I enjoyed your paper on quantification and judgement. Prior to my retirement a few years ago I worked as a Research Director in the Education faculty of an English University. One of my abiding memories was of the futile battles of the 'paradigm war', that is the war between one the one hand, those who favoured quantitative and on the other, those favouring qualitative methodologies. The former argued that their approach was less likely to be appropriated by ideologically driven researchers whose primary objective was to shape education in the favour or one or another set of values. Quantitative researchers saw themselves as more detached and objective in their research. The resolution to this dispute lay in the recognition that education is invariably an ideological driven kind of activity whatever kinds of data are extracted, and that both qualitative and quantitative data, are necessary to constructing useful and consensual perspectives on what education is and how we should be engaging learners in it.
It seems to me that most research is driven towards useful, consensual and coherent conclusions and that these are the main, perhaps the only criteria for truth that we have.
Thank you for your insightful comment and for sharing your experiences, Mike. I absolutely agree with your conclusion.
Quantitative research is of course extremely valuable, but it is a mistake to think that its role can be to entirely remove subjectivity and values from judgements. One motivation for me in writing this piece was to show how if you dig down to the most fundamental levels of mathematics, you do not find objectivity and absolute certainty but the same ambiguity that we find in all of life. For me, this really drives home how the qualitative, human element cannot be erased from research or decision-making no matter how rigorously we apply quantitative methods. Instead, the role of quantitative methods must always be to inform and inspire new ideas that can allow us to work towards approaches that best satisfy our common human values.
I referenced Daniel Kahneman's book "Thinking Fast and Slow" in the piece which, while not explicitly about education, I think is a great example of research that makes extensive use of quantitative methods to motivate and test hypotheses without obscuring or losing touch with the humanity underneath, and so achieves the sort of useful and coherent conclusions you reference. No-one could mistake it for pure deductive logic, but it nonetheless finds truth by the criteria you suggest, as all great research does.
I don't think that we look for objectivity in the fundamental levels of mathematics. If a mathematical analysis is sound then it will be easy to persuade other mathematicians to the objectivity of its conclusions. The concepts and procedures are governed by clear rules and there is not space for subjective guesses or intuitions. I am not saying that intuition does not play a part in mathematical investigation but it is not its intuitive qualities that make an insight objective. It is the ability to demonstrate the process to others, to make it clear that there are no mistakes. The idea of mathematics I would suggest is to avoid ambiguity, to define all the terms of the process and ensure that we have a common understanding of what is going on.Kant referred to this as analytic knowledge in which propositions are true by definition and reference to their interconnection with other propositions within the system. They are necessary truths. The could not be anything but true. They do not refer to anying outside the system.
Absolute certainty is not quite the same thing. Even if I have demonstrated a mathematical proof to others I might yet entertain some uncertainty about the process. I may have made a mistake.
Certainty belongs in the court of more commonplace knowledge, for example I am certain that I live in Scotland and that my wife's name is Jane. If I discovered that I was mistaken, then the whole raft of my understandings would fall apart. I would think that I was going insane. Certainty is a kind of psychological security around with everything hangs, Certainty is also a feature of faith based knowledge.
I don't see any place for the qualitative in the formal sciences, mathematical and otherwise. Qualitative research consists in drawing evidence from interviews, questionnaires, observations and documentary evidence. The knowledge that these procedures deliver is contingent. It could have been different if I had been given different information in my data gathering. Kant refers to this as synthetic knowledge. If synthesises difference areas of knowledge and generated new understandings. It is not subjective. Others can critically review my work and if all the processes and data hang together we might be able to say that it is objectively produced.
At the end of the World WarII a group of traditional historians was asked to write a multi volumed history detailing the whole perdiod of the war, the weaponry involved, the successes and losses, etc. This rather dry set of volumes was usefull for other historians but contained nothing of the experience of the war. It was a purely quantitative presentation. Another historian (sorry I can't remember the names) that took up the challenge of a complete qualtitative account. What was it like to be in a battle, in a sinking ship, landing on a beach and fighting ones way in land. What did it feel, look, or sound like? He could only decide this from diaries (documentary evidence) films, and extensive interviews with veterans.
Sorry to bang on like this. I have plenty of free time and you have raised the issue very well.
Regards MIke (p.s. excuse the typos - there are always some!)
The distinction you draw between certainty and objectivity is a useful one, and it's true that I had conflated them in my comment. You're right to identify that the uncertainty in mathematics comes primarily from the potential for errors rather than from the mathematics itself and that we should separate this out as you have.
I am not sure I agree that there is not space for subjectivity in mathematics, however. There is a famous joke in mathematics: "The Axiom of Choice is obviously true, The Well-Ordering Theorem is obviously false and who can tell about Zorn's Lemma?" The point of the joke is that the Axiom of Choice, The Well-Ordering Theorem and Zorn's Lemma have been proven to all be logically equivalent, so obviously either all must be true or all must be false. The problem is that the axiom of choice cannot be proven to be either true or false using only the traditional (Zermelo-Frankel set theory) axioms of set theory, so it has been up to mathematicians to subjectively decide whether to include it as another (true) axiom or not. On its own terms, this does not threaten the objectivity of mathematics as the mathematician can simply declare the full set of axioms their proofs rely on, and the proved results will be objectively for that set of axioms. However, this full objectivity is only achieved if the mathematician does depend explicitly on the foundational levels of mathematics. As we have seen, this foundational level is undermined by Godel's incompleteness theorem.
As you say, real mathematical analysis does not generally depend on appealing to these foundations, but this necessarily makes it subjective. Mathematical statements depend on a set of words (or symbols) that are either left undefined, or rely on a chain of definitions of other words (symbols). If this hierarchy of definitions is not traced back to the foundational axioms, then at some point the meaning of the statement is dependent on the subjective understanding of words and symbols that the reader has in their own head. There is no guarantee that this understanding is identical with that of the person making the statement. Mathematicians deal with this by asking questions about, and discussing (or arguing) with each other, the meaning of mathematical results until they are content that they have a shared understanding. However, as with all shared understandings, this is subjective - they have simply found enough common ground in their own understandings to move forward.
My view is that all research - mathematical or otherwise - is ultimately synthetic rather than analytic. The concepts a mathematician (or any other researcher) chooses to define, where they draw lines between concepts, and what they omit, is all informed by their full set of experiences, opinions and values that extend far beyond mathematics. Only an infinitesimal fraction of all possible mathematical results or formalisms will ever be researched and presented, and it is this fraction that constitutes mathematics as a real discipline rather than an abstract idea. If particular mathematicians had had different school teachers or supervisors, or been in a better or worse mood when first hearing about a particular area of mathematics, then the mathematics they created could have been different, and mathematics could be other than what it is.
Mathematical logicians hoped to partition off mathematics as an exceptional area of human knowledge that could be analytic by dealing only within the scope of a set of axioms, but Gödel's incompleteness theorems undermined this by showing that a provably consistent set of axioms would necessarily be too trivial to allow for any useful mathematics. Instead, mathematical research, as with all other research, depends on the researcher bringing their own broader understanding and human judgement to their work.
Youyr knowledge of mathematics is far beyond mine and I defer to what you say, although I am still not sure how far I would see the 'choice of axioms' in the development of a theoretical argument, as subjective. It seems to me that axioms are the basic tools that make the argument possible, neither objective nor subjective, but certainly necessary. The objectivity of the argument does not depend on the axioms chosen, but rather on the publically verified process by which it is constructed. Normally I would understand axioms not as a preliminary part of building an argument, but rather assumptions that can be discovered when the argument is deconstructed. Euclid had been doing geometry for some time before he discovered the set of axioms that was guiding his argument.
Very interesting reading
Hi Paul,
I enjoyed your paper on quantification and judgement. Prior to my retirement a few years ago I worked as a Research Director in the Education faculty of an English University. One of my abiding memories was of the futile battles of the 'paradigm war', that is the war between one the one hand, those who favoured quantitative and on the other, those favouring qualitative methodologies. The former argued that their approach was less likely to be appropriated by ideologically driven researchers whose primary objective was to shape education in the favour or one or another set of values. Quantitative researchers saw themselves as more detached and objective in their research. The resolution to this dispute lay in the recognition that education is invariably an ideological driven kind of activity whatever kinds of data are extracted, and that both qualitative and quantitative data, are necessary to constructing useful and consensual perspectives on what education is and how we should be engaging learners in it.
It seems to me that most research is driven towards useful, consensual and coherent conclusions and that these are the main, perhaps the only criteria for truth that we have.
Thank you for your insightful comment and for sharing your experiences, Mike. I absolutely agree with your conclusion.
Quantitative research is of course extremely valuable, but it is a mistake to think that its role can be to entirely remove subjectivity and values from judgements. One motivation for me in writing this piece was to show how if you dig down to the most fundamental levels of mathematics, you do not find objectivity and absolute certainty but the same ambiguity that we find in all of life. For me, this really drives home how the qualitative, human element cannot be erased from research or decision-making no matter how rigorously we apply quantitative methods. Instead, the role of quantitative methods must always be to inform and inspire new ideas that can allow us to work towards approaches that best satisfy our common human values.
I referenced Daniel Kahneman's book "Thinking Fast and Slow" in the piece which, while not explicitly about education, I think is a great example of research that makes extensive use of quantitative methods to motivate and test hypotheses without obscuring or losing touch with the humanity underneath, and so achieves the sort of useful and coherent conclusions you reference. No-one could mistake it for pure deductive logic, but it nonetheless finds truth by the criteria you suggest, as all great research does.
I don't think that we look for objectivity in the fundamental levels of mathematics. If a mathematical analysis is sound then it will be easy to persuade other mathematicians to the objectivity of its conclusions. The concepts and procedures are governed by clear rules and there is not space for subjective guesses or intuitions. I am not saying that intuition does not play a part in mathematical investigation but it is not its intuitive qualities that make an insight objective. It is the ability to demonstrate the process to others, to make it clear that there are no mistakes. The idea of mathematics I would suggest is to avoid ambiguity, to define all the terms of the process and ensure that we have a common understanding of what is going on.Kant referred to this as analytic knowledge in which propositions are true by definition and reference to their interconnection with other propositions within the system. They are necessary truths. The could not be anything but true. They do not refer to anying outside the system.
Absolute certainty is not quite the same thing. Even if I have demonstrated a mathematical proof to others I might yet entertain some uncertainty about the process. I may have made a mistake.
Certainty belongs in the court of more commonplace knowledge, for example I am certain that I live in Scotland and that my wife's name is Jane. If I discovered that I was mistaken, then the whole raft of my understandings would fall apart. I would think that I was going insane. Certainty is a kind of psychological security around with everything hangs, Certainty is also a feature of faith based knowledge.
I don't see any place for the qualitative in the formal sciences, mathematical and otherwise. Qualitative research consists in drawing evidence from interviews, questionnaires, observations and documentary evidence. The knowledge that these procedures deliver is contingent. It could have been different if I had been given different information in my data gathering. Kant refers to this as synthetic knowledge. If synthesises difference areas of knowledge and generated new understandings. It is not subjective. Others can critically review my work and if all the processes and data hang together we might be able to say that it is objectively produced.
At the end of the World WarII a group of traditional historians was asked to write a multi volumed history detailing the whole perdiod of the war, the weaponry involved, the successes and losses, etc. This rather dry set of volumes was usefull for other historians but contained nothing of the experience of the war. It was a purely quantitative presentation. Another historian (sorry I can't remember the names) that took up the challenge of a complete qualtitative account. What was it like to be in a battle, in a sinking ship, landing on a beach and fighting ones way in land. What did it feel, look, or sound like? He could only decide this from diaries (documentary evidence) films, and extensive interviews with veterans.
Sorry to bang on like this. I have plenty of free time and you have raised the issue very well.
Regards MIke (p.s. excuse the typos - there are always some!)
The distinction you draw between certainty and objectivity is a useful one, and it's true that I had conflated them in my comment. You're right to identify that the uncertainty in mathematics comes primarily from the potential for errors rather than from the mathematics itself and that we should separate this out as you have.
I am not sure I agree that there is not space for subjectivity in mathematics, however. There is a famous joke in mathematics: "The Axiom of Choice is obviously true, The Well-Ordering Theorem is obviously false and who can tell about Zorn's Lemma?" The point of the joke is that the Axiom of Choice, The Well-Ordering Theorem and Zorn's Lemma have been proven to all be logically equivalent, so obviously either all must be true or all must be false. The problem is that the axiom of choice cannot be proven to be either true or false using only the traditional (Zermelo-Frankel set theory) axioms of set theory, so it has been up to mathematicians to subjectively decide whether to include it as another (true) axiom or not. On its own terms, this does not threaten the objectivity of mathematics as the mathematician can simply declare the full set of axioms their proofs rely on, and the proved results will be objectively for that set of axioms. However, this full objectivity is only achieved if the mathematician does depend explicitly on the foundational levels of mathematics. As we have seen, this foundational level is undermined by Godel's incompleteness theorem.
As you say, real mathematical analysis does not generally depend on appealing to these foundations, but this necessarily makes it subjective. Mathematical statements depend on a set of words (or symbols) that are either left undefined, or rely on a chain of definitions of other words (symbols). If this hierarchy of definitions is not traced back to the foundational axioms, then at some point the meaning of the statement is dependent on the subjective understanding of words and symbols that the reader has in their own head. There is no guarantee that this understanding is identical with that of the person making the statement. Mathematicians deal with this by asking questions about, and discussing (or arguing) with each other, the meaning of mathematical results until they are content that they have a shared understanding. However, as with all shared understandings, this is subjective - they have simply found enough common ground in their own understandings to move forward.
My view is that all research - mathematical or otherwise - is ultimately synthetic rather than analytic. The concepts a mathematician (or any other researcher) chooses to define, where they draw lines between concepts, and what they omit, is all informed by their full set of experiences, opinions and values that extend far beyond mathematics. Only an infinitesimal fraction of all possible mathematical results or formalisms will ever be researched and presented, and it is this fraction that constitutes mathematics as a real discipline rather than an abstract idea. If particular mathematicians had had different school teachers or supervisors, or been in a better or worse mood when first hearing about a particular area of mathematics, then the mathematics they created could have been different, and mathematics could be other than what it is.
Mathematical logicians hoped to partition off mathematics as an exceptional area of human knowledge that could be analytic by dealing only within the scope of a set of axioms, but Gödel's incompleteness theorems undermined this by showing that a provably consistent set of axioms would necessarily be too trivial to allow for any useful mathematics. Instead, mathematical research, as with all other research, depends on the researcher bringing their own broader understanding and human judgement to their work.
Youyr knowledge of mathematics is far beyond mine and I defer to what you say, although I am still not sure how far I would see the 'choice of axioms' in the development of a theoretical argument, as subjective. It seems to me that axioms are the basic tools that make the argument possible, neither objective nor subjective, but certainly necessary. The objectivity of the argument does not depend on the axioms chosen, but rather on the publically verified process by which it is constructed. Normally I would understand axioms not as a preliminary part of building an argument, but rather assumptions that can be discovered when the argument is deconstructed. Euclid had been doing geometry for some time before he discovered the set of axioms that was guiding his argument.
Super interesting read mate! Keep it up
Thanks Damon! I'm really glad you enjoyed it. Please share it with anyone else you think might find it interesting!