"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.

Quantitative variables are made of numerical measurements that have meaningful units attached to them. Categorical variables take on values that are categories or labels.

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.

Here is the text box definition, pulled from one of my statistics course books. — Jeremiah

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.

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. — 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.

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. — 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.

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. — 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:

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. — 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.

Although mathematics is commonly associated with quantity, it is more broadly the application of necessary reasoning to hypothetical or ideal states of affairs. — aletheist

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.

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. — 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"?

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. — 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.

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. — Cabbage Farmer

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.

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. — 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.

What is "necessary reasoning"? — Cabbage Farmer

Deriving conclusions from information that is already present in the premisses. Also known as deductive reasoning.

What sort of necessary reasoning is commonly associated with quantity? — Cabbage Farmer

Arithmetic is an obvious example, such as 2+2=4.

What sort of necessary reasoning is not commonly associated with quantity? — Cabbage Farmer

Syllogisms are an obvious example, such as "All men are mortal, Socrates is a man, therefore Socrates is mortal."

In what broad sense is mathematics "the application of necessary reasoning"? — 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.

Can we apply reasoning to real states of affairs, or only to hypothetical and ideal states of affairs? — 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.

Can we apply necessary reasoning to real states of affairs, or only to hypothetical and ideal states of affairs? — Cabbage Farmer

Only to ideal states of affairs, since we can never be absolutely sure that real states of affairs are completely deterministic.

Do you mean to say ... — Cabbage Farmer

Your paraphrase seems about right.

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"? — 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.

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? — 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.

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"? — 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.

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"? — 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.

Have you signed up for special troubles associated with dualism? — 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."

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. — Cabbage Farmer

That is fine. Hopefully these additional responses have helped clarify my thoughts for you.

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

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.

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? — 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.

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? — Cabbage Farmer

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?

Such capacities are prior to sophisticated techniques of precise comparison, measurement, and enumeration, and are independent of the concept of number. — Cabbage Farmer

Yet, inherent in them is the concept of quantification/number.

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. — Metaphysician Undercover

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.

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 — Metaphysician Undercover

This also applies to the English language and to formal logic. Should we ban all three from moral discourse?

(because you actually can't use mathematics in moral reasoning) — 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.

and also be convinced that it is right answer because math was used. — 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.

Well this is my whole point. We cannot use mathematics to make moral judgements. You seem to be arguing that we can. — Metaphysician Undercover

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.

But now you've qualified that to say that we would have to have moral agents, to supply moral values. — 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.

This implies that the moral judgement has already been made by the moral agent. — 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.

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. — 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.

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. — 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.