We all know what anger is, and we've all felt it: whether as a fleeting annoyance or as full-fledged rage.

Anger is a completely normal, usually healthy, human emotion. But when it gets out of control and turns destructive, it can lead to problems—problems at work, in your personal relationships, and in the overall quality of your life. And it can make you feel as though you're at the mercy of an unpredictable and powerful emotion. This brochure is meant to help you understand and control anger.

The Nature of Anger

Anger is "an emotional state that varies in intensity from mild irritation to intense fury and rage," according to Charles Spielberger, PhD, a psychologist who specializes in the study of anger. Like other emotions, it is accompanied by physiological and biological changes; when you get angry, your heart rate and blood pressure go up, as do the levels of your energy hormones, adrenaline, and noradrenaline.

Anger can be caused by both external and internal events. You could be angry at a specific person (Such as a coworker or supervisor) or event (a traffic jam, a canceled flight), or your anger could be caused by worrying or brooding about your personal problems. Memories of traumatic or enraging events can also trigger angry feelings.

Expressing Anger

The instinctive, natural way to express anger is to respond aggressively. Anger is a natural, adaptive response to threats; it inspires powerful, often aggressive, feelings and behaviors, which allow us to fight and to defend ourselves when we are attacked. A certain amount of anger, therefore, is necessary to our survival.

On the other hand, we can't physically lash out at every person or object that irritates or annoys us; laws, social norms, and common sense place limits on how far our anger can take us.

People use a variety of both conscious and unconscious processes to deal with their angry feelings. The three main approaches are expressing, suppressing, and calming. Expressing your angry feelings in an assertive—not aggressive—manner is the healthiest way to express anger. To do this, you have to learn how to make clear what your needs are, and how to get them met, without hurting others. Being assertive doesn't mean being pushy or demanding; it means being respectful of yourself and others.

Anger can be suppressed, and then converted or redirected. This happens when you hold in your anger, stop thinking about it, and focus on something positive. The aim is to inhibit or suppress your anger and convert it into more constructive behavior. The danger in this type of response is that if it isn't allowed outward expression, your anger can turn inward—on yourself. Anger turned inward may cause hypertension, high blood pressure, or depression.

Unexpressed anger can create other problems. It can lead to pathological expressions of anger, such as passive-aggressive behavior (getting back at people indirectly, without telling them why, rather than confronting them head-on) or a personality that seems perpetually cynical and hostile. People who are constantly putting others down, criticizing everything, and making cynical comments haven't learned how to constructively express their anger. Not surprisingly, they aren't likely to have many successful relationships.

Finally, you can calm down inside. This means not just controlling your outward behavior, but also controlling your internal responses, taking steps to lower your heart rate, calm yourself down, and let the feelings subside.

As Dr. Spielberger notes, "when none of these three techniques work, that's when someone—or something—is going to get hurt."

The goal of anger management is to reduce both your emotional feelings and the physiological arousal that anger causes. You can't get rid of, or avoid, the things or the people that enrage you, nor can you change them, but you can learn to control your reactions.

Are You Too Angry?

There are psychological tests that measure the intensity of angry feelings, how prone to anger you are, and how well you handle it. But chances are good that if you do have a problem with anger, you already know it. If you find yourself acting in ways that seem out of control and frightening, you might need help finding better ways to deal with this emotion.

Why Are Some People More Angry Than Others?

According to Jerry Deffenbacher, PhD, a psychologist who specializes in anger management, some people really are more "hotheaded" than others are; they get angry more easily and more intensely than the average person does. There are also those who don't show their anger in loud spectacular ways but are chronically irritable and grumpy. Easily angered people don't always curse and throw things; sometimes they withdraw socially, sulk, or get physically ill.

People who are easily angered generally have what some psychologists call a low tolerance for frustration, meaning simply that they feel that they should not have to be subjected to frustration, inconvenience, or annoyance. They can't take things in stride, and they're particularly infuriated if the situation seems somehow unjust: for example, being corrected for a minor mistake.

What makes these people this way? A number of things. One cause may be genetic or physiological: There is evidence that some children are born irritable, touchy, and easily angered, and that these signs are present from a very early age. Another may be sociocultural. Anger is often regarded as negative; we're taught that it's all right to express anxiety, depression, or other emotions but not to express anger. As a result, we don't learn how to handle it or channel it constructively.

Research has also found that family background plays a role. Typically, people who are easily angered come from families that are disruptive, chaotic, and not skilled at emotional communications.

Is It Good To "Let it All Hang Out?"

Psychologists now say that this is a dangerous myth. Some people use this theory as a license to hurt others. Research has found that "letting it rip" with anger actually escalates anger and aggression and does nothing to help you (or the person you're angry with) resolve the situation.

It's best to find out what it is that triggers your anger, and then to develop strategies to keep those triggers from tipping you over the edge.

Relaxation

Simple relaxation tools, such as deep breathing and relaxing imagery, can help calm down angry feelings. There are books and courses that can teach you relaxation techniques, and once you learn the techniques, you can call upon them in any situation. If you are involved in a relationship where both partners are hot-tempered, it might be a good idea for both of you to learn these techniques.

Some simple steps you can try:

• Breathe deeply, from your diaphragm; breathing from your chest won't relax you. Picture your breath coming up from your "gut."
• Slowly repeat a calm word or phrase such as "relax," "take it easy." Repeat it to yourself while breathing deeply.
• Use imagery; visualize a relaxing experience, from either your memory or your imagination.
• Nonstrenuous, slow yoga-like exercises can relax your muscles and make you feel much calmer.

Practice these techniques daily. Learn to use them automatically when you're in a tense situation.

Cognitive Restructuring

Simply put, this means changing the way you think. Angry people tend to curse, swear, or speak in highly colorful terms that reflect their inner thoughts. When you're angry, your thinking can get very exaggerated and overly dramatic. Try replacing these thoughts with more rational ones. For instance, instead of telling yourself, "oh, it's awful, it's terrible, everything's ruined," tell yourself, "it's frustrating, and it's understandable that I'm upset about it, but it's not the end of the world and getting angry is not going to fix it anyhow."

Be careful of words like "never" or "always" when talking about yourself or someone else. "This !&*%@ machine never works," or "you're always forgetting things" are not just inaccurate, they also serve to make you feel that your anger is justified and that there's no way to solve the problem. They also alienate and humiliate people who might otherwise be willing to work with you on a solution.

Remind yourself that getting angry is not going to fix anything, that it won't make you feel better (and may actually make you feel worse).

Logic defeats anger, because anger, even when it's justified, can quickly become irrational. So use cold hard logic on yourself. Remind yourself that the world is "not out to get you," you're just experiencing some of the rough spots of daily life. Do this each time you feel anger getting the best of you, and it'll help you get a more balanced perspective. Angry people tend to demand things: fairness, appreciation, agreement, willingness to do things their way. Everyone wants these things, and we are all hurt and disappointed when we don't get them, but angry people demand them, and when their demands aren't met, their disappointment becomes anger. As part of their cognitive restructuring, angry people need to become aware of their demanding nature and translate their expectations into desires. In other words, saying, "I would like" something is healthier than saying, "I demand" or "I must have" something. When you're unable to get what you want, you will experience the normal reactions—frustration, disappointment, hurt—but not anger. Some angry people use this anger as a way to avoid feeling hurt, but that doesn't mean the hurt goes away.

Problem Solving

Sometimes, our anger and frustration are caused by very real and inescapable problems in our lives. Not all anger is misplaced, and often it's a healthy, natural response to these difficulties. There is also a cultural belief that every problem has a solution, and it adds to our frustration to find out that this isn't always the case. The best attitude to bring to such a situation, then, is not to focus on finding the solution, but rather on how you handle and face the problem.

Make a plan, and check your progress along the way. Resolve to give it your best, but also not to punish yourself if an answer doesn't come right away. If you can approach it with your best intentions and efforts and make a serious attempt to face it head-on, you will be less likely to lose patience and fall into all-or-nothing thinking, even if the problem does not get solved right away.

Better Communication

Angry people tend to jump to—and act on—conclusions, and some of those conclusions can be very inaccurate. The first thing to do if you're in a heated discussion is slow down and think through your responses. Don't say the first thing that comes into your head, but slow down and think carefully about what you want to say. At the same time, listen carefully to what the other person is saying and take your time before answering.

Listen, too, to what is underlying the anger. For instance, you like a certain amount of freedom and personal space, and your "significant other" wants more connection and closeness. If he or she starts complaining about your activities, don't retaliate by painting your partner as a jailer, a warden, or an albatross around your neck.

It's natural to get defensive when you're criticized, but don't fight back. Instead, listen to what's underlying the words: the message that this person might feel neglected and unloved. It may take a lot of patient questioning on your part, and it may require some breathing space, but don't let your anger—or a partner's—let a discussion spin out of control. Keeping your cool can keep the situation from becoming a disastrous one.

Using Humor

"Silly humor" can help defuse rage in a number of ways. For one thing, it can help you get a more balanced perspective. When you get angry and call someone a name or refer to them in some imaginative phrase, stop and picture what that word would literally look like. If you're at work and you think of a coworker as a "dirtbag" or a "single-cell life form," for example, picture a large bag full of dirt (or an amoeba) sitting at your colleague's desk, talking on the phone, going to meetings. Do this whenever a name comes into your head about another person. If you can, draw a picture of what the actual thing might look like. This will take a lot of the edge off your fury; and humor can always be relied on to help unknot a tense situation.

The underlying message of highly angry people, Dr. Deffenbacher says, is "things oughta go my way!" Angry people tend to feel that they are morally right, that any blocking or changing of their plans is an unbearable indignity and that they should NOT have to suffer this way. Maybe other people do, but not them!

When you feel that urge, he suggests, picture yourself as a god or goddess, a supreme ruler, who owns the streets and stores and office space, striding alone and having your way in all situations while others defer to you. The more detail you can get into your imaginary scenes, the more chances you have to realize that maybe you are being unreasonable; you'll also realize how unimportant the things you're angry about really are. There are two cautions in using humor. First, don't try to just "laugh off" your problems; rather, use humor to help yourself face them more constructively. Second, don't give in to harsh, sarcastic humor; that's just another form of unhealthy anger expression.

What these techniques have in common is a refusal to take yourself too seriously. Anger is a serious emotion, but it's often accompanied by ideas that, if examined, can make you laugh.

Changing Your Environment

Sometimes it's our immediate surroundings that give us cause for irritation and fury. Problems and responsibilities can weigh on you and make you feel angry at the "trap" you seem to have fallen into and all the people and things that form that trap.

Give yourself a break. Make sure you have some "personal time" scheduled for times of the day that you know are particularly stressful. One example is the working mother who has a standing rule that when she comes home from work, for the first 15 minutes "nobody talks to Mom unless the house is on fire." After this brief quiet time, she feels better prepared to handle demands from her kids without blowing up at them.

Some Other Tips for Easing Up on Yourself

Timing: If you and your spouse tend to fight when you discuss things at night—perhaps you're tired, or distracted, or maybe it's just habit—try changing the times when you talk about important matters so these talks don't turn into arguments.

Avoidance: If your child's chaotic room makes you furious every time you walk by it, shut the door. Don't make yourself look at what infuriates you. Don't say, "well, my child should clean up the room so I won't have to be angry!" That's not the point. The point is to keep yourself calm.

Finding alternatives: If your daily commute through traffic leaves you in a state of rage and frustration, give yourself a project—learn or map out a different route, one that's less congested or more scenic. Or find another alternative, such as a bus or commuter train.

If you feel that your anger is really out of control, if it is having an impact on your relationships and on important parts of your life, you might consider counseling to learn how to handle it better. A psychologist or other licensed mental health professional can work with you in developing a range of techniques for changing your thinking and your behavior.

When you talk to a prospective therapist, tell her or him that you have problems with anger that you want to work on, and ask about his or her approach to anger management. Make sure this isn't only a course of action designed to "put you in touch with your feelings and express them"—that may be precisely what your problem is. With counseling, psychologists say, a highly angry person can move closer to a middle range of anger in about 8 to 10 weeks, depending on the circumstances and the techniques used.

What About Assertiveness Training?

It's true that angry people need to learn to become assertive (rather than aggressive), but most books and courses on developing assertiveness are aimed at people who don't feel enough anger. These people are more passive and acquiescent than the average person; they tend to let others walk all over them. That isn't something that most angry people do. Still, these books can contain some useful tactics to use in frustrating situations.

Remember, you can't eliminate anger—and it wouldn't be a good idea if you could. In spite of all your efforts, things will happen that will cause you anger; and sometimes it will be justifiable anger. Life will be filled with frustration, pain, loss, and the unpredictable actions of others. You can't change that; but you can change the way you let such events affect you. Controlling your angry responses can keep them from making you even more unhappy in the long run.

How to Handle Rude People: Ten Useful Tips

Ideas on How to Respond If Someone is Rude to You

s.e. Jones, Yahoo! Contributor Network
Sep 18, 2009 "Share your voice on Yahoo! websites. Start Here."

http://voices.yahoo.com/how-handle-rude-people-ten-useful-tips-4267543.html?cat=44

Everyone has to deal with a rude person at one time or another. Whether it's having someone cut in front of you in line, talking loudly during a movie, or saying things to or about you that needn't have been spoken. The thing is though, in dealing with someone being rude to or around you, it's more about how you handle the situation than whatever that other person may be doing, because you can't control their behavior, you can only control your own. That's where this list of ten useful tips on how to handle rude people comes in.

1 - Don't let them rile you. Rude people are either oblivious or don't care how you feel about what they are doing, thus, getting upset only serves to ruin your own mood, and perhaps that of those around you. Don't let other people being rude mess up your day.

2 - Stand up to them when you're right. Sometimes, you pretty much have to do something when someone is being rude; if the people behind you are talking during a movie perhaps, or someone is swearing around your children, or whatever. In these instances, it's best to state your case directly to them, ask them to stop and then disengage. If they continue with the rude behavior, seek other avenues of help, such as calling seeking out an usher, security guard or even the police.

3 - Get away from them. Probably the best thing you can do in most instances when encountering someone that is being rude, is to just get away from them as quickly and smoothly as possible. The reason for this is because there is little chance that anything you say or do will have much impact on the rude person. They are not going to change who they are simply because it annoys you.

4 - Send them an anonymous note. To avoid a confrontation, sometimes it might be appropriate to send the rude person an anonymous note telling them how you see their rude behavior and how it's affected you. In this way, you will feel that you have been heard, and the rude person will have been told, and who knows, maybe will change their future behavior.

5 - Whisper. If you must confront someone that is being rude, it is almost always best to speak in a low voice so as to not call attention from others as you say what you have to the rude person. Doing so will help keep the rude person off the defensive if they feel everyone around them is watching.

6 - Report them. Sometimes people being rude is actually illegal; cutting you off in traffic for example or driving dangerously can lead to serious consequences. Calling the police and reporting them is certainly an option you should consider.

7 - Ignore them. While it may be difficult to ignore the words or actions of rude people, more often than not it will be your best move. This is because rude people are used to being confronted and have a ready response, and because they are not afraid of confrontation.

8 - Take the high road. It's easy to find yourself falling down to the level of someone who has been rude to you, and to respond in kind. The problem with this though is afterwards, you won't feel that good about yourself. Try to keep in mind when someone is being rude to you, that you are better than that and don't have to get ugly to make your point.

9 - Keep it short and simple. When dealing with someone who has been rude to you, it is almost always best to keep your response as short and simple as possible. And once you've made your point, to avoid adding anything else. This is because the thing you don't want is a confrontation or argument. And the reason you don't want an argument is because you can't win. If someone chooses to be rude, there is truly nothing you can do but tell the how you feel. After that, it's just repetition.

10 - Don't engage them. Finally, don't allow yourself to be pulled in by rude people. Don't engage them in conversation or try to convince them of their errant ways. It's futile. The best thing you can do is get away from them, and barring that, to make your feelings known and then clam up.

These ten useful tips on how to handle rude people are for everyone to use when they are next finding themselves at the mercy of a rude person. If you do use one or more of these tips the next time it happens to you, I hope things work out better for you than perhaps they might have otherwise. Good luck.

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How to React When Someone Is Rude

Being polite doesn't mean you have to be a pushover. Here's how to deliver effective comebacks to even the most ill-mannered.

By Peggy Post

http://www.goodhousekeeping.com/family/etiquette/react-someone-rude-oct04

So Rude!

A speeder tailgates you on the freeway. An acquaintance asks how much money you make. A man you've just met makes suggestive remarks about your figure. The woman at the cash register fails to even acknowledge your presence, much less say thank you. For a word of only four letters, rude pops up in a wide variety of circumstances. Here's what you need to know so that the next time someone is disrespectful, you don't have to just stand there fuming.
The Retort Report

Can you rise above other people's bad behavior? Good. If not, here's what to say.

While shopping...
What happened: You're next in line at a counter where service is as slow as molasses. When a new line opens up, the person behind you rushes to get there first.
What you say: "I'm sorry, but I was ahead of you in line. I believe I'm next." If the person doesn't retreat, don't push the issue further — just speak to the manager. And suggest that when a cashier opens a new register, she should say, "May I help the next person in line?"

What happened: The person ahead of you in the supermarket express lane has a cart full of items.
What you do: Wait patiently for your turn, then find the manager and ask why the store's staff doesn't turn away rule breakers. He may have instructed them that "the customer is always right," but the policy might change if enough people complain.

At work...
What happened: You've left several messages for the director of an organization, but he hasn't called back.
What you do: Try once more. This time, say, "Mr. Johnson, this is Jane Stewart, following up regarding the information I've requested. I'm worried about meeting my deadline, so could you please give me a call?" If Mr. Johnson still doesn't respond, contact someone else at the organization, politely explain the problem, and ask for help.

What happened: In the lunchroom, you overhear a coworker trashing a workmate she doesn't like.
What you do: Speak up only if she's talking loudly enough for others to hear (as an eavesdropper, you have no grounds to complain). Say, "Sue, I don't mean to butt in, but if you're going to speak badly of Ann, could you do it someplace more private?" If her accusations are false, you could point that out as well.

Out in public...
What happened: You're walking down a street in your neighborhood when a man and his young son throw their empty paper cups to the ground, just a few feet from a trash bin.
What you do: Pick up the cups and deposit them in the receptacle — your way of teaching by example. The silent message you're sending is strong enough to make your point.

What happened: A passenger on the commuter train is taking up three seats with his bags. You're tired and desperately need to sit.
What you say: "Excuse me — would you mind if I sat down?" Many seat hoggers will comply if challenged. But if he refuses to make room, don't force the issue — he may be trying to provoke a confrontation. The last thing you want to do is rile up someone like that.

Power of Being Positive

When someone's behavior irks you, size up the situation: Will fighting back accomplish anything? I suspect you know the answer — but if you really feel the need to take a stand, then the best response is to kill the perpetrator with kindness. Good behavior is catching: The more you display it, the more it spreads. Next time your patience is tested, try these tips.

Stay calm. Take a deep breath, then count to 10 — that way, when you speak, you deliver a clear message rather than an emotional reaction.

Give the offender the benefit of the doubt. Though you may be tempted to take the rudeness personally, try to imagine what else the person might be going through — a family problem, trouble at work, etc.

Offer empathy. If a child behind you on a plane is kicking your seat, calmly say to the parent, "I know space is tight and kids will be kids, but your son has been kicking my seat since we boarded. I'd appreciate it if you'd ask him to stop."

Encourage a positive response. "Shut up! You're driving everyone crazy!" will raise the person's hackles. More likely to make him stop: "Many of us are trying to read...do you think you could please lower your voice?" said with a smile on your face.

Use humor when you can. If a friend comments, "You look awful!" counter with a sarcastic "How kind of you to say so!" Or just laugh it off.

Take it to a higher level. If you've been treated rudely by an employee of a business, tell the manager or write a letter to the company.

The Dirty Dozen

Based on mail from GH readers and on surveys conducted by the Emily Post Institute, these are the 12 most offensive behaviors ever.

• Using obscenities in public, especially around children
• Telling racist jokes
• Doing the "cell yell": the phone conversation loud enough for others to hear every word
• Treating service providers (salespeople, food servers, etc.) as if they're beneath you
• Letting kids run wild in public places
• Endangering others on the road by playing NASCAR wannabe
• Yelling at the referee, coach, players or other parents at a youth sporting event
• Leaving spit, trash or pet poop on the sidewalk
• Staying seated on a crowded bus or train when an elderly, pregnant or disabled person boards
• Charging thoughtlessly through crowds — especially when you're skating, biking or pushing a stroller
• Jumping the line (e.g., at the supermarket checkout)
• Lighting up around nonsmokers without asking first

Mathematical Knowledge: A Dilemma

Rui Vieira thinks of a number.

What exactly are the objects of mathematics, and how do they relate to our knowledge of them? Since Plato (427 BC–347 BC) such questions have been central to the philosophy of mathematics. Plato realized that mathematics seems to involve perfect circles, triangles, and so on. But as Plato also noticed, there are no perfect circles or triangles to be found in the world, only imperfect approximations. Imagine a polygon [shape] with an ever-increasing number of equal sides. As the number of sides approaches infinity, the polygon will become a circle. Thus a perfect circle may be conceived of as a regular polygon with an infinite number of infinitesimally small sides. So no matter how accurate our or our computer’s rendering of a circle may be, it will only be an imperfect approximation. Finite humans and their computers cannot create objects with infinite mathematical features, such as the infinite sides of our ideal circle. Plato concluded that since there are no perfect mathematical objects to be found in the world, the objects of mathematics – perfect circles, triangles, and indeed numbers themselves – must somehow exist as eternal abstract entities beyond space and time in some otherworldly Platonic heaven called the world of Forms (or Ideas). Plato’s particular type of mathematical realism (ie, of attributing objective reality to mathematical objects), has been one of the most prevalent views of mathematics among both philosophers and mathematicians ever since.

But for many philosophers after Plato – particularly the empiricists, who view all knowledge as ultimately acquired through experience – this transcendental view of mathematics was implausible. Thus, since Plato’s time, the history of the philosophy of mathematics has also been replete with anti-Platonist accounts attempting to bring mathematics back down to earth from its heavenly perch. One such early Twentieth Century account, called conventionalism, viewed mathematical statements as ‘analytic’ or true by convention. An analytic statement is any statement that’s true solely because of the conventional meaning of its terms – such as “All bachelors are unmarried”. Similarly, mathematical statements were to be viewed as true solely because of the conventional definition of their terms. Also in the early Twentieth Century, mathematician David Hilbert was pre-eminent in developing formalism, the view that mathematics is nothing more than the rule-based manipulation of symbols whose meanings are largely unimportant. Around the same time, mathematician L.E.J. Brouwer developed intuitionism, which viewed mathematical statements as being constructed through our subjective intuitions. These accounts of mathematics all rejected Platonism by avoiding any reference to mind-external mysterious abstract mathematical objects.

However, in an important paper, ‘Mathematical Truth’, in the Journal of Philosophy, Vol. 70 (1973), philosopher Paul Benacerraf argued that anti-Platonist accounts of mathematics deprive mathematical statements of their objective truth in the everyday popular sense, ie of the idea that mathematical truths are true whether anyone thinks about them or not. Objective truth is a property of mathematics which for most of us is obvious, but anti-Platonist accounts render mathematics subjective (although Benacerraf’s argument is directed at conventionalism and formalism, I don’t think intuitionism fares any better).

In his paper, Benacerraf shows that mathematical statements such as “There are at least three perfect numbers greater than 17” have the same grammatical and logical form as non-mathematical statements such as “There are at least three large cities older than New York”. Thus, Benacerraf contends that if mathematical statements are to be true in the same objective way that statements are true of everyday objects or facts, then the objects of mathematical statements must similarly exist independent of our subjective conventions or forms. That is, just as true statements about real cities must be true of objectively-existing cities, so must our true mathematical statements about numbers, perfect circles and the like, be true of objectively-existing numbers, perfect circles and the like. It would seem, therefore, that Platonism is the only mathematical account which would allow for the objective truth of mathematics.

How Do We Know Mathematical Truths?

However, a problem apparently besetting Platonism is also posed by Benacerraf. If mathematical objects such as numbers, perfect circles exist as abstract entities beyond space and time, neither affected by nor causing anything in our everyday world, as Platonism asserts, then how is it that we can come to have knowledge of them? If we must posit the existence of Platonic mathematical objects to make mathematical statements objectively true, then we are left with a dilemma: either to say that mathematics is subjective, or to explain how we ever have knowledge of inaccessible non-spatiotemporal objects.

I think only particularly anti-Platonist and whollyempiricist accounts of mathematics can fruitfully respond to this epistemological dilemma. In his influential book The Nature of Mathematical Knowledge (1984), philosopher Philip Kitcher proposed just such an account by updating the mathematical empiricism of John Stuart Mill. For Kitcher, mathematics consists not of abstract Platonic entities, but of generalized human empirical operations performed on physical objects: such as the collecting, correlating or segregating of, for example, pebbles. For instance, the mathematical statement 2 + 3 = 5 would, according to Kitcher, first refer to the collecting operation performed on objects called ‘making two’, then to the collecting operation of ‘making three’, and then to the final combining operation of ‘making five’. Higher mathematics would then be developed from these primitive perceptual beginnings, eventually allowing us to perceive the world’s empirical structures through mathematics. As Kitcher argues, we no longer even need to perform the empirical operations physically; we can simply perform them in our minds. Even so, the mental operations represented by our mathematical notations would still be traceable ultimately to empirical, that is, to physical, operations.

Given this empiricist account, ‘mathematical truth’ in the popular objective sense would be retained. Mathematical statements would be true in virtue of the objective reality of the empirical operations to which mathematical statements ultimately refer. Moreover, these operations are very much located in space and time, thus avoiding Benacerraf’s epistemological difficulties. We would have knowledge of mathematics not from knowing about abstract objects, but from knowing about the empirical operations on which mathematics is ultimately based.

An Infinite Maths Lesson

A common criticism of the empiricist account is that mathematics is unavoidably about infinities, such as infinite sets. We might rightly wonder how the mathematics of the infinite could ever have been developed under an empiricist account, since it is not possible for limited humans to perform infinite empirical operations on physical objects. One possible solution, suggested by Kitcher, would be for us to view the infinite domains of mathematics as idealized infinite empirical operations. We are to imagine a sort of ideal agent with infinite abilities somehow completing infinite empirical operations, thus providing us with an empirical account of infinities. For Kitcher, this conceptual device would be akin to useful idealizations in physics, such as ideal gases and frictionless surfaces.

Some have responded that ideal agents are themselves too similar to the very Platonic abstractions we’re trying to avoid. Others have argued that problematic idealizations regarding infinities need not be invoked if our empiricist countenances a potential infinity only, and not an actual infinity (a distinction first made by Aristotle). An actual infinity is a completed infinite totality, such as the completed set of natural numbers, or the completed decimal expansions of irrational numbers like pi. By contrast, a potential infinity is an unending, always incomplete process – such as counting the natural numbers, or calculating potentially infinite decimal expansions. Actual infinities, if viewed as empirical operations, would require the ideal agents proposed by Kitcher for their completion. But an empiricist could instead construe the infinities of mathematics as always incomplete – only involving potentially infinite operations. For example, our empiricist could construe pi as a conceptually-unending empirically-based operation or calculation, generating a potentially infinite but never completed decimal expansion. So although our generation of pi’s decimal expansion would be potentially infinite, it would always remain finite at any point in time, due to human limitation. This idea avoids the need for ideal agents. For our empiricist, there are no actual infinites, nor any actual infinite operations somehow ‘completed’ by ideal agents – there are only potential infinities.

As a further example, let’s return to the earlier-mentioned perfect Platonic circle which was conceived as an infinitely-sided polygon. This could also be construed by our empiricist not as an actual infinity, but as a potential infinity only – as a conceptually unending, always incomplete empirical operation. Contrary to Platonism, then, for the empiricist there exist no abstract and eternal Platonic numbers, perfect circles, triangles and the like, neither on earth nor in Platonic heaven: there are only empirical operations, and imperfect finite approximations.

Can Mathematical Truth Be Contingent?

A final concern for mathematical empiricism is that if mathematics is indeed ultimately reducible to observable operations, then that makes the truth of mathematical statements contingent, since empirically-derived statements are contingently true for the most part. But it seems to most of us that on the contrary, mathematics is necessarily true, independent of any contingency. A quick empiricist retort would be that just because mathematics seems to be necessarily true, this doesn’t make it so. As with John Stuart Mill, our empiricist could quite happily accept mathematics as only contingently true.

To be sure, empiricist accounts of mathematics do make understanding higher mathematics more difficult, and thus less appealing to many. Nevertheless, empiricist accounts at least prevent the unnecessary bloating of our metaphysics, since they avoid the need to assume the existence of abstract Platonic mathematical objects, and a realm for them to exist in.

© Rui Vieira 2010

Rui Vieira is a graphic designer living in Mississauga, Ontario.

http://philosophynow.org/issues/81/Mathematical_Knowledge_A_Dilemma