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Reader responses to "Is Mathematics a Science?"

*Selected reader responses to "Is Mathematics a Science?" will be posted here.
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An Open Question I

I found your website through your excellent article ("Is Psychology a Science?"), but I did want to write a small note about one of your replies to the "practicing psychotherapist" regarding the topic of mathematics as a science.

You said:*Physics is a science because mathematics is a science. Were the latter not true, the former would not be true either, and you have just shot yourself in the foot.*

This isn't clear to me. Why would the falsehood of one imply the falsehood of the other? That's easy. In order for a particular field to be scientific, each element of its practice must also be scientific.

If I am a biologist and if I rely on microscopes to gather evidence, then the microscopes must reliably magnify the true shapes and colors of the subjects of study. If instead the microscopes distort the images of cell nuclei and cannot render them with clarity and fidelity, this means they cannot used to reliably gather data. No reliable data, no evidence, no falsification, no science.

If I am a physicist and if I rely on particle accelerators to gather evidence, the accelerators must reliably generate, detect and record particle interactions. For example, the magnets in the accelerator must have particular properties and orientations. If a single perfectly functional magnet were to be oriented in the reverse of its intended placement, it would falsely identify electrons as positrons or the reverse, and the resulting data would not be reliable. No reliable data, no evidence, no falsification, no science.

If I am in a scientific field that relies on mathematics (that's all of them) and if the mathematics is so badly constructed as to produce incorrect evaluations of my data, then my field will be undermined by the defect in the underlying mathematics, just as though the mathematics was a defective microscope or a misaligned magnet.

In a well-known story of this kind, a researcher discovered a planet orbiting a distant star with a period similar to that of earth. As is often the case in studies of this kind, the detection was based on shifts in the parent star's spectral lines. Just before an important conference in which the researcher was to present his findings, he discovered to his horror that he had failed to subtract earth's own motion from the data — he had actually been reading earth's motion toward and away from the star rather than that of a hypothetical planet. The researcher's math was defective. He was obliged to publicly retract his discovery on the ground of bad mathematics. Whatever definition of Science/Mathematics/Physics you're using (and there are many), No, among scientists there is just one. Your claim of many definitions takes the post-modernist position (more on this below), and to allow post-modernism into this (or any) discussion is to fatally undermine it. one field may very well offer certain things which make it more 'science-y' than another. Except that there are no flavors of science. All scientific fields depend on proper use of mathematics. By contrast, if we improve the mathematics of astrology, it's still not scientific.

In the final analysis, a field either does or does not produce reliable evidence, and either does or does not provide a basis for falsifying its own theories. This means there aren't different flavors of science, only one, and it is a question of degree, not kind. He said:*(In passing, I'll observe that mathematics is not a science either, contrary to your claim in one of your replies to replies...)*

You replied:*Mathematics is the queen of all sciences. It is the most rigorous of all sciences, and it is governed by evidence and proof to a greater degree than any of the other sciences.*

The debate as to whether or not mathematics is a science is an old one. Among scientists, there is no debate about whether mathematics is a science. Among philosophers, different story. If philosophers were to suddenly outlaw all onanistic debates, they would have to find honest work. Don't hold your breath for this development. Maybe the best example of the boundaries between science and not-science is the 'discovery' (or is it creation?) of Non-Euclidean geometry. This was a discovery, not a creation. I call it a discovery because the universe is non-Euclidean. Were this not the case, there would be no gravity. According to present theory, gravity isn't a force but instead results from the large-scale curvature of spacetime. To see whether this assertion is supported by observation, Google for "Einstein rings," visible examples of large-scale spacetime curvature that violates classical Euclidean geometry.

Modern physics depends intimately on the acceptance of Non-Euclidean geometry, so your example is perfect in a way you didn't foresee. Non-Euclidean geometries describe systems where (for example) triangles don't add up to 180 degrees.

Now Science tells us that the day-to-day geometry we use is Euclidean. Contemporary "science" makes no such assertion, quite the opposite. The Euclidean assumption is only true to a first approximation on a small scale like a tabletop. On a very large scale, it's true again (the overall curvature of the universe seems close to "flat"). But at intermediate scales, the scale of the solar system or any galaxy, the Euclidean assumption is false. See "Einstein rings" above. Mathematics, however, does not make this distinction. To a mathematician, the acceptance of Euclidean geometry really boils down to the acceptance of an axiom (Euclid's 5th postulate). That's it. For us, it's as simple as that. To whom does "us" refer? Because the large-scale structure of the universe is curved in four dimensions, Euclid's Fifth Postulate doesn't apply in that domain. As a toy problem, I can disprove Euclid's Fifth Postulate by assuming a priori that the earth is flat and carrying out the measurements in, say, Kansas. At the end of the experiment, I have two choices: I can say that (1) Euclid's Fifth Postulate is false, or (2) the earth is not flat. With respect to the universe, the outcome is the same. According to your definition, any field which is considered Science must contain results that are falsifiable. Are results in mathematics falsifiable? Yes, certainly. My article [Is Mathematics a Science?] contains several examples of falsifiable mathematical assertions — the stated conjecture about Euler's prime generator quickly fails, and the Monty Hall conjecture (that changing doors will have no effect) fails as well. It's not clear to me. Nor was it clear to many philosophers (like Popper) — which is why I was surprised the issue was so clear to you. I didn't just assert this — I gave examples in my article. There are any number of similar examples. What's my own viewpoint? Mathematics is a spectrum, which contains the pure mathematicians at one end (clumped with the logicians) and the applied mathematicians at the other end. The difference between pure and applied mathematics comes down to the tautology of application. A statement about a universe with ten dimensions is applied mathematics only if the universe really has ten dimensions (an open question), otherwise it's pure (meaning it has no essential connection with reality). I would be willing to admit that applied mathematics can be considered a science (it depends on WHAT the applied mathematician is studying and what definition for 'science' we're using), There is only one definition of science. Philosophers can have a field day redefining science over cups of tea, but in the actual practice of science, there is precisely one definition. It is not surprising that philosophers prefer the post-modernist thesis (the idea that there are no shared truths, all knowledge is subjective, and everything is personal opinion) — they have nothing to lose and everything to gain. By adopting the post-modernist outlook, they can publish any number of words without risk of arriving at a useful conclusion. but it's a bit of a stretch for me to consider a logician as a scientist. There are logical statements that can be, and that are, falsified. The Monty Hall problem is a logical statement that leads to a falsification, one so tricky that about 10,000 readers of the magazine that published it, including many academics, wrote to object that the published result was false. The readers were mistaken, but that's not the matter under discussion.

Logician Raymond Smullyan famously said, "Recently, someone asked me if I believed in astrology. He seemed somewhat puzzled when I explained that the reason I don't is that I'm a Gemini." If you think about this, you will realize Smullyan is clearly saying formal logic includes the possibility of falsification and contradiction. I'll end with a beautiful quote by Jean Dieudonne. Deidonne was a French mathematician noted for his work on abstract algebra and functional analysis — the latter being extremely pure.

*"On foundations we believe in the reality of mathematics, but of course when philosophers attack us with their paradoxes ...*
Okay, this quote either refers directly to Gödel's Incompleteness Theorems or it anticipates them. Gödel's Incompleteness Theorems don't mean that mathematics cannot produce science. It means there are limits to the domain of mathematics, just as there are limits to the domain of Euclidean geometry (which necessarily excludes the large-scale geometry of the universe).

The essential point of Gödel's Incompleteness Theorems is that mathematics cannot fully evaluate itself — this would lead to a paradox of self-reference. But mathematics is able to evaluate any number of topics other than itself with perfect reliability.*... we rush to hide behind formalism and say, "Mathematics is just a combination of meaningless symbols," and then we bring out Chapters 1 and 2 on set theory. Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real."*

(Nevermind is mathematics even a science! Dieudonne questions whether or not it's even 'real'!) Dieudonne's philosophical position doesn't hinder the successful application of mathematics to science, in the same way that Euclid's Fifth Postulate doesn't hinder the successful application of geometry to the universe, and for the same reason — selection of an appropriate domain.

You said:

This isn't clear to me. Why would the falsehood of one imply the falsehood of the other? That's easy. In order for a particular field to be scientific, each element of its practice must also be scientific.

If I am a biologist and if I rely on microscopes to gather evidence, then the microscopes must reliably magnify the true shapes and colors of the subjects of study. If instead the microscopes distort the images of cell nuclei and cannot render them with clarity and fidelity, this means they cannot used to reliably gather data. No reliable data, no evidence, no falsification, no science.

If I am a physicist and if I rely on particle accelerators to gather evidence, the accelerators must reliably generate, detect and record particle interactions. For example, the magnets in the accelerator must have particular properties and orientations. If a single perfectly functional magnet were to be oriented in the reverse of its intended placement, it would falsely identify electrons as positrons or the reverse, and the resulting data would not be reliable. No reliable data, no evidence, no falsification, no science.

If I am in a scientific field that relies on mathematics (that's all of them) and if the mathematics is so badly constructed as to produce incorrect evaluations of my data, then my field will be undermined by the defect in the underlying mathematics, just as though the mathematics was a defective microscope or a misaligned magnet.

In a well-known story of this kind, a researcher discovered a planet orbiting a distant star with a period similar to that of earth. As is often the case in studies of this kind, the detection was based on shifts in the parent star's spectral lines. Just before an important conference in which the researcher was to present his findings, he discovered to his horror that he had failed to subtract earth's own motion from the data — he had actually been reading earth's motion toward and away from the star rather than that of a hypothetical planet. The researcher's math was defective. He was obliged to publicly retract his discovery on the ground of bad mathematics. Whatever definition of Science/Mathematics/Physics you're using (and there are many), No, among scientists there is just one. Your claim of many definitions takes the post-modernist position (more on this below), and to allow post-modernism into this (or any) discussion is to fatally undermine it. one field may very well offer certain things which make it more 'science-y' than another. Except that there are no flavors of science. All scientific fields depend on proper use of mathematics. By contrast, if we improve the mathematics of astrology, it's still not scientific.

In the final analysis, a field either does or does not produce reliable evidence, and either does or does not provide a basis for falsifying its own theories. This means there aren't different flavors of science, only one, and it is a question of degree, not kind. He said:

You replied:

The debate as to whether or not mathematics is a science is an old one. Among scientists, there is no debate about whether mathematics is a science. Among philosophers, different story. If philosophers were to suddenly outlaw all onanistic debates, they would have to find honest work. Don't hold your breath for this development. Maybe the best example of the boundaries between science and not-science is the 'discovery' (or is it creation?) of Non-Euclidean geometry. This was a discovery, not a creation. I call it a discovery because the universe is non-Euclidean. Were this not the case, there would be no gravity. According to present theory, gravity isn't a force but instead results from the large-scale curvature of spacetime. To see whether this assertion is supported by observation, Google for "Einstein rings," visible examples of large-scale spacetime curvature that violates classical Euclidean geometry.

Modern physics depends intimately on the acceptance of Non-Euclidean geometry, so your example is perfect in a way you didn't foresee. Non-Euclidean geometries describe systems where (for example) triangles don't add up to 180 degrees.

Now Science tells us that the day-to-day geometry we use is Euclidean. Contemporary "science" makes no such assertion, quite the opposite. The Euclidean assumption is only true to a first approximation on a small scale like a tabletop. On a very large scale, it's true again (the overall curvature of the universe seems close to "flat"). But at intermediate scales, the scale of the solar system or any galaxy, the Euclidean assumption is false. See "Einstein rings" above. Mathematics, however, does not make this distinction. To a mathematician, the acceptance of Euclidean geometry really boils down to the acceptance of an axiom (Euclid's 5th postulate). That's it. For us, it's as simple as that. To whom does "us" refer? Because the large-scale structure of the universe is curved in four dimensions, Euclid's Fifth Postulate doesn't apply in that domain. As a toy problem, I can disprove Euclid's Fifth Postulate by assuming a priori that the earth is flat and carrying out the measurements in, say, Kansas. At the end of the experiment, I have two choices: I can say that (1) Euclid's Fifth Postulate is false, or (2) the earth is not flat. With respect to the universe, the outcome is the same. According to your definition, any field which is considered Science must contain results that are falsifiable. Are results in mathematics falsifiable? Yes, certainly. My article [Is Mathematics a Science?] contains several examples of falsifiable mathematical assertions — the stated conjecture about Euler's prime generator quickly fails, and the Monty Hall conjecture (that changing doors will have no effect) fails as well. It's not clear to me. Nor was it clear to many philosophers (like Popper) — which is why I was surprised the issue was so clear to you. I didn't just assert this — I gave examples in my article. There are any number of similar examples. What's my own viewpoint? Mathematics is a spectrum, which contains the pure mathematicians at one end (clumped with the logicians) and the applied mathematicians at the other end. The difference between pure and applied mathematics comes down to the tautology of application. A statement about a universe with ten dimensions is applied mathematics only if the universe really has ten dimensions (an open question), otherwise it's pure (meaning it has no essential connection with reality). I would be willing to admit that applied mathematics can be considered a science (it depends on WHAT the applied mathematician is studying and what definition for 'science' we're using), There is only one definition of science. Philosophers can have a field day redefining science over cups of tea, but in the actual practice of science, there is precisely one definition. It is not surprising that philosophers prefer the post-modernist thesis (the idea that there are no shared truths, all knowledge is subjective, and everything is personal opinion) — they have nothing to lose and everything to gain. By adopting the post-modernist outlook, they can publish any number of words without risk of arriving at a useful conclusion. but it's a bit of a stretch for me to consider a logician as a scientist. There are logical statements that can be, and that are, falsified. The Monty Hall problem is a logical statement that leads to a falsification, one so tricky that about 10,000 readers of the magazine that published it, including many academics, wrote to object that the published result was false. The readers were mistaken, but that's not the matter under discussion.

Logician Raymond Smullyan famously said, "Recently, someone asked me if I believed in astrology. He seemed somewhat puzzled when I explained that the reason I don't is that I'm a Gemini." If you think about this, you will realize Smullyan is clearly saying formal logic includes the possibility of falsification and contradiction. I'll end with a beautiful quote by Jean Dieudonne. Deidonne was a French mathematician noted for his work on abstract algebra and functional analysis — the latter being extremely pure.

The essential point of Gödel's Incompleteness Theorems is that mathematics cannot fully evaluate itself — this would lead to a paradox of self-reference. But mathematics is able to evaluate any number of topics other than itself with perfect reliability.

(Nevermind is mathematics even a science! Dieudonne questions whether or not it's even 'real'!) Dieudonne's philosophical position doesn't hinder the successful application of mathematics to science, in the same way that Euclid's Fifth Postulate doesn't hinder the successful application of geometry to the universe, and for the same reason — selection of an appropriate domain.

An Open Question II

It's not the position you take in your arguments which bothers me. It's the vehement manner in which you make your claims.
These aren't my claims. My writing is derived entirely from standard definitions. You're entitled to dispute my manner, as long as you don't confuse style with substance.
*No, among scientists there is just one* [definition of science/mathematics/physics]. *Your claim of many definitions takes the post-modernist position (more on this below), and to allow post-modernism into this (or any) discussion is to fatally undermine it.*

Very well. Give me the definition of 'mathematics'. http://en.wikipedia.org/wiki/Mathematics While you're at it, you might as well give me the definition of the word "science". http://en.wikipedia.org/wiki/Science

A quote from the above article:

*"Critical rationalism instead holds that unbiased observation is not possible and a demarcation between natural and supernatural explanations is arbitrary; it instead proposes falsifiability as the landmark of empirical theories and falsification as the universal empirical method."*

Conclusion? I didn't offer my definition of science, I offered the commonly accepted definition. It's strange because sure, I hang around scientists and mathematicians. You're the first person I've 'met' who has been so sure that there is only one definition. That's because I'm not a philosopher, and I share Richard Feynman's disdain for philosophers. Perhaps I've been hanging around imposters. Philosophers aren't impostors — they truly, sincerely, perpetually cannot accomplish anything or come to a useful conclusion. There is no hint of imposture — they really are what they seem.

It is essential to a philosopher's professional survival to pretend that nothing is agreed on and anything is open to debate — except, of course, for his privilege to debate anything.

Very well. Give me the definition of 'mathematics'. http://en.wikipedia.org/wiki/Mathematics While you're at it, you might as well give me the definition of the word "science". http://en.wikipedia.org/wiki/Science

A quote from the above article:

Conclusion? I didn't offer my definition of science, I offered the commonly accepted definition. It's strange because sure, I hang around scientists and mathematicians. You're the first person I've 'met' who has been so sure that there is only one definition. That's because I'm not a philosopher, and I share Richard Feynman's disdain for philosophers. Perhaps I've been hanging around imposters. Philosophers aren't impostors — they truly, sincerely, perpetually cannot accomplish anything or come to a useful conclusion. There is no hint of imposture — they really are what they seem.

It is essential to a philosopher's professional survival to pretend that nothing is agreed on and anything is open to debate — except, of course, for his privilege to debate anything.

An Open Question III
*"Critical rationalism instead holds that unbiased observation is not possible and a demarcation between natural and supernatural explanations is arbitrary; it instead proposes falsifiability as the landmark of empirical theories and falsification as the universal empirical method."*

*Conclusion? I didn't offer my definition of science, I offered the commonly accepted definition.*

Again with this "commonly accepted" definition. Commonly accepted by whom? Wikipedia? Apart from your argument's distinctly sophomoric quality, you are obviously unaware that to allow multiple definitions of science is to allow the Discovery Institute (you know, the creationists) to claim that science doesn't require either evidence or falsifiability, and on that basis to demand acceptance of their own variety of non-falsifiable "science." By the way, they've been making this particular demand or years.

You are trying to open the door to multiple definitions of science, and you clearly don't understand the implications. There is a deeper problem I see, and that problem is that the most difficult judgments are made at the fuzzy boundaries of "commonly accepted definitions". They aren't fuzzy, except to a postmodernist. The requirements for evidence on which impartial observers can agree, and falsifiability, do not constitute a ball of lint. We can all agree that Chemistry and Biology are Sciences. I don't see how someone can state so definitively that Mathematics (and at the other end, Psychology, I suppose) are or are not Sciences. You have just conjoined mathematics and psychology in a sentence about scientific standing, as though they were comparable. No two fields more distinct from each other could possibly have been chosen. On that basis, it seems you have an emotional investment in staking out fuzzy boundaries. Once you've reached the fuzzy boundary, it IS important how you define some things. Not if you can juxtapose mathematics and psychology in the same sentence, as though there is some basis for comparison. "Commonly accepted" just doesn't cut it. Look, if you want to go down the postmodernist road, you are on your own, and that is by definition. You need to realize that postmodernism is self-contradicting, just like your insistence that there are no firm definitions and for the same reason — the position is fatally self-referential. Similarly, how does one differentiate between an applied mathematician who studies fluid dynamics and an engineer or physicist who does the same? Scientists discover things, engineers apply them. A scientist questions the basis of a theory, but an engineer may only apply the theory. This is a perfectly clear dichotomy. Again, you've reached the boundary. Not any real-world boundary.*That's because I'm not a philosopher, and I share Richard Feynman's disdain for philosophers.*

Nor am I. I remember the first mathematics class I ever took: the professor asked us if any of us could define what "mathematics" meant. We admitted we could not. He admitted he could not, either. And life went on. A classic academic debate. Not knowing where boundaries are is usually not problematic. We all know what baldness means, despite the fact we can't define exactly how many hairs on the head a non-bald person can lose before going bald. Now you are literally splitting hairs. Ultimately, I don't see this discussion going anywhere. Once you wandered off into postmodernism, you sounded the death knell for any meaningful discourse. So we'll just agree to disagree. We aren't in disagreement. You haven't taken a position that can survive its own philosophical foundation. You clearly don't understand that if everything is a fuzzy boundary, then there is no basis for asserting a position or expecting to be heard or understood. It's the ultimate escape from reality.

*Postscript: At the outset I misjudged this correspondent's age and experience, but by the end of the exchange it was clear that he expected to live by different rules than the rest of us. I was expected to be rational and consistent and supply evidence for my position, but he wasn't. Without even bothering to examine the copious evidence I presented that mathematics is a science, he proceeded to argue that it isn't, using as his authority the opinions of his friends. As to debating issues, he only felt the need to say, "I don't accept your definitions, besides definitions don't mean anything." Fine, kid, knock yourself out.*

Again with this "commonly accepted" definition. Commonly accepted by whom? Wikipedia? Apart from your argument's distinctly sophomoric quality, you are obviously unaware that to allow multiple definitions of science is to allow the Discovery Institute (you know, the creationists) to claim that science doesn't require either evidence or falsifiability, and on that basis to demand acceptance of their own variety of non-falsifiable "science." By the way, they've been making this particular demand or years.

You are trying to open the door to multiple definitions of science, and you clearly don't understand the implications. There is a deeper problem I see, and that problem is that the most difficult judgments are made at the fuzzy boundaries of "commonly accepted definitions". They aren't fuzzy, except to a postmodernist. The requirements for evidence on which impartial observers can agree, and falsifiability, do not constitute a ball of lint. We can all agree that Chemistry and Biology are Sciences. I don't see how someone can state so definitively that Mathematics (and at the other end, Psychology, I suppose) are or are not Sciences. You have just conjoined mathematics and psychology in a sentence about scientific standing, as though they were comparable. No two fields more distinct from each other could possibly have been chosen. On that basis, it seems you have an emotional investment in staking out fuzzy boundaries. Once you've reached the fuzzy boundary, it IS important how you define some things. Not if you can juxtapose mathematics and psychology in the same sentence, as though there is some basis for comparison. "Commonly accepted" just doesn't cut it. Look, if you want to go down the postmodernist road, you are on your own, and that is by definition. You need to realize that postmodernism is self-contradicting, just like your insistence that there are no firm definitions and for the same reason — the position is fatally self-referential. Similarly, how does one differentiate between an applied mathematician who studies fluid dynamics and an engineer or physicist who does the same? Scientists discover things, engineers apply them. A scientist questions the basis of a theory, but an engineer may only apply the theory. This is a perfectly clear dichotomy. Again, you've reached the boundary. Not any real-world boundary.

Nor am I. I remember the first mathematics class I ever took: the professor asked us if any of us could define what "mathematics" meant. We admitted we could not. He admitted he could not, either. And life went on. A classic academic debate. Not knowing where boundaries are is usually not problematic. We all know what baldness means, despite the fact we can't define exactly how many hairs on the head a non-bald person can lose before going bald. Now you are literally splitting hairs. Ultimately, I don't see this discussion going anywhere. Once you wandered off into postmodernism, you sounded the death knell for any meaningful discourse. So we'll just agree to disagree. We aren't in disagreement. You haven't taken a position that can survive its own philosophical foundation. You clearly don't understand that if everything is a fuzzy boundary, then there is no basis for asserting a position or expecting to be heard or understood. It's the ultimate escape from reality.

Science is Just Another Religion

All the ardent validations and engrossing intellectual forays notwithstanding, the actuality is that - at it's foundation - science is also very largely a belief system similar to religion. As a matter of fact it can be shown that science is simply the most popular religion of this era.
Only for people who don't understand science. Among scientists and educated people, science has its position because it works, and the fact that it works is an empirically testable and falsifiable proposition.

Religion is founded entirely on untested claims. Science tests its own claims as surely and as regularly as it tests scientific theories.

In the 1950s, at the height of the polio epidemic, Sister Kenny opened a now-famous clinic to treat polio by massaging the limbs of its sufferers. At the same time, using the methods of science, Jonas Salk created a vaccine. There is no more apt comparison of religion and science. To put this in contemporary terms, if you have cancer, would you prefer a massage or a vaccine?

Religion is based on perfect confidence and no evidence. Science is based on perpetual skepticism of everything including science itself, and the only reason science prevails is because it meets the requirements of the most skeptical observer.

Obviously for a disenchanted religious believer who wants to jump ship, science looks like another belief system. In the same way, to a hammer, everything looks like a nail. But to a scientist, constitutionally inclined to doubt everything, science delivers something religion cannot provide — results.

Religion is founded entirely on untested claims. Science tests its own claims as surely and as regularly as it tests scientific theories.

In the 1950s, at the height of the polio epidemic, Sister Kenny opened a now-famous clinic to treat polio by massaging the limbs of its sufferers. At the same time, using the methods of science, Jonas Salk created a vaccine. There is no more apt comparison of religion and science. To put this in contemporary terms, if you have cancer, would you prefer a massage or a vaccine?

Religion is based on perfect confidence and no evidence. Science is based on perpetual skepticism of everything including science itself, and the only reason science prevails is because it meets the requirements of the most skeptical observer.

Obviously for a disenchanted religious believer who wants to jump ship, science looks like another belief system. In the same way, to a hammer, everything looks like a nail. But to a scientist, constitutionally inclined to doubt everything, science delivers something religion cannot provide — results.

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