No

Most mathematical truths cannot be proved. The overwhelming majority of mathematical relations cannot be known. The overwhelming majority of numbers can't be represented. Only a tiny fraction of mathematical functions can be computed.

And, the above restrictions are imposed by physics!

I'm afraid to say it but math is or seems PERFECT. Yet we all know there can be no such thing as perfection. There's always a negative that comes with every positive.

My question then is:

Is mathematics flawless? — TheMadFool

Do you mean to distinguish a concept of perfection from a concept of flawlessness?

I'm not sure how I'd distinguish one from the other, and I'm not sure how to apply either term to particular cases without some arbitrary standard we might use to distinguish things called perfect or flawless in one respect from things that called imperfect or flawed in the same respect.

One such arbitrary standard leads us to say: Each thing is perfectly and flawlessly just what it is. I suppose this truism applies to mathematics as well as to anything else.

We might construct another arbitrary standard by specifying a range of the spectrum of visible light as "red", and carving out a small sub-range as "perfectly red", and on this basis distinguish perfect red, imperfect red, and not red.

But apart from the specification of some such standard or criterion, I'm not sure what it means to say a thing is "perfect", and I don't see how it helps the picture to substitute "flawless" for "perfect".

Most mathematical truths cannot be proved. The overwhelming majority of mathematical relations cannot be known. The overwhelming majority of numbers can't be represented. Only a tiny fraction of mathematical functions can be computed. — tom

tom's critique cuts right to the heart of a question about the limits of mathematics.

But to show the limits of a thing is not necessarily to show that it is "imperfect" in any given respect. The respect in which we're supposed to say whether "mathematics is flawless" has not yet been specified.

A chef's knife is good for cutting food, but not for cutting brick or steel, nor for heating or cooling water, nor for an infinite range of other purposes. Shall we call the knife "imperfect" or "an imperfect knife" on these grounds?

Should we assume that the "provability", "knowability", "representability", and "computability" you indicate are among the most relevant criteria according to which we might judge whether mathematics is "perfect" or "flawless"?

I'm sorry, I think I haven't been clear enough.

Math is now a universally applicable tool, finding its way into almost every subject worth studying. Implicit in this is the premise that math is the tool of preference. In other words it is perfect and we believe as true the results of mathematical calculations/manipulations; math, invariably, improves or underscores the credentials of any study worth its money.

My question is is math deserving of this respect and trust? Could it not be flawed? What does a mathemstical analysis of a given subject deprive us of? Are there some areas of study where math is harmful instead of beneficial?

My question is is math deserving of this respect and trust? Could it not be flawed? What does a mathemstical analysis of a given subject deprive us of? Are there some areas of study where math is harmful instead of beneficial? — TheMadFool

In that case, the mathematics is perfect - its use however is not.

As systems become more complicated, in order to make them tractable, it is inevitable that certain simplifying assumptions are made. You might be able to write down an exact set of equations, but can you solve them? More often, the equations themselves are approximations.

My question is is math deserving of this respect and trust? Could it not be flawed? What does a mathemstical analysis of a given subject deprive us of? — TheMadFool

I assume you understand that to have a sound logical conclusion requires that you start from sound premises. Mathematics, being a form of logic, is no exception. This means that how we apply the numbers, how we evaluate things to be expressed mathematically (form the premises) is a crucial component of the reliability of the mathematical conclusions.

A chef's knife is good for cutting food, but not for cutting brick or steel, nor for heating or cooling water, nor for an infinite range of other purposes. Shall we call the knife "imperfect" or "an imperfect knife" on these grounds? — Cabbage Farmer

I think the question which TheMadFool is asking now, is what type of things is mathematics not good for. We could start with morality. I think that most people would agree that mathematics is not very good for solving moral issues. If we move from morality into social studies, we will find some areas where mathematics becomes useful, through the use of statistics, probabilities, and such things. I think that we might find a grey area here, between social sciences and moral philosophy, where some might argue mathematics is useful and others might argue that mathematics is not useful. If one is convinced that the mathematics is useful, when it is not, then the use of mathematics would be harmful.

Are there some areas of study where math is harmful instead of beneficial? — TheMadFool

Suppose that an individual is using mathematics according to some falsely determined principles of application, or in a subject where mathematics in not applicable. That person might be convinced, simply because the mathematics was applied, and the mathematics produced conclusions, that these conclusions were truths. But if these aren't real truths, then the use of mathematics here is harmful. It is not "mathematics" itself which is harmful, but it is the person's attitude toward mathematics which is harmful. Likewise, if one uses deductive logic without adequately judging the propositions accepted as premises, and believes in the truth of the conclusions produced by the logic, we have the same problem. It is not the logic itself which is the problem, it is the way that it is used.

We could go on, and examine the principles of logic itself, to see if the principles are sound. If unsound principles are accepted by logicians, and find there way into the logical system, through common convention, then we have a problem within the logic itself. Modern logic is very complicated with modal logic, fuzzy logic, operators, and such things. All of these principles need to be thoroughly examined to understand whether they are acceptable, and in which instances they are applicable. We have the same situation in mathematics, there are many new concepts and principles such as imaginary numbers, which are constantly being produced. In all of these new concepts, we need to be understood and determine sound applicability. If there is no sound applicability then this mathematics itself has a problem.

I agree with you. Math is a useful tool. But we need to be careful where we apply it.

mplicit in this is the premise that math is the tool of preference. — TheMadFool

You need to calculate the load capacity of a bridge; thrust required for take-off; amount required to settle your tax bill; flour required for your cake recipe.

Can you suggest a better 'tool'?

Can you suggest a better 'tool'? — Wayfarer

Nope.

I just wanted to know if math had shortcomings of its own. Metaphysician Undercover said math cannot be used in morality.

Math is now a universally applicable tool, finding its way into almost every subject worth studying. — TheMadFool

Some applications of mathematics are applications to contexts that do not involve "studying a subject" in the ordinary sense of that phrase. For instance, I might use algebra to figure out how much I owe a creditor each month, how much cement I should purchase to replace the front wall of my porch, or how many hours I might spend each week in the practice of philosophical discourse without neglecting other obligations.

On the other hand, mathematics does not seem "applicable" or relevant to some pursuits worth undertaking, though mathematics may be applied, one way or another, even to a study of those pursuits.

Implicit in this is the premise that math is the tool of preference. — TheMadFool

I wouldn't put it that way. Mathematics is rightly held to be a reliable tool, but not "the tool of preference" in every endeavor. For instance, I suppose we can apply mathematics to the study of poetry, but mathematics is not "the preferred tool" of most poets, most audiences of poets, or most scholars of poetry. Math is one tool among others, with various roles in various sorts of endeavor, central in some, peripheral in others, practically irrelevant in yet others.

More specifically, mathematics is useful when we are able and willing to count and measure, and to analyze quantitative relationships in what has been counted or measured. For instance, the ratio of soldiers in two armies, or the number of hours it would take to travel from one city to another at various average rates of speed. The use of numbers is part of the art of the military general, but only one part; a good mathematician is not necessarily a good general or a good strategist.

In other words it is perfect — TheMadFool

Again you resort to the use of this word, "perfect", and I have no idea why.

Now it seems you suggest the role you impute to math as "the universal tool of preference" is what warrants the claim about its "perfection". I've already suggested that the role of mathematics is rather limited or even negligible in many fields of endeavor; perhaps this is enough to put talk of "perfection" to rest?

and we believe as true the results of mathematical calculations/manipulations; — TheMadFool

The results of mathematical calculations are purely mathematical and pertain to nothing but number in general, except insofar as we coordinate them with something else in the world, for instance by way of measurement.

If a scale says I weigh 240 pounds and I say "I don't believe it", normally it means I think that something's wrong with the scale, not that something's wrong with the concept of pounds, or the concept of weight, or the concept of number. Suppose I test the scale by weighing dumbbells I have good reason to believe I know the weights of, and thus confirm that I weigh 240 pounds. Suppose, further, that I'd prefer to weigh 200 pounds. How many pounds would I need to lose in order to reach my target weight? What could it mean to disbelieve the results of the arithmetical calculation by which I determine that I have 40 pounds to lose?

It's not the numbers and the calculations that we take issue with in such cases, but rather the measurements or the extra-mathematical use of those measurements.

The calculations speak for themselves and stand on their own, outside the context of any such application.

math, invariably, improves or underscores the credentials of any study worth its money. — TheMadFool

Again I disagree, along the same lines as before. Arguably an undue use of mathematics degrades the study of poetry, while even an appropriate use of mathematics adds little value to that study.

Of course there's hardly any money in poetry nowadays, but I suppose we can find examples analogous in the relevant respect.

My question is is math deserving of this respect and trust? — TheMadFool

I'm still not sure what kind of respect and trust you imagine mathematics to have in our society.

It surely deserves our respect as a reliable and useful set of conventions for operating with numbers in general, and I suppose as the best example of formalized and formalizable rational thought.

That doesn't mean that every application of mathematics is worthy of the same respect and trust. Much as the respect and trust we might have for the English language doesn't extend to every single use of the English language.

Could it not be flawed? — TheMadFool

Calculations, measurements, applications, attempted proofs can be flawed, and often are. But could "mathematics itself" be flawed? What is this question supposed to ask?

Do you mean, could it be the case that we've got it all wrong, and that 2 + 2 = 5? I'm not sure what this would mean, either.

What does a mathemstical analysis of a given subject deprive us of? — TheMadFool

Nothing at all, unless we're so biased in favor of mathematical analysis that we forget about other relevant features of a given task. Moderation in all things.

Are there some areas of study where math is harmful instead of beneficial? — TheMadFool

None, if it's applied in the right way for the right reasons.

I think the question which TheMadFool is asking now, is what type of things is mathematics not good for. — Metaphysician Undercover

I'll say, mathematics is useful in a given context insofar as counting and measuring and analyzing quantitative relationships are useful in that context.

We could start with morality. I think that most people would agree that mathematics is not very good for solving moral issues. — Metaphysician Undercover

I expect the majority of utilitarians might object to that claim.

If we move from morality into social studies, we will find some areas where mathematics becomes useful, through the use of statistics, probabilities, and such things. I think that we might find a grey area here, between social sciences and moral philosophy, where some might argue mathematics is useful and others might argue that mathematics is not useful. If one is convinced that the mathematics is useful, when it is not, then the use of mathematics would be harmful. — Metaphysician Undercover

I'll say, if one finds mathematics useful in his own moral thinking, let him use it; and likewise with every other field of endeavor. And if two people disagree about the utility or aptness of some particular application of mathematics, let them work out their disagreement or part ways.

It may be that some uses of mathematics will be of interest to moral philosophers, for instance in assigning "weights" to each "value" in a moral model; or in the collection and analysis of big data pertaining to moral behavior, norms, and intuitions.

To say that moral thinking or moral phenomena can be "quantified" in this way, and that there may be some use for such quantitative approaches, is not to suggest that such practices could displace ordinary moral reasoning and intuitions, or should be required for responsible moral discourse.

My question is is math deserving of this respect and trust? Could it not be flawed? What does a mathemstical analysis of a given subject deprive us of? Are there some areas of study where math is harmful instead of beneficial? — TheMadFool

Maths is a model of reality as a perfect syntactical mechanism. It predicts the patterns that will be constructed as the result of completely constrained processes. So if reality is also spontaneous and vague in some fundamental way, maths can't "see" that. It presumes an absolute lack of indeterminism to give a solid basis to its story of determinism.

This isn't a big problem because humans using maths as a tool can apply it with "commonsense". And when humans are actually building "machines" - as they mostly are in maths dominated activities - then the gap between the model and the world being created is hardly noticeable.

The key issue when it comes to applying commonsense is the making of measurements. We have to use our informal judgement when plugging the numbers meant to represent states of the world into our models or systems of equations. So it is outside the actual maths how much we round numbers up, how we spread our sampling, etc, etc. Garbage in, gabage out, as they say.

The flipside of all this is then when we are dealing with a world that is complex and it is not absolute clear what to measure. Or worse still, the world may be actually spontaneous or vague and relatively undifferentiated, and so every definite-sounding measurement will be dangerously approximate.

So the issue is no that maths simply fails to apply to some aspects of life. If you are talking ethics or economics for example, game theory gives some completely exact models which can be used. However then they have to presume a world of machine humans - perfectly rational actors. Thus judgement then has to come in about how much one can rely on this particular modelled presumption. Can the actual model work around the issue by adding some further stochastic factor or is the real world variance in some way "untameable".

So maths works well when the world is made simple - as when building machines. And then complexity can cause fatal problems for this mechanistic modelling when the complexity makes good measurement impractical. For a chaotic system, it may be just physically impossible to measure the initial state of the world with enough accuracy.

Then where the metaphysical strength issues really bite is if the world is actually spontaneous or vague at a fundamental level - as quantum physics says it is.

The final source of indeterminism is the semiotic one - the issue of semantically interpreting a sign or mark. We can both see a word like "honesty" or "beauty" written on a page as a physical symbol. But how do we ever completely co-ordinate our understandings or reactions to the word?

So clearly, to the extent that human lives revolve around the common understanding of systems of signs, there is an irreducible subjectivity that makes maths a poor tool for modelling what is going on. That would be why philosophers would put aesthetics and morality in particular beyond the grasp of such modelling.

However as with the probabilisitc modelling of chaotic and quantum processes, that is not to say maths couldn't be applied to semiosis. Instead, it may be the case that we just haven't really got going on trying. It is not impossible there would be a different answer here in another 100 years.

I expect the majority of utilitarians might object to that claim. — Cabbage Farmer

I'm sure they might, but I would argue that this is an example of where the use of mathematics is harmful, when one thinks that mathematics is useful, but it is not. This person produces conclusions believed to be right, with the certitude associated with mathematics, which might actually be wrong.

I'll say, if one finds mathematics useful in his own moral thinking, let him use it; and likewise with every other field of endeavor. — Cabbage Farmer

Why would you say this? If you saw an individual applying logic to false premises, and proceeding to act on the conclusions, wouldn't you feel obliged to inform that person that the conclusions are false? And if that person was acting immorally because the mathematics told him to, do you think that this is ok? Maybe the mathematics told him that if he robbed a bank he would have more money and more money would allow him to buy more things, and having more things would allow him to me more generous. So he thought that robbing the bank would improve his moral character.

To say that moral thinking or moral phenomena can be "quantified" in this way, and that there may be some use for such quantitative approaches, is not to suggest that such practices could displace ordinary moral reasoning and intuitions, or should be required for responsible moral discourse. — Cabbage Farmer

The issue though, is what would be the case if moral issues cannot be quantified in this way. If they cannot, then the person who uses mathematics in this way will inevitably go wrong. But by assuming that mathematics can be used in this way, that person will be convinced by the mathematics, that he or she is right, and will proceed to act in the wrong way, claiming to be right. So before one proceeds to use mathematics this way, one ought to demonstrate that moral issues can be quantified in this way.

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I'll say, mathematics is useful in a given context insofar as counting and measuring and analyzing quantitative relationships are useful in that context. — Cabbage Farmer

Although mathematics is commonly associated with quantity, it is more broadly the application of necessary reasoning to hypothetical or ideal states of affairs. As such, the usefulness of its conclusions is entirely dependent on how well its initial assumptions capture the significant aspects of reality - not just the model itself, but the representational system that governs its subsequent transformations. It is thus highly suitable for analyzing natural phenomena, since the habits of matter are largely entrenched; but not so much for analyzing human behavior, since the habits of mind are much more malleable.

Thanks for the replies.

I'd just like to point out that math is central to everything there is.

The simple reason is the ''ER'' and ''EST' words.

BettER, HeaviEST, saddER, whitER, etc.
The above words are comparison words and as such all are an attempt to quantify or in other words all want to use math (the ultimate quantifying tool).
How can we compare two or more things without quantification (use of math) knowing that quantification is necessary in that arena?

How can we compare two or more things without quantification (use of math) knowing that quantification is necessary in that arena? — TheMadFool

Quantification is NOT necessary for comparison. If my son and daughter stand next to each other, anyone can observe that my son is taller than my daughter, without quantifying anything at all. We make these kinds of qualitative comparisons all the time, and mathematics plays no role in them whatsoever.

You have forgotten to consider how we compare one distinct quality to another. All of your examples are of how we take one quality, one category, and compare within that category. Now you need to consider how we compare apples to oranges, one category to another. This is what we do in morality, we compare distinct types of activities to see which is better. But as I said before, better is not quantifiable. So as much as it may be true that all quantifications are modes of comparing qualities, it is not true that all qualities may be compared through quantification.

We make these kinds of qualitative comparisons all the time, and mathematics plays no role in them whatsoever. — aletheist

What I think is mathematics brings precision into the matter and through this a finely nuanced apprehension of the situation.

In your example of the height comparison of boy and girl math helps us to answer who is and by how much taller between the two.

Doesn't the exactitude of math help us fine-tune our knowledge of our universe?

But as I said before, better is not quantifiable — Metaphysician Undercover

I think the words of comparison we use (better, more beautiful, most ugly, etc) betray an innate desire to quantify things.

Quantification helps us make better decisions.

In your example of the height comparison of boy and girl math helps us to answer who is and by how much taller between the two. — TheMadFool

But that is a different question than the merely qualitative comparison of which one is taller than the other, which requires no measurement - and therefore no math - as long as they are standing together.

Doesn't the exactitude of math help us fine-tune our knowledge of our universe? — TheMadFool

The exactitude of math is only possible because it deals with purely hypothetical or ideal states of things. Exact analyses of mathematical models can only serve as approximate analyses of actual situations.

I think the words of comparison we use (better, more beautiful, most ugly, etc) betray an innate desire to quantify things. — TheMadFool

As much as I don't believe that, it still may be the case. But how would you back up that claim? How do you justify it? For instance, I like hockey better than football, so for me hockey is better than football. How does this indicate to you that I have an innate desire to quantify things? I have no desire to quantify these things, I simply prefer watching hockey over football.

We've found that quantification is a good way of judging things. The fact that we quantify many things indicates that we have a desire to judge things. So the logical premise should go this way, "if one has the desire to quantify, then one has the desire to judge, quantifying being a means for judging. You want to commit the logical fallacy called affirming the consequent. You want to say that since we have the desire to judge, therefore we have the desire to quantify

But that is a different question than the merely qualitative comparison of which one is taller than the other, which requires no measurement - and therefore no math - as long as they are standing together. — aletheist

Qualitative comparisons are subjective. Quantitative comaprisons are objective.

I think there's a instinctive preference for the latter. With precise measurements come precise decisions.

The exactitude of math is only possible because it deals with purely hypothetical or ideal states of things. Exact analyses of mathematical models can only serve as approximate analyses of actual situations. — aletheist

Mathematical models are an approximation yes but they work - they grasp at the key players in any situation, sweeping aside the irrelevant, the redundant, the red herrings, etc.

How does this indicate to you that I have an innate desire to quantify things? I have no desire to quantify these things, I simply prefer watching hockey over football. — Metaphysician Undercover

I believe words of comparison like ''more'', ''most'', ''least'', ''greater'', ''braver'', etc. are indications of the need to quantify all aspects of experience.

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I believe words of comparison like ''more'', ''most'', ''least'', ''greater'', ''braver'', etc. are indications of the need to quantify all aspects of experience. — TheMadFool

But why? You have not provided evidence for this. All these words indicate is that we like making relationships between things. Saying, "I like hockey more than football" is just an expression of preference: it indicates no desire to quantify pleasure in any numerical sense. We also know that there are aspects of experience that cannot be quantified, neither in any specific sense (emotions) and even in theory: I cannot quantify "blue".

But why? You have not provided evidence for this. — Chany

The words I mentioned are the evidence. They make sense only in a quantified universe.

''I can't quantify blue''

The words I mentioned are the evidence. They make sense only in a quantified universe. — TheMadFool

What is the quantity of happiness, how do we go about measuring it, and how can we see the accuracy of quantification? Again, I feel no need to pull out a calculator when someone says "I like hockey more than football".

Also, not seeing anything with color.

Qualitative comparisons are subjective. Quantitative comaprisons are objective. — TheMadFool

Nonsense. My son is objectively taller than my daughter. Yellow is objectively lighter in color than indigo. A pillow is objectively softer than a stone. There are ways to assign quantities to each of these qualities, but the unit of measurement in each case will be arbitrary.

On the other hand, people make "quantitative" comparisons that are subjective all the time. This movie is four stars, that one is only two. This essay gets a grade 93%, that one gets only 88%. This soccer player has an 83 rating in the video game, that one only has a 79.

Mathematical models are an approximation yes but they work - they grasp at the key players in any situation, sweeping aside the irrelevant, the redundant, the red herrings, etc. — TheMadFool

They can be used successfully, but not always or by just anyone. It takes good judgment developed through experience to ascertain the significant aspects of an actual situation that warrant being included in a mathematical model.

"I like hockey more than football". — Chany

That's quantification if ever I saw one.

My son is objectively taller than my daughter. Yellow is objectively lighter in color than indigo — aletheist

There are times when one isn't sure of who is taller e.g. when two people are very close in height. In such cases math lends the precision to validate the ''who is taller'' judgment.

I think no other human invention has that much depth and breadth of application as mathematics. — TheMadFool

Spoken language and written language. Even math depends on these two.

That's quantification if ever I saw one. — TheMadFool

No, it is not. That is qualitative.