Perfection and Math
In any conversation one doesn't have to wait long before numbers come into the picture. Games, cooking, reading, laundry, shopping, peeing, doing number 2, dancing, music, science, you name it. It's all numbers, numbers and more numbers.
I think no other human invention has that much depth and breadth of application as mathematics.
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? Yes/no please give reasons.
I think no other human invention has that much depth and breadth of application as mathematics.
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? Yes/no please give reasons.
Comments (42)
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!
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".
Quoting 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"?
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?
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.
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.
Quoting 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.
Quoting 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.
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'?
Nope.
I just wanted to know if math had shortcomings of its own. [B]Metaphysician Undercover[/b] said math cannot be used in morality.
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.
Quoting 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.
Quoting 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?
Quoting 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.
Quoting 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.
Quoting 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.
Quoting 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.
Quoting 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.
Quoting TheMadFool
None, if it's applied in the right way for the right reasons.
I'll say, mathematics is useful in a given context insofar as counting and measuring and analyzing quantitative relationships are useful in that context.
Quoting Metaphysician Undercover
I expect the majority of utilitarians might object to that claim.
Quoting 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.
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'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.
Quoting 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.
Quoting 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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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?
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.
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?
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.
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.
Quoting 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.
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
Qualitative comparisons are subjective. Quantitative comaprisons are objective.
I think there's a instinctive preference for the latter. With precise measurements come precise decisions.
Quoting 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.
I believe words of comparison like ''more'', ''most'', ''least'', ''greater'', ''braver'', etc. are indications of the need to quantify all aspects of experience.
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".
The words I mentioned are the evidence. They make sense only in a quantified universe.
''I can't quantify blue''
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.
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.
Quoting 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.
That's quantification if ever I saw one.
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.
Spoken language and written language. Even math depends on these two.
No, it is not. That is qualitative.
"I like hockey more than football"
Can you clarify
It is categorical (or sometimes called qualitative) because things such a like and dislike are categories. There are no objective standards to measuring degrees of likeness. Even though you can assign numbers to the categories, they are still categories.
Here is the text book definition, pulled from one of my statistics course books.
Like and dislike as far as math is concerned are categorical labels and not numerical measurements. If I say I like something more than something else then that is its label; that is not a numerical measurement.
That would be why probability ranges from 0 to 1 then? Categorical differences are measured relatively in fact?
Probability ranges 0 to 1 because you can never have something with a higher probability of 100% or a probability lower then 0%
Statistically probability is the proportion of possible outcomes from the repeated exercise of a random event.
Categorical differences can be a number of things from colors, does a medicine make your feel better, is it night or day, etc.
So I am not entirely sure what you mean by "measured relatively", as it might depend on what you are trying to find out. But for an example if I wanted to know if flowers grow more in day or night, then I would have to compare the two, and that would actually be a study that used both categorical and quantitative variables.
Quoting Metaphysician Undercover
If the person drawing conclusions is careful to distinguish the measurements and calculations on the one hand, from assumptions and inferences about what has been measured on the other, then I don't see what harm there is in it.
The certitude associated with mathematics does not extend to sloppy inferences drawn on the basis of careful measurements and valid calculations.
A person who jumps to conclusions is likely to err, whether or not he uses mathematics to support his arguments.
Quoting Metaphysician Undercover
I say it because I see no such unreasonable act implied by the general idea of using mathematics in association with moral thinking.
You seem convinced that any such use of mathematics must be illogical, immoral, or harmful, but you haven't made it clear to me how this view of yours is warranted in your discourse.
The bank robber in your example needn't have used arithmetic to conclude that he would have more money after a successful robbery than he had before. It's his moral reasoning that is wrong, not his rough quantitative judgment. Getting hung up on the culprit's correct use of quantitative reasoning distracts from the real problem in this case.
We can use correct math or correct English to do good or to do harm, to make valid or invalid arguments, to speak truth or to speak falsehood. That doesn't tell us anything about whether it's right or wrong in general to use math or English.
Quoting Metaphysician Undercover
What counts as a "moral issue" for the sake of this conversation? I've already given at least two examples of the way that mathematics can be applied to moral thinking:
Quoting Cabbage Farmer
Do you have some reason to suppose that such uses are illogical, immoral, or impossible? I think it's clear enough that mathematics can conceivably be used for such purposes.
Notice that in such examples, mathematics cannot determine moral models or judgments all by itself. We'd still have to rely on moral agents to supply moral values, moral intuitions, and so on. Math can't tell us what is good or bad, but it can conceivably play some role in helping us sort out our thinking and observations about morality.
Likewise, math is useful for physical science, but doesn't determine physical models or judgments all by itself.
It may be I'm unacquainted with your idiom.
What is "necessary reasoning"?
What sort of necessary reasoning is commonly associated with quantity?
What sort of necessary reasoning is not commonly associated with quantity?
In what broad sense is mathematics "the application of necessary reasoning"?
Can we apply reasoning to real states of affairs, or only to hypothetical and ideal states of affairs?
Can we apply necessary reasoning to real states of affairs, or only to hypothetical and ideal states of affairs?
I suppose I'm willing to agree that mathematics is the "science of formal systems". This suggests one way to flesh out the meaning of a phrase like "necessary reasoning": Necessary inferences in a given formal system are valid moves according to the rules of the system. Of course the signs and statements in a formal system are meaningless gibberish, empty form, until they're interpreted one way or another.
Concepts of number are prior to the formalization of number-concepts and number-relations by way of abstract signs. Concepts of distance are prior to the formalization of geometric concepts and relations by way of abstract signs. Concepts of order, time, modality, or inference are prior to the formalization of ordinal, temporal, modal, or inferential concepts and relations by way of abstract signs. And so on.
An agreement to interpret a particular formal system as representing, say, number-concepts and number-relations in general, does not settle questions about how to apply the system to specific cases.
Quoting aletheist
Do you mean to say: The usefulness of conclusions [obtained by an application of necessary reasoning to hypothetical or ideal states of affairs] is entirely dependent on how well the initial assumptions [of the application] capture all aspects of reality relevant [to any formally expressible judgment pertaining to the hypothetical or ideal states of affairs]. This includes assumptions that determine the model, as well assumptions that determine the representational system that governs its subsequent transformations.
Or how would you correct that paraphrase?
What is the difference between the "model" and the "representational system" that governs subsequent transformations of the model? How are each of these terms related to the definition of "hypothetical or ideal states of affairs"?
How do we isolate "assumptions" that guide the definition of model, representational system, and states of affairs? Is it the assumptions, or the whole package, that determines the aptness of the conclusions obtained?
What role does measurement play in your account? Or more generally: How are "aspects of reality" translated into formal signs in the model, or into bits of "necessary reasoning"?
Quoting aletheist
It's not clear how this follows from anything you've just said about "necessary reasoning".
Isn't human behavior part of nature? Aren't human behaviors "natural phenomena"?
One of the ways that minds are malleable is that they can be changed by producing changes in brains. I presume we agree that brains have the "habits of matter". Have you signed up for special troubles associated with dualism?
Moreover, there's a difference between "analyzing" a particular phenomenon, and analyzing many particular phenomena of the same kind to arrive at a generalization with proven predictive power. I suppose a mathematician can use his art to "analyze" trajectories of water or smoke in a video recording of a belly-flop or a forest fire; but an analysis of a large collection of such analyses isn't necessarily useful for predicting trajectories in future dives or fires with great precision, because of the complexity of the underlying phenomena.
Likewise, a mathematician can use his art to analyze video recordings in which various human animals take various pathways through city traffic, or respond in various ways to fuzzy puppies in a park, or to humans of superior rank in an office. I don't see any reason not to call this an analysis of human behavior, and to acknowledge that the predictive power of such analyses resembles the predictive power of analyses of other complex observable phenomena in nature.
On the basis of such analyses, we develop more or less reliable models with more or less predictive power. There's a statistical and probabilistic character to such models, whether we're talking about human beings or molecules of smoke or water.
Accordingly, I'm not sure what difference you've gestured at here, and what relevance it may have for our conversation about the uses of mathematics.
OK, his "moral reasoning" is wrong. But that's the whole point, that mathematics cannot be used for moral reasoning. The issue here is how can one use mathematics in performing moral reasoning. I think that it can't be done. And the further point is that if one does think that there is a way to use math in moral reasoning, that individual could very easily have a wrong answer (because you actually can't use mathematics in moral reasoning), and also be convinced that it is right answer because math was used.
Quoting Cabbage Farmer
Well this is my whole point. We cannot use mathematics to make moral judgements. You seem to be arguing that we can. But now you've qualified that to say that we would have to have moral agents, to supply moral values. This implies that the moral judgement has already been made by the moral agent. So the mathematics is not going to be used to make any moral judgement, this is already supplied by the moral intuition of the moral agent. What is the mathematics to be used for then? If the moral agent supplies the moral values, then the moral questions of what is right and wrong, has already been answered, prior to applying the math.
Deriving conclusions from information that is already present in the premisses. Also known as deductive reasoning.
Quoting Cabbage Farmer
Arithmetic is an obvious example, such as 2+2=4.
Quoting Cabbage Farmer
Syllogisms are an obvious example, such as "All men are mortal, Socrates is a man, therefore Socrates is mortal."
Quoting Cabbage Farmer
I am following Charles Sanders Peirce in suggesting that all necessary reasoning is fundamentally mathematical reasoning. He defined mathematics as the science of drawing necessary conclusions about ideal states of affairs by means of diagrams, which are representations that embody the significant relations among the parts of their objects.
Quoting Cabbage Farmer
We can apply reasoning to real states of affairs, but typically we do so by modeling them as ideal states of affairs. We have to identify the significant parts and relations of the actual situation and create a diagram accordingly within an appropriate representational system, whose rules govern our transformations of the diagram.
Quoting Cabbage Farmer
Only to ideal states of affairs, since we can never be absolutely sure that real states of affairs are completely deterministic.
Quoting Cabbage Farmer
Your paraphrase seems about right.
Quoting Cabbage Farmer
The representational system is a set of rules, such as Euclid's postulates for geometry. It is ideal because it may or may not accurately capture aspects of reality; for example, non-Euclidean geometry is more appropriate in certain cases. The model is a diagram constructed and manipulated in accordance with those rules, such as a sketch of a triangle and any auxiliary elements that must be added in order to carry out a particular proof. It is ideal because the actual drawing includes features that are irrelevant to the problem at hand, such as the thickness of the lines and their deviation from being perfectly straight.
Quoting Cabbage Farmer
Isolating assumptions can be quite a challenge, especially for more complex situations, such as a computer model of a structure that I analyze in accordance with the principles of mechanics in order to ascertain whether all of the members and connections are adequately designed for the forces to which they might be subjected. It is the whole package that validates the conclusions - the representational system and its assumptions, the individual model and its assumptions, and their correspondence (in some sense) to the actual state of affairs. As I like to put it, engineers solve real problems by analyzing fictitious ones, which involves simulating contingent events with necessary reasoning.
Quoting Cabbage Farmer
The representational system is often grounded in past inductive investigations; i.e., science. We have learned from collective experience that making certain assumptions and applying certain rules generally produces results that are useful. Learning how to create appropriate models is part of the personal experience that is required to develop competence in a particular field, since it often involves exercising context-sensitive judgment, not just following prescriptive procedures. Again, the modeler must be able to ascertain which parts and relations within the actual situation are significant enough to warrant inclusion in the model.
Quoting Cabbage Farmer
The behavior of matter much more closely conforms to exceptionless laws of nature than the behavior of people, even taking their habits into account. As such, necessary reasoning is much more likely to be useful and effective in modeling and predicting the behavior of material things than the behavior of intelligent and willful people, who are quite capable of deviating from their habits at any time.
Quoting Cabbage Farmer
No, Peirce vigorously rejected both dualism and materialism/physicalism; he wrote, "The one intelligible theory of the universe is that of objective idealism, that matter is effete mind, inveterate habits becoming physical laws."
Quoting Cabbage Farmer
That is fine. Hopefully these additional responses have helped clarify my thoughts for you.
I'm not sure what this means.
Is math "central" to the Sun, or is it central to our perception of the Sun, or is it central to a scientific understanding of the Sun -- or is it merely a tool that has proven to be extremely useful in cultivating empirical knowledge of natural phenomena, including the Sun?
It seems to me we'd know nothing at all about the Sun if we relied on nothing but mathematics to inform our views on the Sun.
Accordingly, I'd say it's not mathematics that's "central" to our understanding of the Sun, but rather our observations of the Sun, and thus the Sun itself, that's central to our understanding of the Sun. And likewise with all other observable phenomena.
Of course mathematics is extremely useful in the analysis of such observations.
Quoting TheMadFool
Once we have the capacity to recognize various items as of the same sort, we're on the road to number. This man, another man, and another.... This spear, another spear, and another....
A second ability enables us to judge that one group is greater or lesser than another, has more or fewer members. This many men, not enough spears to equip each man.
In some applications, exercises of the second ability, judgments of relative quantity, are based on easy eyeballing. In others, they're based on the exercise of a third ability, a careful sorting that establishes what we call a one-to-one correspondence between two groups of objects, say men and spears.
Having any of these three abilities is independent of having a fourth ability, the skill of enumerating. Acquiring this fourth ability involves acquiring a concept of number, as well as a practice of counting that establishes a one-to-one correspondence between a series of numbers and a series of objects enumerated.
In that regard, at least, the fourth ability, and some applications of the second, seem to depend on or to entail the third. The second, third, and fourth all depend on and entail the first, the ability to recognize various items as of the same sort, to bring a group of objects under the same concept.
The concept of spear comes along with its own unit of counting. But the concept of water or porridge does not. The concept of man comes along with its own unit of counting, but the concepts of a man's height and weight and speed do not.
Nevertheless, we may say we have a fifth ability, analogous to the second noted above, to recognize relative differences in the volume of accumulations of water or porridge, and likewise to recognize differences in magnitudes of other "properties" of observable things, such as height, weight, and speed.
This fifth ability is exercised in some cases by eyeballing, in some cases by the application of an arbitrary standard or unit of measurement without enumeration, and in some cases by the application of an arbitrary standard or unit of measurement with enumeration.
Sometimes it's easy to tell there's more porridge in one batch than in another batch. In controversial cases, we may dole each batch into small bowls of equal size, and compare the collections of bowls, with or without enumerating. For close calls, we can arrange the bowls from each batch in a one-to-one correspondence to make the comparison carefully, with or without enumerating.
Sometimes it's easy to tell that one thing is taller than another, just by looking. In tougher cases, we may carefully compare two heights, or any linear distances, without enumerating them, by using a piece of rope or wood that's longer than either of the objects to be compared, marking off the height of each object on that measuring tool, and noting which is longer. Given those two measurements, we can enumerate them by expressing each as a fraction of the whole length of the measuring tool. We might instead take a smaller length of rope or wood, and let this stand as the unit of measure, by using it to mark uniform intervals of length on longer objects.
It takes a bit more time, technology, and conceptual sophistication, but we find similar ways to assign arbitrary units in the careful measurement of weight and speed. For instance, by using a scale or a water clock.
A concept of number is a product of culture that emerges among animals like us at some point or another in history, and depends on our ability to bring different objects under a single concept. A practice of enumerating is another such product, that not only depends on a concept of number and on the ability to sort one-to-one, but also requires a more or less sophisticated system of signs to serve as names for each number in a counting procedure. It seems reasonable to expect, and evidence suggests, that the concept of number and a primitive ability to count emerged among us before our system of number signs had progressed very far.
The tools and techniques that equip us to assign arbitrary units by which to measure and compare volumes, lengths, weights, speeds, and other measurable "properties" of phenomena we encounter in the world are likewise products of culture.
Before the emergence of such tools and techniques, before the emergence of a system of number signs, before the emergence of a concept of number, there is the ability to recognize the difference between many and few, much and little, greater and lesser. We have the ability to make quantitative comparisons of groups of men or spears, of accumulations of water or porridge, of heights and weights and speeds of observable things.
We have a concept of weight because some things feel heavy. We make comparisons of weight because some things feel heavier than others. We have a concept of brightness because some things look bright; we make comparisons of brightness because some things look brighter than others. Some things feel hotter than others. Some things move faster than others. And so on.
The capacity to make such comparisons seems to come along with the capacity to apply the relevant concept in each case. If you can recognize that some things feel heavy, you can recognize that some things feel heavier than others; or, at least, the latter ability is not far off from the former, upon which it depends.
Such capacities are prior to sophisticated techniques of precise comparison, measurement, and enumeration, and are independent of the concept of number.
I imagine that for math to work for the sun or anything else there must be a mathematical principle already in play. That is to say we discover math in the sun/anything else. You speak as if we invent math. If the math didn't already exist in our observations no amount of mathematics will work, right?
Quoting Cabbage Farmer
Yet, inherent in them is the concept of quantification/number.
You've said this many times, but so far as I can tell, you haven't given any reasons to warrant the claim. Worries that some people might use math incorrectly, that some people might make claims about math without good reason, give us no reason to suppose that math "cannot be used".
People use the English language to make invalid arguments, to utter falsehoods, to lie and deceive, but this fact does not support the claim that the English language "cannot be used" to make arguments, nor does it support the claim that the English language "should not be used" to make arguments.
Why should the same susceptibility to abuse lead us to rule out math, but not to rule out the English language? I've made this point before and you've yet to respond to it.
Quoting Metaphysician Undercover
This also applies to the English language and to formal logic. Should we ban all three from moral discourse?
Quoting Metaphysician Undercover
This is blatant question begging.
Of course I agree the individual's argument could possibly be wrong, because people err in reasoning, whether or not they use math to help them think things through.
Quoting Metaphysician Undercover
This seems to be the hub of your worries about the use of math in moral reasoning: The mere fact that math is used as part of the reasoning that supports a moral judgment, might persuade some people that the whole piece of reasoning must be correct, since the math it includes is correct.
Compare: "This plan to repair a wall must be correct, since the math we used to calculate the exact amount of concrete is correct." Suppose the calculation is correct. It's based on measurements. What if the measurements are wrong? The calculation is also based on information about the ratio of dry mortar to sand, and of dry mix to square meters; what if this information is wrong? What if the wall or design plan has changed since measurements were made? What if the cement is of the wrong sort, or we're misinformed about how much water to add, or cement acts funny at current temperature or altitude…? None of this is taken into account in the calculation, which plays a limited but crucial role in the plan.
Here someone has made an assumption that, since math is involved in a piece of reasoning about repairing a wall, the whole construction plan must be correct, since the math is correct. But that's a strikingly unreasonable assumption.
Should we conclude that math "cannot be used" or "ought not be used" in reasoning about construction projects?
It seems to me you're arguing in exactly the same way about the use of mathematics in moral reasoning.
It's as if you're concerned that everyone with an inflated sense of the value or prestige of math -- an attitude arguably exemplified in the OP in this thread -- will be easily deceived by incorrect arguments that make use of correct mathematics.
Perhaps there are some people who would be so fooled. But that's no reason to say that math cannot or should not be used in moral reasoning. Again, the fact that people can abuse a tool or a language is not a reason to say that the tool or language cannot or should not be used.
I am arguing that mathematics can be used in moral reasoning.
I'm not sure that's the same as arguing that math "can be used to make moral judgments". Pure mathematics, like pure logic, informs us of nothing. You can't draw moral conclusions, or even begin to frame moral questions, on the basis of math alone. Just like you can't apply mathematics to any real problem without supplying some information from outside of math -- say, facts about the wall we want to reface, facts about the ratio of cement to square meters.
Quoting Metaphysician Undercover
Any application of mathematics requires something to which mathematics is applied. Pure math is empty form. We always need something else to fill in the blanks -- something outside that form to inform our calculations, to make them calculations about something or other.
Quoting Metaphysician Undercover
Do you say that a premise is the same thing as a conclusion?
In some cases a moral judgment is made by moral reasoning, or is justified (or criticized) by moral reasoning.
Having a moral value is not the same thing as reasoning in support of a particular moral judgment.
Quoting Metaphysician Undercover
Not so. Moral values, moral intuitions, and moral maxims, are not the same as moral judgments informed by moral values, intuitions, or maxims, nor the same as exercises of reasoning that support or justify such judgments.
Mathematics cannot supply the facts about moral values and moral intuitions, just like it can't supply the facts about concrete or the Sun. Supplying facts about the world is not the business of mathematics. That doesn't mean that mathematics has no role in analyzing collections of facts.
Quoting Metaphysician Undercover
What is a moral value, according to you? How does a moral value "answer" all questions of right and wrong? Is there such a thing as "reasoning" about moral questions, or does the mere possession of a "moral value" do all the work for us?
It's beginning to seem as though you don't distinguish moral values, moral judgments, and moral reasoning from each other, as if there were no difference between these three terms.
Perhaps this accounts for your worries about the claim that "math can be used in moral reasoning" -- as if anything that plays a part in moral reasoning must therefore be something like, or something as informative as, a moral value.