterson]

As a former Physics major, I am of the opionion that photography is art, not science or math...

[Dennis Ng]

Your later statements about the logic fault for Maths and the refutation for science criteria is agreeable. However, I was very hard to follow your earlier statement and I am just not sure.

Around 192x (deliberately not to check wiki as it is hurt), one guy has proved surprisingly that every math systems that is "bigger" than arithmetic would have clauses in the system that cannot be proved within the system. One interpretation of this is that you cannot have self-consistent maths. A shock to Russia/Whitehead and 50 yrs later me when I start my major in Philosophy with no one told me that my initial idea was disapproved in 192x. In other words, Math is not more-or-less consistent, it is math impossible to be consistent. It is just that what is foundation or really any foundation of maths is in doubt since then. The philosophy of maths is very fascinating but also meant that your use of that to fit into photography may be on a shaky ground that your thought. But may be you are right (if you adopt one school of thought among the three I knew of). May be you are not. No one knew what Maths really is. Drawing parallel from that is now impossible since 192x. Good luck to those who has not lost the innocent of their pic about Maths.

For science, whilst what you said post-Popper is right but can one also said that the Queen of Science like Physics (and Economics for that matter as Queen of Social Science) has another issues based on past experience. It is hard to disapprove these days for many Physics model (and especially the economists which has more than two hands). In fact, scientific development are more conjectured (and more social as Kuhn would said) than you think. One wonder how Earth is flat can be easily uphold for so long as like all your photography argument there. I hope you are not burnt in stick or house arrested like one famous scientist whilst you are maintaining the "truth". Even to his death, Einstein cannot believe God play dice even though he got his Nobel Price for QM. It is more social than you think and hence those people who argue DoF may be in a deeper pit than you can help them out.

Whilst I sort of agree with your assertion later as said, I am afraid I cannot follow your logic because of the philosophical background noise and bokeh is too distracting for the subject to come up.

[mark]

I think you're being too kind to the pundits who believe their opinions are somehow more scientific if they throw in a few numbers.

[Noons]

Indeed! It reminded me of an imbecile theory of a few years ago:
'The way to see if a scanned film image has "detail" is to resize it to half the size, then resize it again to original: if the ending size was markedly different from original, then the scan had lost "detail". If not, then the photo had no "detail" to start with!'

The whole lot was done starting from a jpg file! Now: what is the most distinguishing feature of jpg compression? Hint: it's called "LOSSY" compression, for a VERY specific reason...

So what happened? Well the end product had the same size as the starting one.
Of necessity! Thank maths for that!

Repeat the exercise with tif files of the SAME scan and the result is night-and-day different.

Did that stop any of the imbecile lunatics using the process with jpgs from caliming it as "proof of lack of detail in film scans"? Of course not!

[Mark M]

Yes, but…

Theory has its place because you don't always have the ability to test every possible situation and science, for all its wonder, can only prove things false.

To relate it into photography, we all know that stopping down a lens increases the depth of field right? How do you know it always works? Because it always has? That's fine, but have you tested it on the moon or under water? All it would take is one instance of a shallow depth of field at f/22. But we don't worry about that and its not because we've tested enough cases, but rather because we we think the theory is sound.

Theory is what we find useful day-to-day. It is easily explained by optics and easily transferred to all the situations that we can't test. We don't need more than a couple empirical tests, and in fact most people would find it pretty unsatisfying to have only induction without the explanatory theory.

What I find more common among photographers is the insistence on applying objective measures to subjective material. Sharpness is a perfect example. You can't possibly argue with the theory behind diffraction. It's real, correct, and will stand up to any empirical test you want to throw at it. Until, that is, you ask for a subjective appraisal, such as, 'is this degraded by diffraction?' or 'is this too soft?' This is a much more common situation—people overestimate the subjective importance of objective fact in photography.

[William]

Data is merely a record of the state of nature during the time of measurement.

Data has no value until humans place it in context by using their prior experience. The conversion of data into knowledge can be ad hoc or rigorous. Either way humans impose a model on the data. If the difference between the model and the data is mathematically devoid of signal, then the model is valid. Competing models are evaluated by determine which residual has the least amount of signal. The most probable model becomes a hypothesis whose primary value is its usefulness to predict the future.

The above is precisely what happened when relativity overturned Newtonian physics as the complete model for physics. That is science.

[ctein]

Dear Zalman and Hugh,

Oh, very well said! I could not have expressed it better. The underlying problem, indeed, lies not in theory but in overly simplistic theory that doesn't model the real world well. One sees many examples of this in the discussion. There's the the “weakest link in the chain” fallacy (if you're not attacking the most deficient component of some aspect of image quality then you won't see any improvement-- untrue) and the “maximizing data” fallacy (that the best-looking photograph will be one that retains the most original data as measured by information theory). (In case some readers are wondering, I will not be writing columns about this in the near future, because I've written about them in the past.)

~~~~~

Dear John Woods,

Yeah, I'm pretty much in Eddington's camp.

~~~~~

Dear Drew and Mark,

Dick had a peculiar view of experimentation. He really was a theorist at heart. While he understood the theories weren't worth spit if the data didn't confirm them, my distinct impression is he didn't really think experimentation could drive theory. His infamously wrong comment about the uselessness of particle accelerators, for example. He seem to feel the major breakthroughs only occurred in the mind, not in the laboratory.

~~~~~~

Dear Noons,

Viz. what bill wrote...

I hadn't run across that particular bogus experiment. Oh my.

One I encounter frequently is folks trying to estimate print permanence by sticking their photographs in a sunny window and seeing how long they last. That trick never works. Deterioration is subject to reciprocity failure–– if you increase the illuminance by a factor of 10, the deterioration rate doesn't go up by a factor of 10. With a few photographic materials, it doesn't go up at all. Two different prints that have the same sunlight fade rates will have different fade rates under normal indoor light levels. You can even have a situation where the print that is more stable under sunlit levels is much less stable under indoor light levels.

pax \ Ctein
[ Please excuse any word-salad. MacSpeech in training! ]
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-- Ctein's Online Gallery http://ctein.com 
-- Digital Restorations http://photo-repair.com 
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[ctein]

Dear Tregix,

Heh! The problem isn't theory vs. experiment (which is why I didn't title the column "theory vs. experiment"), it's as Zalman and Hugh stated–– bad modeling and inadequate understandings and applications of theory. The underlying point still stands: if a “theoretical photographer” comes up with a prediction about how photography should work that good experiments don't support, it's not the experimenter's problem to explain why the theory failed.

I am a much better experimentalist than theoretician. Actually, I am a “phenomenologist. “ Well, that's what they were starting to call it 45 years ago when I was in college; I'm sure they have a better term for it now. Probably some reader can even tell me what it is! Anyway, what I'm very good at is understanding theories (in almost any field) and designing experiments that can test them.

~~~~~

Dear Jan and Jakob,

Hey, I'm not getting profound about mathematics and science, I'm taking a dig at certain kinds of photographic arguments. This is shallow end of the pool stuff, and you can't read too deeply into the pronouncements about mathematics and science. Consider them as talking points rather than deep philosophical insights, because they're definitely not intended as the latter.

I'm very interested in the philosophy of science and mathematics. This is definitely not a column about them.

pax \ Ctein
[ Please excuse any word-salad. MacSpeech in training! ]
======================================
-- Ctein's Online Gallery http://ctein.com 
-- Digital Restorations http://photo-repair.com 
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[Jason B. Hill]

I'm a doctoral candidate in mathematics (University of Colorado Boulder) and a photographer. Consider this: The formal axiomatic structures of mathematics portray explicit, trimmed down views of the real world (both what we see and what we don't) and focus on specific aspects or phenomena, ignoring everything else that normally complicates the situation. A good photograph does the same thing, in my opinion, portraying some environment or abstraction or emotion or whatever, while not including other factors present when the photo was taken.

You can't just say that mathematics is building and building and building of formal axiomatic structures without an end. In my research (permutation group theory), finding the intersection between observations of symmetries in the real world and the axiomatic consequences that relate to those structures IS the goal. I will agree that many mathematicians are farther to the "pure/theoretical" side, especially in the U.S., but just because they aren't observing conclusions empirically, and instead do so entirely logically, doesn't mean they're not doing science. Their results can be challenged, reproduced, or modified in anyone else's lab/office.

Personally, I used to want to keep my photography and mathematics separate. Now, I think that's a bad idea... I'm good at both and better when I'm creative with my technical and mathematical abilities. An example is here: http://www.silentlandscapes.com/toronto-stereographic-projection

I think photography is an interesting area, namely because it requires some amount of technical and artistic ability. A mathematician or scientist who is only capable technically won't go very far. Creativity and artistic ability lends itself well to our disciplines and to problem solving in general.