Thurston on Confusion

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Mathematics is a process of staring hard enough with enough perseverance at at the fog of muddle and confusion to eventually break through to improved clarity. I’m happy when I can admit, at least to myself, that my thinking is muddled, and I try to overcome the embarrassment that I might reveal ignorance or confusion. Over the years, this has helped me develop clarity in some things, but I remain muddled in many others. I enjoy questions that seem honest, even when they admit or reveal confusion, in preference to questions that appear designed to project sophistication.
—William Thurston in his MathOverflow self-summary

Being a Good Person Does Not Depend On Perfection

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The difficulty lies, not in the new ideas, but in escaping from the old ones, which ramify, for those brought up as most of us have been, into every corner of our minds.
John Maynard Keynes

Ah, being a good person. Consider the following.

A man who only restrains from murdering people most of the time will not be considered a good man. He’s a murderer, even though he doesn’t always murder the people he meets. Slip up one time and bam, you’re a murderer. In contrast, saving one life isn’t enough to make a man a saint.

The point is this: in general, to be a bad person, you only have to be bad some of the time, but to be a good person, you have to be good all of the time. Consider: you can be regarded as a thief even if you do not usually steal, but to be regarded as an honest man you can never steal. To be faithful to your wife means that you are faithful all of the time, while you only have to be unfaithful some of the time to be regarded as unfaithful.

There is an asymmetry here, then. To be good requires perfect goodness, while being bad does not require perfect badness.

This is absurd. Abandon the notion that you need to be perfectly good all of the time. It’s impossible. You need a healthier relationship with the good or you’ll never be able to think straight.

What does this have to do with thinking straight? Most people believe themselves to be good people. This is part of their identity. As I’ve pointed out above, this entails — usually implicitly — that they are perfectly good, or pretty close. If they are confronted by a new idea about what it means to be good, then, and they do not conform to that idea, they will be motivated to reject that idea because it threatens their self-image.

Speaking to someone about renunciation is like hitting a pig on the nose with  a stick. He doesn’t like it at all.

–Tibetan proverb

Let’s make it concrete. When I talk with people and point out that a harm of omission is still a harm, they don’t like this at all, even though it’s pretty straightforward. Here are a few scenarios:

  • A man is going to die unless you press a button. Is it good to press the button?
  • A man is drowning. You can save the man. Is it good to save the man?
  • A man is starving. You can afford to feed the man. Should you feed the man?
  • A man will die of malaria in Africa because he cannot afford a insecticide-treated mosquito net. You could, instead of spending $20 at Starbucks each week, donate to the Against Malaria Foundation and save the man. Should you save the man?

The answer to all of these is yes. If it’s not mind-numbingly obvious to you, you are confused. Seriously. There’s nothing to explain. It’s better to save people than to not save people, even if you have to go without your latte.

The trouble with allowing for harms of omission is that it doesn’t allow you to preserve the notion that to be good means you are perfectly good. If you define being good as not actively harming others, being perfectly good is manageable. If failing to help someone counts as harming, it’s no longer possible to be perfectly good.

Most people respond by arguing against harms of omission. Not because this is the weak link in the chain, but because it’s right there in consciousness, while intuitive beliefs about goodness requiring perfection are lurking in the background.

If you abandon the notion that none of us are perfectly good people — that perfect goodness is too exacting a standard — most of the motivation to reject harms of omission disappears.

Let’s go even further. Let’s say that to be a good person, you have to be perfectly good. We then come to a choice: either, you can define good to make it possible for people to be perfectly good or you can accept the notion that none of us can be called good. But this is missing the point!

Why do we care about what is good? What’s the point of being good? It’s action. It’s to go out there in the world and improve it. It’s not about labels. It’s not about who’s good and who’s bad. It’s about helping.

Further Reading

  • The notion that there is an asymmetry between good and bad events is the main  theme of the paper “Bad Is Stronger Than Good.” I’ve found it a useful concept   when thinking about many different things, from blog comments to dog  training.
  • One of the criticisms of utilitarianism is that it’s too demanding, that  no one can live up to its standards. This argument appeals to the intuition   that to be a good person requires perfect goodness and, as such, perfect goodness must be manageable. See here for an overview.
  • It also seems a strange criticism to argue that a normative theory is too demanding. The rules of multiplication don’t change for large numbers, even though humans have a hard time with them.
  • Paul Graham has an essay on the difficulties of thinking straight about things that are part of your identity.

Is belief a choice?

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‘Snow is white’ is true if and only if snow is white.
—Alfred Tarski

Is belief a choice? Let me ruin the surprise: Yes, you get to choose what to believe. If you want to believe that you can fly, you are free to believe that. Reality, however, is a hostile place. It does not care about what you believe.

Jump from a cliff and you will not fly, no matter how much you wish it to be so. Beliefs do not change what is. They do not change what is true. You can choose to believe true things or false things, but your belief does not change what is real, what is actual.

If you choose to believe only nice things, you will end up believing many false things. Reality is not nice. There are not only nice things out there in the world. While the idea of an eternal afterlife is nice, the niceness of the idea says nothing about whether or not it is true. It would be nice if serial killers were just pretending and their supposed victims actually went to live in the tropics, but this doesn’t make it so.

I don’t mean only to attack nice beliefs as false. The opposite holds as well. Something that is painful to think is not true just by virtue of being painful to think. Negative, pessimistic beliefs are not true because of their negativity. They are true if and only if they correspond to reality. Gary Kasparov might think to himself, “I’m no good at chess,” and he might feel bad after thinking it, but that doesn’t make it true.

What is real, what is actual, what is true, all of these things are already so. You can believe whatever you like, but this doesn’t change what is already so. When people believed that the sun rotated around the earth, this didn’t make it so. Belief is a choice, but truth is not. You don’t choose what is true. It has already been decided. Belief feels like a choice because it is a choice, but do not confuse belief with truth. Believing something does not make it so. Truth already is.

Depressed? Try Plastic Surgery

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Ohlsen, Ponten, and Hamburt (1978) noted that twenty-five of seventy-one women in their study were receiving psychiatric treatment prior to a breast augmentation procedure, whereas only three continued to do so after the operation.1 Klassen and his colleagues (1996) also found substantial reductions in psychiatric symptomatology among people receiving plastic surgery.2 Cole and his colleagues (1994) reported that 73 percent of their patients reported a higher quality of life after cosmetic surgery, compared to only 6 percent who reported a lower quality of life.3 The largest gains were for cosmetic breast surgery (both reductions and enlargements), with slightly smaller gains for abdominoplasty (tummy tucks) and only slight gains for rhinoplasty (nose jobs).
—Well-Being: Foundations of Hedonic Psychology

Looking for something less invasive? Try Botox.

Ten depressed patients were treated with botulinum toxin A, and 9 of 10 patients were no longer depressed 2 months after treatment. The tenth patient had an improvement in mood.4

Sources


1. Ohlsen, Lennart, Bengt Ponten, and Gunnar Hambert. “Augmentation mammaplasty: A surgical and psychiatric evaluation of the results.” Annals of plastic surgery 2.1 (1979): 42-52.

2. Klassen, A., et al. “Patients’ health related quality of life before and after aesthetic surgery.” British journal of plastic surgery 49.7 (1996): 433-438.

3. Cole, Richard P., et al. “Measuring outcome in low-priority plastic surgery patients using quality of life indices.” British journal of plastic surgery 47.2 (1994): 117-121.

4. Finzi, Eric, and Erika Wasserman. “Treatment of depression with botulinum toxin A: a case series.” Dermatologic Surgery 32.5 (2006): 645-650.

Let Them Eat Lobster

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Prior to [the mid-19th century], lobster was considered a mark of poverty or as a food for indentured servants or lower members of society in Maine, Massachusetts, and the Canadian Maritimes, and servants specified in employment agreements that they would not eat lobster more than twice per week. Lobster was also commonly served in prisons, much to the displeasure of inmates.

—Wikipedia’s lobster page

Is veganism for women?

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I recently tried my hand at making this bean-based, vegan taco filling (recommended), and noticed something curious while reading the recipe’s comments:

I made this for my vegetarian daughter and it was so good I served it to the whole family for dinner.

This stuff is the bomb. Even my carnivore husband loved it!

My daughter is vegan, while the rest of the family isn’t. My husband usually can’t stand non meat recipes, and loved this one!

See the pattern? Men are carnivores, women herbivores. This is from a sample of seven comments. Half of the reviews show this trend. I also have the vague sense that women, generally, are more interested in veganism and are more open to talking about it.

And since we live in the future, I don’t have to speculate. Are women more likely to be vegans? Yes. There seem to be somewhere between two to three woman-vegans for every man-vegan.1

Wherefore art thou vegan?

Tiger got to hunt, bird got to fly; Man got to sit and wonder, “Why, why, why?”
—Kurt Vonnegut, Cat’s Cradle

It’s interesting in and of itself that women are more likely to be vegans, but why are women more likely to adopt a vegan lifestyle? What motivates these strange, alien creatures? Fellow men, let us analyze them. Or something.

By the way, we could reframe this question as, “Why are men not vegans?” instead of asking why women are vegans. I suspect that each frame suggests different answers.

You wouldn’t download a steak

The notion that human life is sacred just because it is human life is medieval.
—Peter Singer

Most vegans adopt a vegan lifestyle out of ethical concerns, or at least that’s what they report. Perhaps women are the fairer sex, then, and are more attuned to ethical issues. Or, perhaps women are more attuned to the suffering of animals and, thus, value reducing it more than men do. This would suggest that women have more empathy for animals or, even, more empathy in general than men.

Is there any data to support this? Wikipedia’s page on sex differences suggests that there may be some difference in reported levels of empathy between men and women. Women also tend to score higher on the Big 5 facet of agreeableness,16 which can be thought of as a measure of empathy.

If it were the case that women are more ethically minded, I would expect them to be more likely to be PETA members, veterinarians, more likely to donate to charity, and more likely to volunteer than men. Is this the case?

The lady doth protest too much

I was not designed to be forced. I will breathe after my own fashion. Let us see who is the strongest.
—Henry David Thoreau, On the Duty of Civil Disobedience

I can’t find any information specific to PETA demographics, but one survey of animal rights activists reported four women for every man,6 and women nearly always outnumber men at animal rights demonstrations.7 The proportion of female to male members of the American Society for the Prevention of Cruelty to Animals is higher than 5 to 1.7 Further, women are more opposed to animal research than men, and this effect is robust across nations.8

The majority of women in animal rights activism is not a recent phenomena, but dates back to at least 1875 with the founding of the Victoria Street Society, an early British animal protection agency, now know as the National Anti-Vivisection Society.9 Indeed, Victorian women were drawn more to animal rights than any other cause, except feminism.10

More, most veterinarian students are women and the proportion of professional women vets is on the rise.2 Women are also more likely to donate to charity in general and are more generous when doing so, and they are especially likely to donate to causes supporting animal welfare.3 According to this report from the Bureau of Labor Statistics, women are also more likely to volunteer in general.

What if we just ask people how they feel about the ethics of meat-eating or animal suffering? At least one survey has done so.5 Men were about 20% more likely to indicate that eating meat is morally justifiable, 72.3% for men versus 54.3% for women. Women were also more likely than men to express agreement with the idea that food should be prepared in a way that minimizes animal suffering, 95.1% versus 84.8%.

Men, however, have the dubious distinction of being more likely to actually have sex with animals.7

Is right or left the feminine side?

To succeed in a domain that violates your intuitions, you need to be able to turn them off the way a pilot does when flying through clouds. You need to do what you know intellectually to be right, even though it feels wrong.
— Paul Graham

It’s interesting to note that while female philosophers are more likely to specialize in applied ethics, they are less likely to switch on the trolley problem. The trolley problem goes like this: there is a train coming down the tracks and it’s going to hit and kill five people. You are standing next to a lever which, if you pull it, will divert the train. However, the diverted train will kill one person. Should you pull the lever?

Women failing to switch seems counter-intuitive. If women are more ethical, they ought to switch on the trolley problem. However, the trolley problem is notable because it’s a problem where intuition clashes with ethics. You can save more lives by diverting the trolley, but it feels wrong. Is there evidence to suggest that women are more likely to trust their intuitions?

Yes. The Rational-Experiential Inventory measures individual differences in “faith in intuition,” and women generally score higher than men on this subscale.11 Lay theories of the differences between men and women reflect the same sentiment. Women are thought of as “emotional creatures,” while men are thought of as cold and rational (or testosterone-fueled savages). That is: folk psychology posits that women operate on feelings and intuition. Indeed, the main thrust of the phrase “in touch with your feminine side” is about emotion.

We can go further. The MBTI personality scale measures differences (among other things) in the dimension of thinking versus feeling. Thinking types endorse statements like, “I make decisions with my head,” while feeling types endorse choosing with their heart. The T/F split is 60/40 for men, while the ratio is reversed for women.12

The belief in a god, an enduring human soul, and that prayer affects the world around us, are all intuitively appealing. To identify as an atheist suggests that one has discarded these intuitive beliefs. Are men more likely to be atheists? According to the Wikipedia page on atheism demographics, men are more likely to be atheists in the United States and the United
Kingdom. Canadian men are also more likely to be atheists than Canadian women,13 and, in one sample, women held more positive views of Christianity across age groups.14 Men are also more likely to be autistic and those with high-functioning autism are much more likely to identify as atheist.17 There is also more atheism in the natural sciences than the social sciences, which tend to be male dominated,20 although note that there seem to be a disproportionate amount of theists among mathematicians.19

Further, women are more likely to believe in the devil, heaven and hell, creationism, ghosts, communication with the dead, ESP, and astrology. Men, however, win out when it comes to belief in aliens.18

Brain imaging studies further support the notion that utilitarian judgments are characterized by increased activation of regions associated with deliberative processing and conflict resolution,21, 22 while deontological judgments (e.g. murder is always wrong) are characterized by reliance on emotional heuristics.23 It also seems to be the case that those with weaker empathic responses have an easier time with consequentialist reasoning.24, 25 This suggests that women fail the trolley problem specifically because they experience stronger feelings of empathy. Men might outperform women on contrived ethical dilemmas not because they trust their feelings less, but rather because they experience a weaker, easier-to-override empathic response.

Static, dynamic, and sensitive typing

Are women really more sensitive than men?

I’ve already reviewed a fair amount of evidence that suggests this is the case. Women are more likely to volunteer, donate to charity, eat in ways that minimize animal suffering, and so on. This is certainly suggestive of the idea that women are more empathetic than men.

It’s not decisive, though. There could be other reasons why women do all of these things. Maybe women volunteer more because they don’t work as much as men. Maybe women volunteer more because their friends are already volunteering. It could be a self-fulfilling prophecy. Or maybe women are more likely to say they care about others not because they actually care more, but because they’re more motivated to look good than men are. All possibilities.

One review of sex differences in empathy found that, when study participants are aware that empathy is being measured, women have an
advantage. When studies are more subtle and people are not aware of what is being measured, the gender differences disappear.26 A more rigorous meta-analysis by Ickes et al. replicated this result.27 The gender differences in empathy seem to result from differences in the amount of work men and women put into appearing empathetic. If you pay participants based on how well they perform at an empathy task, gender differences disappear.28

It would seem, then, that differences in male and female empathy can be wholly attributed to differences in self-concept. If we lived in alternative universe where men were seen as gentle and caring and women brutish and insensitive, men would score higher on measures of empathy, even if each sex’s physiology was exactly the same as it is in this universe. Women are more sensitive than men because they try harder at being more sensitive than men.

Final thoughts

Nature, Mr. Allnut, is what we are put in this world to rise above.
—Rose, The African Queen (1951)

I was going to go on and consider differences in eating disorders, disgust sensitivity, time spent thinking about food, and maybe more, but the evidence seems overwhelming at this point. I’m convinced, anyways, and it seems especially inappropriate to continue beating a dead horse in a post about veganism. Women are more likely to adopt a vegan lifestyle, at least in part, because the average woman is more likely than the average man to self-identify as a kind person and, as such, is more motivated to adopt lifestyle changes in an attempt to reduce animal suffering.

Sources


1. “In US, 5% Consider Themselves Vegetarians.”
Also, see this page.

2. Lofstedt,
Jeanne. “Gender and veterinary medicine.”
The Canadian Veterinary Journal 44.7 (2003): 533.

3. Schnepf, Sylke V., and Greg Piper. “Gender Differences in Charitable Giving.” (2008).

3. Klopp, Sheree A., Cynthia J. Heiss, and Heather
S. Smith. “Self-reported vegetarianism may be a marker for college women at risk for disordered eating.”
Journal of the American Dietetic Association 103.6 (2003): 745-747.

4. Hudson, James I., et
al. “The prevalence and correlates of eating disorders in the National Comorbidity Survey Replication.”
Biological psychiatry 61.3 (2007): 348-358.

5. Beardsworth, Alan, et al. “Women, men and food: the significance of gender for nutritional attitudes and choices.” British Food Journal 104.7 (2002): 470-491.

6. Plous, Scott. “An attitude survey of animal rights activists.” Psychological Science 2.3 (1991): 194-196.

7. Herzog, Harold A. “Gender differences in humananimal interactions: A review.” Anthrozoos: A Multidisciplinary Journal of The Interactions of People & Animals 20.1 (2007): 7-21.

8. Hagelin, Joakim, Hans-Erik Carlsson, and Jann Hau. “An overview of surveys on how people view animal experimentation: some factors that may influence the outcome.” Public Understanding of Science 12.1 (2003): 67-81.

9. Elston, Mary Ann. “Women and anti-vivisection in Victorian England,
1870–1900.” Vivisection in historical perspective (1987): 259-294.

10. French, Richard D., and Richard French. Antivivisection and medical science in Victorian society. Princeton: Princeton University Press, 1975.

11. Riding, Richard J., and Stephen G. Rayner, eds. Cognitive styles. Greenwood Publishing Group, 2000.

12. MBTI manual: a guide to the development and use of the Myers-Briggs type
indicator instrument. CPP, 2003. See also this page.

13. Veevers, Jean E., and D. F. Cousineau. “The heathen Canadians: demographic
correlates of nonbelief.” Pacific Sociological Review (1980): 199-216.

14. Francis, Leslie J., and Carolyn Wilcox. “Religiosity and femininity: Do women really hold a more positive attitude toward Christianity?.” Journal for the Scientific Study of Religion (1998): 462-469.

15. Feltey, Kathryn M., and Margaret M. Poloma. “From sex differences to gender role beliefs: Exploring effects on six dimensions of religiosity.” Sex Roles 25.3-4 (1991): 181-193.

16. Costa Jr, Paul, Antonio Terracciano, and Robert R. McCrae. “Gender differences in personality traits across cultures: robust and surprising findings.” Journal of personality and social psychology 81.2 (2001): 322.

17. Caldwell-Harris, Catherine, et al. “Religious belief systems of persons with high functioning autism.” Annual Meeting of the Cognitive Science Society, Boston, MA. 2011.

18. Rice, Tom W. “Believe it or not: Religious and other paranormal beliefs in the United States.” Journal for the Scientific Study of Religion 42.1 (2003): 95-106.

19. Larson, Edward J., and Larry Witham. “Leading scientists still reject God.” Nature 394.6691 (1998): 313-313.

20. Rosser, Sue V., and Mark Zachary Taylor. “Why Are We Still Worried about Women in Science?.” Academe 95.3 (2009): 7-10.

22. Greene, Joshua D. The secret joke of Kant’s soul. Cambridge, MA: MIT Press, 2007.

21. Greene, Joshua D., et al. “The neural bases of cognitive conflict and control in moral judgment.” Neuron 44.2 (2004): 389-400.

23. Greene, Joshua D., et al. “An fMRI investigation of emotional engagement in moral judgment.” Science 293.5537 (2001): 2105-2108.

24. Wiech, Katja, et al. “Cold or calculating? Reduced activity in the subgenual cingulate cortex reflects decreased emotional aversion to harming in counterintuitive utilitarian judgment.” Cognition 126.3 (2013): 364-372.

25. Bartels, Daniel M., and David A. Pizarro. “The mismeasure of morals: Antisocial personality traits predict utilitarian responses to moral dilemmas.” Cognition 121.1 (2011): 154-161.

26. Graham, Tiffany, and William Ickes. “When women’s intuition isn’t greater
than men’s.” (1997).

27. Ickes, William, Paul R. Gesn, and Tiffany Graham. “Gender differences in
empathic accuracy: Differential ability or differential motivation?.” Personal Relationships 7.1 (2000): 95-109.

28. Klein, Kristi JK, and Sara D. Hodges. “Gender differences, motivation, and empathic accuracy: When it pays to understand.” Personality and Social Psychology Bulletin 27.6 (2001): 720-730.

Great Math Quotes

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I don’t collect many things, who needs that junk? But I do have a collection of great math quotes that I’d like to share with you.

Great Math Quotes

We must not believe those, who today, with philosophical bearing and deliberative tone, prophesy the fall of culture and accept the ignorabimus. For us there is no ignorabimus, and in my opinion none whatever in natural science. In opposition to the foolish ignorabimus our slogan shall be: We must know — we will know!
—David Hilbert

If I feel unhappy, I do mathematics to become happy. If I am happy, I do mathematics to keep happy.
—P. Turan, “The Work of Alfred Renyi”

One must make a start in any line of research, and this beginning almost always has to be a very imperfect attempt, often unsuccessful. There are truths that are unknown in the way that there are countries the best road to which can only be learned after having tried them all. Some persons have to take the risk of getting off the track in order to show the right road to others…. We are almost always condemned to experience errors in order to arrive at truth.1
—Denis Diderot

What is mathematics?

We often hear that mathematics consists mainly of “proving theorems.” Is a writer’s job mainly that of “writing sentences?”
—Gian-Carlo Rota

Last time, I asked: “What does mathematics mean to you?” And some people answered: “The manipulation of numbers, the manipulation of structures.” And if I had asked what music means to you, would you have answered: “The manipulation of notes?”
—Serge Lang

The purpose of computing is insight, not numbers.
—Richard W. Hamming

Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.
—Paul Erdős

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.
—John von Neumann

Analysis

I recoil with fear and loathing from that deplorable evil, continuous functions with no derivative.1
—Charles Hermite, on the Weierstrass function

Combinatorics

Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing; you don’t get a feeling of having done an honest day’s work. Don’t get the wrong idea – combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow undertaking, with sophisticated techniques.
—Gian-Carlo Rota

Analytic Geometry

The introduction of numbers as coordinates is an act of violence.
—Hermann Weyl, Philosophy of Mathematics and Natural Science

Non-Euclidean Geometry

Out of nothing I have created a strange new universe.
—János Bolyai

I have made such wonderful discoveries that I am myself lost in astonishment.
—János Bolyai

For God’s sake, please give it up. Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life.
—Farkas Bolyai, quoted in The Mathematical Experience (very recommended, buy a copy)

The assumption that the angle sum of a triangle is less than 180° leads to a curious geometry, quite different from ours but thoroughly consistent, which I have developed to my entire satisfaction. The theorems of this geometry appear to be paradoxical, and, to the uninitiated, absurd, but calm, steady reflection reveals that they contain nothing at all impossible.1
—Carl Friedrich Gauss

Foundations and Certainty

I shall persevere until I find something that is certain or, at least, until I find for certain that nothing is certain.1
—René Descartes

[This] science is the work of the human mind, which is destined rather to study than to know, to seek the truth rather than to find it.
— Évariste Galois

Persist and faith will come to you.
— Jean le Rond d’Alembert

I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics; it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But.. the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.
—Bertrand Russel, Portraits from Memory

The splendid certainty which I had always hoped to find in mathematics was lost in a bewildering maze….It is truly a complicated conceptual labyrinth.
— Bertrand Russel, My Philosophical Development

But does mathematics need absolute certainty for its justification? In particular, why do we need to be sure a theory is consistent or that it can be derived by an absolutely certain intuition of pure time, before we use it? In no other science do we make such demands. In physics all theorems are hypothetical; we adopt a theory so long as it makes useful predictions and modify or discard it as soon as it does not. This is what happened to mathematical theories in the past, where the discovery of contradictions has led to modification in the mathematical doctrines accepted up to the time of that discovery. Why should we not do the same in the future?
—Haskell B. Curry, Foundations of Mathematical Logic

We are less certain than ever about the ultimate foundations of mathematics and logic. Like everybody and everything in the world today, we have our “crisis…” We have had it for nearly fifty years. Outwardly it does not seem to hamper our daily work, and yet I for one confess that it has had a considerable practical influence on my mathematical life; it directed my interests to fields I considered relatively “safe,” and has been a constant drain on the enthusiasm and determination with which I pursued my research work. This experience is probably shared by other mathematicians who are not indifferent to what their scientific endeavors mean in the context of man’s whole caring and knowing, suffering and creative existence in the world.1
—Hermann Weyl

The method of postulating what we want has many advantages; they are the same as the advantages of theft over honest toil.1
—Bertrand Russel

The recent research on foundations has broken through frontiers only to encounter a wilderness.
—Morris Kline, Mathematics: The Loss of Certainty, (recommended, buy a copy)

We have put a fence around the herd to protect it from the wolves but we do not know whether some wolves were already enclosed within the fence.
Henri Poincaré

Applications

This science [mathematics] does not have for its unique objective to eternally contemplate its own navel; it touches nature and some day it will make contact with it. On this day it will be necessary to discard the purely verbal definitions and not any more be the dupe of empty words.
—Henri Poincaré, The Foundations of Science

It would be necessary to have completely forgotten the history of science not to remember that the desire to understand nature has had on the development of mathematics the most important and happiest influence… The pure mathematician who should forget the existence of the exterior world would be like a painter who knows how to harmoniously combine colors and forms, but who lacked models. His creative power would soon be exhausted.
—Henri Poincare, The Value of Science

The mathematics of our day seems to be like a great weapons factory in peace time. The show window is filled with parade pieces whose ingenious, skillful, eye-appealing execution attracts the connoisseur. The proper motivation for and purpose of these objects, to battle and conquer the enemy, has receded to the background of consciousness to the extent of having been forgotten.
—Felix Klein, Development of Mathematics in the Nineteenth Century

Nature does not offer her problems ready formulated. They must be dug up with pick and shovel, and he who will not soil his hands will never see them.1
—John L. Synge

Platonism

God eternally geometrizes.1
—Plato

The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics.1
—Johannes Kepler

Formalism

A serious threat to the very life of science is implied in the assertion that mathematics is nothing but a system of conclusions drawn from the definitions and postulates that must be consistent but otherwise may be created by the free will of mathematicians. If this description were accurate, mathematics could not attract any intelligent person.1
—Richard Courant

Further Reading

  • There are more math quotes here and here (with comics!). This is a good list, too.
  • For computer science quotes, try here. The nazis at StackOverflow deleted a great collection of quotes, but you can find an archive here.
  • For more general quotes, this page is my favorite.

Sources


1. Quoted in Mathematics: The Loss of Certainty.