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Discussing "The Foundations Of Arithmetic" By Gottlob Frege
In this episode of Canonball we discuss "The Foundations Of Arithmetic," which was written by Gottlob Frege and published in 1884.
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Gottlob Frege was born in 1848 in Wismar, in the German Confederation. Wismar is a port city, it's today in Germany, situated on the north coast, on the coast of the Baltic Sea. Frege was a German philosopher, a logician, and a mathematician. He also worked as a professor at the University of Jena, which is another city in Germany. He is a descendant, through his mother and grandmother, of Philip Melanchthon, who was a German Lutheran reformer, who worked with Martin Luther, and was the first systematic theologian of the Protestant Reformation. And Frege was himself Lutheran. Frege's father wrote a textbook for children ages 9 to 13 on the German language, and the first section of the textbook dealt with the logic and structure of language. And I don't think it would be too much of a reach to imagine that Frege's father's interest in and knowledge of language would affect his son's thinking, which, as we'll see, deals with language and the logic of language, as well as with mathematics. He went on to study math and physics in university, where one of his teachers was Ernst Abbe, who was a notable physicist and optical engineer from that period. Frege then did all of his work, lived a relatively long life, and died in 1925 at the age of 76 in Bad Kleinen. The book of Frege's that we're going to be looking at today is titled Foundations of Arithmetic, and its subtitle is A Logico-Mathematical Inquiry into the Concept of Number. And it's a short, dense book of about 150 pages in which Frege is trying to pin down with some level of rigor what numbers are, and then by extension what arithmetic is, what we're doing when we perform these relatively simple operations with numbers. And before we get into passages from it, it might be useful to get a little orientation about his ideas generally. He argues both in numbers and in propositions in favor of Platonism and against psychologism and formalism. Now, what do those three words mean? Platonism, named after Plato, is the philosophical position that abstract objects exist in some third realm that's neither the internal world of our consciousness, it's not our ideas and our thoughts, and nor is it the sensed world outside of us. That when people say, what is peace in the abstract? There are examples of peace. You can talk about a war ending or a community living peacefully. These are manifestations of an abstraction called peace. But peace itself, I think most people would say, is just a word. That's the phrase that often comes up, and that's called nominalism. Nominalism is the idea that abstractions and universal objects do not actually exist. They're only labels that we use as a kind of placeholder in grammar and also maybe a placeholder in our own thinking. They're a way that we categorize things that we observe either inside ourselves or outside ourselves. And this is in part how Plato comes up with his parable of the cave, because it's not only big words like peace or war, but every kind of abstraction, like the abstract horse or the abstract tree, the Platonist would argue, exists in some third world of forms. Frege never uses that phrase, but that comes up in Plato. And I think most people today would consider themselves nominalists, as I would. But I guess we would have to get into a conversation about what we mean by exist. In what sense exactly do we mean that these abstractions exist in some place that's neither that part of our sensation that we identify as being inside ourselves, as arising internally, our thoughts and our awareness, nor is it the section of our awareness that we say comes from outside of ourselves, that comes from our senses. Though, of course, that line is not totally clear either, since our sense perceptions occur internally, though we attribute them to something external. So again, concerning numbers and propositions, Frege argues in favor of Platonism and against psychologism and formalism. So we just talked about Platonism and then nominalism is the opposite of that. Psychologism is a set of different philosophical positions that assert that certain psychological aspects of humans are central in explaining facts and entities with which we would not normally associate psychology, like for example math. So in this context we would be talking about how number and arithmetic are somehow a feature of human psychology.something external to it that would exist even if there were no humans anywhere, but something that humans project outwards in a certain way because of how their mind works. And Frege opposes that approach. And he also opposes formalism, which is the idea that mathematical and logical statements only apply to the symbols that they relate to. That math is not a set of logical propositions about some abstract section of reality, but instead it's more like a game that it applies to itself, but it doesn't necessarily describe anything real, even though it might be applicable in many contexts. It might map on to many different phenomena, but it's not describing a real thing, even a real abstraction, the way that a photo or a description of a tree is representing that object. That's formalism and Frege opposes that as well. And what he puts forward and what he's sometimes marked as the beginning of is the program of logicism, which is the idea that mathematics is an extension of logic and that some or all of mathematics can be reduced to logic or modeled in it. And while Frege is generally taken as the starting point of it, Bertrand Russell and Alfred North Whitehead in their Principia Mathematica and elsewhere are viewed as its champions. And the German mathematician Richard Dedekind and the Italian Giuseppe Pino developed it as well. But so we just talked about five different views, Platonism, Nominalism, Psychologism, Formalism, and Logicism. And all of these do not only apply to math, but we're looking at them in the context of math. And you'll notice that these are very subtle differences on a very subtle topic. And it's sometimes tempting when you're reading a book to view the author as the hero and his opponents against whom he's arguing as rivals in some way that he needs to overcome. And you're silently cheering him on and doing so. And that might in part be because you're reading a book, you're spending time and energy that you could very well spend on something else, examining this guy's thoughts. And it would be more worthwhile if it turned out that this guy were the correct one and everybody else were wrong. But of those five concepts that I just listed, we could probably find examples of smart and capable people arguing the opposite position that Frege does. And that's not to advocate for a kind of relativism, but I think the only answer is to acknowledge that this is an ongoing discussion and you should side with somebody or a position only to the extent that their arguments make sense. You can't see something that would disprove it. The arguments against it seem weaker. You can't passively imbibe these positions from other people. You have to get in there and understand what they're saying for yourself and try to see if it makes sense to you. But Frege opened up this new direction of thinking about mathematics as an extension of logic. And another thing that he developed to that end was his Begriffsschrift or concept writing, which is his own formal system of logical notation, which we categorize as second order logic. There is propositional logic, then first order logic, then second order logic, and then higher order logic. And his is second order, which is a type of formal system that has certain characteristics that I'm not going to go into. But if you're interested, you can go and check that out. And it was interesting for me to see that because I was familiar with George Boole in investigation of the laws of thought doing something similar, except he seemed more focused on syllogistic logic, on Aristotelian logic. Whereas from what I could gather, Frege's system, his concept writing is a bit broader than that. And it sometimes has been pointed out that Aristotle's logic as laid out in the Organon cannot represent more complex mathematical statements like Euclid's theorem, which states that there are infinitely many prime numbers, for example. Aristotle's logic can't represent that, whereas Frege's concept writing can. And further, apparently, the logical formalization that led to Bertrand Russell and Alfred North Whitehead preparing the Principia Mathematica and also Kurt Gödel's incompleteness theorems can reasonably be traced back to having grown out of Frege. And Bertrand Russell apparently later found a major flaw in Frege's work, which is now called Russell's Paradox, but was then resolved later. And Frege acknowledged the flaw just as a later book was going to print and apparently wrote an appendix about it. But that's the way that the development of this kind of knowledge goes as it's refined over time by smart people. And Russell was a quartercentury younger than Frege and even if he found a flaw in what Frege did, Russell benefited a lot from Frege's work and then built on it. At least that's what it looks like from the outside. This is very technical stuff. If you want to see what I mean, you can search Principia Mathematica Bertrand Russell and look at the images. This is a massive three-volume work that's written basically in a different language, in a logico-mathematical language that I certainly don't understand, though I would like to. But the impression from people who do understand it is that Russell was influenced by and built on Frege's work, at least to some degree. And that's how you would hope these things would go, is you start something and then the next generation improves on it. But if you're somebody who's interested in epistemology and exact thinking and trying to pin down what you think you know and establish that on very firm footing and you start to say, well, how do I really know such and such that this follows from that? Some people at that point throw up their hands and say, it's impossible to know. And some people like Frege and others coming up on 150 years ago in 1879 roll up their sleeves and try to identify and address each problem that they encounter. At each inference, at each logical step, saying, is that an intuition? Am I doing this synthetically? Is it analytic? Is that step valid? If it's an intuition, I should state it as such and describe it. And one of the results of one smart guy trying to do that is Frege's concept writing. And this book we're going to be looking at today, in particular, The Foundations of Arithmetic, is seen as the major text in logicism. And Michael Dummett, a professor of logic at Oxford who wrote a lot about Frege in particular and made his own contributions before he died in 2011, points to Frege and this book as the origin of what's now called the linguistic turn, which is the reorientation of a lot of philosophy in the 20th century toward language. And as we'll see, Frege deals a lot with language in his writing and in his philosophy. But this book was published in 1884 when Frege was about 36 years old. And as I said, he's trying to get at the philosophical foundations of number and arithmetic. And he goes over a lot of other theories of number and refutes them or finds problems with them and then lays out his own theory. And some of the ideas that come up in this book and for which Frege is known are the distinction between concept and object. And Frege's view basically is that a sentence comprises an object and a predicate, and the predicate signifies the concept. So if we take the sentence, Socrates is a philosopher, then we have the object, Socrates, and the predicate or concept is a philosopher. And those two things together make up the thought, Socrates is a philosopher. And this is distinct from what had been taken up to that point, which is term logic, in which instead of thinking of an object and a concept, you think of two objects connected by a copula, an is. So according to term logic, the statement Socrates is a philosopher would have the two objects, Socrates and philosopher, connected by is or is a. And the significance of this distinction in viewing it this way is not immediately obvious, but when he's trying to figure out exactly what a number is, what one is, what the difference between one and a unit is, is the number one a unit? When you're dealing at that level with something foundational like number, this kind of distinction becomes important. And apparently also viewing it in this way helped to clarify certain ideas in set theory, because it's a different way to articulate the element and the set and the relationship between those two things. But one of the things that he's known for is this distinction between concept and object, which is almost certainly going to come up in the passages that we'll look at later. And he's also known for what's called the context principle, which is an idea in the philosophy of language that a philosopher should never ask for the meaning of a word in isolation, but only in the context of a proposition. This idea that a word can only be defined in context. It can't be defined in the abstract. And the Harper and brothers translation of this, which was published in 1960, and which I think is the only translation of this available in English, includes a few pages at the beginning called an analysis of contents. And I'm not sure if Friege wrote this or if the publisher prepared it or the translator. I'm inclined to think from the way that it's written that Friege wrote it, but any German speakers out there who are familiar with the original German of this can let me know whether there is a section at the beginning titled something like analysis of contents that lays out what he's talking about in all the different sections and subsections. But the way that it's written makes me think that Friege wrote it. But if the publishers wrote it.it still credit to them because either way these few pages at the beginning help a lot in reading this book because you can then very easily keep track of what he's talking about and why. Where does what he's talking about on a given page fit into his larger philosophy? Why is this important? And you couldn't understand the book only from reading these few pages, but when you're reading the full text, it's useful to sometimes go back and refer to that. And it would not surprise me at all if Frege prepared it because it seems to fit very comfortably into his mode of thinking, this very exact and organized way that he considers whatever is the object of his focus and attention. But he opens the book by talking about how at that time there had been a lot of work done in trying to establish rigor in various areas of math. And this is something that came up both in looking at Heisenberg and at Poincaré. And he points out that there hadn't been work done or as much work done on arithmetic and on the very basic notion of number, but that logically that's where this kind of investigation is going to end up. And so he starts walking in that direction and he goes over the controversy of whether the laws of number are analytic or synthetic and whether they're a priori or a posteriori. And these are, of course, Kant terms if you're familiar with them. And if you're not, it's worth it to go over them very briefly. Kant and other philosophers have used the term analytic to refer to knowledge that is based on the laws of logic and the term synthetic to refer to knowledge that is based on intuition. And intuition refers to basically sensed information and our understanding of time and space and the relations between various objects because of the knowledge of our senses. Then a priori refers to knowledge that is derived only from reason, not from experience or empirical evidence. It's prior knowledge. It's before any kind of experience or evidence because it's based on reason. And a posteriori refers to knowledge that is based on experience and evidence. So to just keep those terms in your mind very roughly because they will probably come up again, synthetic and a posteriori kind of go together. Synthetic having to do with intuition, a posteriori having to do with experience and evidence and analytic and a priori kind of go together because they both have to do with logic. That's just a very quick summary of those terms. And a lot of this book is about into which of those categories numbers and the laws of numbers fall. And one very nice aspect of this short book is how Frege goes through a number of different thinkers and he's not bringing them in for no reason. Each one of them, he cites them specifically to talk about some statement or view of theirs. He talks about a number of them of the more household philosophical names. He talks about Kant, Leibniz, Locke, Berkeley, Hobbes, Hume, Descartes, and John Stuart Mill. Of the less famous but still significant mathematical thinkers, he mentions Rudolf Lipschitz, Hermann Henkel, William Stanley Jevons, Ernst Schroeder, Oskar Schlomilch, Karl Johannes Thome, Otto Hess, and somebody named Bauman. And I couldn't figure out who he was talking about. So let me know if you can find a 19th century philosopher, mathematician named Bauman, B-A-U-M-A-N-N. Maybe he's better known in Germany. And his approach in looking at each of these thinkers and their views on number is a really nice example of this kind of investigation because he shows how you can benefit and advance by finding problems in other people's positions. And he really seems to be very sincerely trying to get at the truth. And every time he goes over an argument and finds a problem with it, you feel like with him you're getting one step closer to wherever you're going. He's not just saying, this guy's wrong and that guy's wrong. He's saying, here's exactly why that guy is wrong, or here is why what he is saying doesn't quite make sense. And so then he can take that stepwise refinement and put it into his overall picture. But Kant is, of course, one of the people who comes up the most. And he talks about how in looking at whether numerical formulas, something like 8 plus 3, whether they are provable, he says Kant says that they're not. That no matter how much you look at 8 and 3, you can't get to 11 from those two numbers using logic. You need an intuition, a sensed experience of having 8 and putting 3 more into it and then counting how many you have and you have 11. And Frege says you can't possibly have an intuition of very big numbers of 7,240,365, for example. We don't have an intuition of that number. And if you added a similar number to that number and you came up with some other number as a sum, you wouldn't have an intuition of that either. And he talks about howHenkel calls this a paradox and then he goes over Leibniz's proof of 2 plus 2 equals 4 and says that it has a gap in it and that proof also came up when we were looking at one of Poincaré's books. He talks about another guy I forgot to mention Herman Grossman comes up a few times and the person on whom Frege seems to come down the hardest is John Stuart Mill and he's never polemical but he sometimes seems a tiny bit impatient with Mill's view of number and our knowledge of number as being entirely empirical and that the way that we know what a number is is only by observed facts and along the same lines he objects to Mill calling arithmetical truths laws of nature and he says that in doing so he's confusing them with their applications. Frege says that the laws of arithmetic are not themselves laws of nature they're not derived from experience and evidence but rather they are applied to nature and he looks at the question of is number a property of external things if you have two apples is the two a property of the apples is it an adjective that's attached to them the apples are red and they are two and he looks at the question of whether number is subjective or not and he looks at whether the number word one whether that expresses a property of objects and that's a little bit different from a property of an external thing and one of the things he talks about is that not all numbers are the same especially zero one and all the other numbers are different though he might say that three and four are more similar to each other than one is to three but one is the basic I was about to say unit but you absolutely can't say unit he goes into a very big long thing about the word unit and why one cannot be considered a unit he talks about the problem with the word unit talks about whether units are identical with one another and you would think that they are that the definition of a unit is many iterations of the same thing that's sort of how we would think about what a unit is that if you were to take away this inch and put another inch in its place whatever that would mean it would be exactly the same as a unit of measurement but then he talks about how if they are exactly the same if all of the inches were exactly the same then they would even occupy the same space and then they would be one they wouldn't be many and that doesn't mean that in order to have more than one inch they have to all be slightly different measurements they're all exactly the same length they're all the same inch but if they were truly exactly the same in every aspect including their position they would all be occupying the same space and there would only be one of them so by definition in order to have units and have them be functional they can't be completely exactly the same in every aspect because they at least have to occupy different areas of space and so then they are not identical then he talks about how one is a proper name while unit is a concept word one is a name like George it refers to one thing and to a great extent that's what a lot of this book is about is trying to figure out what that one thing is when we use the word one what are we talking about because when we use the word unit you might think that when we say the word one we're referring to a unit but he says the word unit refers to a concept whereas one refers to a proper name and he says that number cannot be defined as units and all of this problem of reconciling how units are supposed to be identical with their distinguishability is hidden in the ambiguity of this word units that we don't usually think about that closely and he talks about how statements of number are statements of fact explained by the objectivity of concepts what the heck does that mean this is touching on that object concept distinction that we were talking about earlier but he's saying that statements of number there is one Apple our relation of object and concept so in that case there's two things going on there's the number one and then there is the quantity of apples and he says you can rearrange the sentence to say one is the number of apples that there is nobody would ever talk like that because you'd look like a crazy person but if we're trying to understand number statements this is one way to organize it and see it kind of clearly that we have an object and a concept as we were talking about earlier and in that context the number is the object for the concept one is the number of apples that there is and on a related note he later says that not or zero is the number that belongs to a concept under which nothing falls so if our concept is dragons flying over the earth today nothing falls under that concept and so the appropriate number is zero zero is the number which belongs to a concept under which nothing falls and he points out how some philosophers and thinkers when they say we're unable to imagine such a thing they use that to mean we can't proceed in that direction if we were to follow this conclusion it would mean this and who can imagine such a thing and Frege says just because we cannot imagine an object we are not to be debarred from investigating it so you can try to investigate something even if you can't imagine it. Defying the imagination does not mean that it doesn't exist. And remember, he's writing decades ahead of the breakthrough into quantum physics that we looked at in reading Heisenberg. But here, less so than Poincaré, because Poincaré, again, was writing a little bit later, you sometimes get the sense that Frege is anticipating something, or at least that he was thinking about things that would have had him going in a particular direction that others were going in at that time, and which led to these breakthroughs. And he talks about the notion of numerical identity. How do you know that two values are equal? And the idea of one-to-one correlation comes up a lot. He quotes somebody else, I don't remember who it is, it might be Leibniz, but talking about how identity is when for every unit in value A, there's an equal unit in value B, that if you were to match them up one at a time, you were to pair them off, then at the end, there would be none left over on either side. That means the two are identical or equal. And his main disagreement with Kant, and one of the things he's arguing throughout the whole book, is that Kant argues that numerical statements are synthetic a priori, whereas Frege says that they are analytic a priori. And now that we've looked at some of his general ideas, we can now get into passages from Gottlob Frege's Foundations of Arithmetic. The first passage that we're going to read is the very beginning of the book. This is how Frege lays out the question that he's investigating. And as a reminder, because it comes up in this passage, the definite article in grammar is the, and the indefinite article is a or an. So we could say the apple or an apple, and that's the definite article and the indefinite article. Frege writes, quote, when we ask someone what the number one is, or what the symbol one means, we get as a rule the answer, why, a thing. And if we go on to point out that the proposition, the number one is a thing, is not a definition, because it has the definite article on the one side and the indefinite on the other, or that it only assigns the number one to the class of things, without stating which thing it is, then we shall very likely be invited to select something for ourselves, anything we please, to call one. Yet, if everyone had the right to understand by this name whatever he pleased, then the same proposition about one would mean different things for different people. Such propositions would have no common content. Some, perhaps, will decline to answer the question, pointing out that it is impossible to state either what is meant by the letter a, as it is used in arithmetic, and that if we were to say a means a number, this would be open to the same objection as the definition one is a thing. Now, in the case of a, it is quite right to decline to answer. a does not mean some one definite number which can be specified, but serves to express the generality of general propositions. If, in a plus a minus a equals a, we put for a some number, any we please, but the same throughout, we get always a true identity. This is the sense in which the letter a is used. With one, however, the position is essentially different. End quote. So, let's stop there for a second before we keep going. So, he says that if you ask what the number one is, people say it's a thing, but the problem is if you say the number one is a thing, then you're mixing the definite and the indefinite article, and at best what you've done is that you've said the number one belongs to this category called things. It's sort of like saying what's a tiger? It's an animal. So, you've said the category that it belongs to, but you haven't said anything significant about it, and then he says some people might answer this by saying we also use the letter a in algebra or x or y, and that also is undefined. It could be any number that you could think of, and now he's getting into why that's different. That doesn't apply to trying to define number or one and doing it by saying it's a thing, that if you say a plus a minus a equals a, that's true no matter what number you put in, but that's not true of particular numbers. Getting back to the text, he goes on, quote, can we, in the identity one plus one equals two, put for one in both places some one in the same object, say the moon? On the contrary, it looks as though whatever we put for the first one, we must put something different for the second. Why is it that we have to do here precisely what would have been wrong in the other case? End quote. So, let's stop again. So, he's saying that with a plus a minus a, if you were to put in anything, not just any number, if you were to put in an apple and you had apple plus apple minus apple, you would still end up with equals apple, because you have one apple, you add another apple to it, you take one apple away, and then you have one apple left over, and then he says if number is similar to variables, that it's just anything, you should be able to put in an object in the same way, like the moon. So, if you take one plus one equals two, and then you go moon plus moon,Equals two to be honest I'm not totally sure but I think what he's saying is that in order to get to two in order to get to something that's Fundamentally different from the first one from the first object You couldn't have two of the same object two of the same object will never get you to a different kind of object So rather than one acting like a variable where you can put the same object in place of the number every time it appears Like with the variable a you'd have to put in something different to get to two which is Fundamentally different from one anyway He's illustrating how thinking of one as a kind of variable as a placeholder for other things Doesn't work back to the text quote again Arithmetic cannot get along with a alone But has to use further letters besides B C and so on in order to express in general form relations between different numbers It would therefore be natural to suppose that the symbol one also if it served in some way to confer Generality on propositions would not be enough by itself yet Surely the number one looks like a definite particular object with properties that can be specified For example that of remaining unchanged when multiplied by itself and quote. Let's stop again because there I think he's saying that you could say that if you were working with letters You would say a plus a equals B the same way that you would say 1 plus 1 equals 2 but still the number 1 has properties that variables don't have a variable is a kind of Mystery box that outside of the function that it appears in you don't have any information about it Whereas he's saying one has certain qualities Continuing on quote in this sense a has no properties that can be specified Since whatever can be asserted of a is a common property of all numbers Whereas 1 to the power of 1 equals 1 asserts nothing of the moon nothing of the Sun nothing of the Sahara Nothing of the peak of Tenerife for what could be the sense of any such assertion? Questions like these catch even mathematicians for that matter or most of them unprepared with any satisfactory answer yet Is it not a scandal that our science should be so unclear about the first and foremost among its objects? And one which is apparently so simple small Hope then that we shall be able to say what number is if a concept fundamental to a mighty science gives rise to difficulties Then it is surely an imperative task to investigate it more closely until those difficulties are overcome Especially as we shall hardly succeed in finally clearing up negative numbers or fractional or complex numbers So long as our insight into the foundation of the whole structure of arithmetic is still defective Many people will be sure to think this is not worth the trouble Naturally, they suppose this concept is adequately dealt with in the elementary textbooks where the subject is settled once and for all who can? Believe that he has anything still to learn on so simple a matter So free from all difficulties the concept of positive whole number held to be that an account of it fit for children can be both Scientific and exhaustive and that every schoolboy without any further reflection or acquaintance with what others have thought knows all there is to know About it. The first prerequisite for learning anything is thus utterly lacking I mean the knowledge that we do not know and quote a bit later He is laying out his approach and what he's doing by looking at other people's positions And he says that he does this to show that his position is not just another one alongside the others He has pulled together all of the viewpoints on this and explained why they don't fit as we've talked about and that His view will hopefully settle the question in light of all those other deficiencies Frieger writes quote in order then to dispel the solution that the positive whole numbers really present no difficulties at all But the universal concord reigns about them I have adopted the plan of criticizing some of the views put forward by mathematicians and philosophers on the questions involved It will be seen how small is the extent of their agreement so small that we find one dictum precisely Contradicting another for example Some hold that units are identical with one another others that they are different and each side supports its assertion with arguments that cannot be Rejected out of hand my object in this is to awaken a desire for stricter inquiry at the same time this preliminary Examination of the views others have put forward should clear the ground for my own accounts by convincing my readers in advance that these other Paths do not lead to the goal and that my opinion is not just one among many all equally tenable and in this way I hope to settle the question finally at least in essentials I realized that as a result I have been led to pursue arguments more philosophical than many mathematicians may approve But any thorough investigation of the concept of number is bound always to turn out rather philosophical It is a task which is common to mathematics and philosophy and quote later in a general comment on epistemology he writes quote never let us take a description of the origin of an idea for a Definition or an account of the mental and physicalon which we become conscious of a proposition for a proof of it. A proposition may be thought, and again, it may be true. Let us never confuse these two things. We must remind ourselves, it seems, that a proposition no more ceases to be true when I cease to think of it than the sun ceases to exist when I shut my eyes." End quote. Later, stating his position very clearly against what we now call psychologism and in favor of what we now call logicism, he says, quote, "...no less essential for mathematics than the refusal of all assistance from the direction of psychology is the recognition of its close connection with logic. I go so far as to agree with those who hold that it is impossible to affect any sharp separation of the two." End quote. And the two that he's referring to there are logic and mathematics, not logic and psychology. He's saying, "...I go so far as to agree with those who hold that it is impossible to affect any sharp separation between logic and mathematics." Back to the text, quote, "...this much everyone would allow, that an inquiry into the cogency of a proof or the justification of a definition must be a matter of logic. But such inquiries simply cannot be eliminated from mathematics, for it is only through answering them that we can attain to the necessary certainty." End quote. And talking about a standard of rigor, he says that it's not enough to have the definitions that we're using only be justified as an afterthought. You throw in a definition, you don't see any obvious contradiction, so you assume that that's good. He says that's not enough. And if you do that, you will only have empirical certainty. And he's using empirical certainty, having to do with observed evidence, as a secondary form of certainty. It's not logical certainty. It's not rigorous. And if you treat your definitions this way, you assume that they're good because you don't see any obvious contradiction. If you don't stress test them, you don't examine them, you don't try to tear them apart, you don't try to find problems with them, you could at any time encounter something that blows up your whole explanation. And then he talks about three principles that he has used in his investigation. Friege writes, quote, "...it must still be borne in mind that the rigor of the proof remains an illusion, even though no link be missing in the chain of our deductions, so long as the definitions are justified only as an afterthought, by our failing to come across any contradiction. By these methods, we shall at bottom never have achieved more than an empirical certainty, and we must really face the possibility that we may still in the end encounter a contradiction which brings the whole edifice down in ruins. For this reason, I have felt bound to go back rather further into the general logical foundations of our science than perhaps most mathematicians will consider necessary. In the inquiry that follows, I have kept to three fundamental principles, always to separate sharply the psychological from the logical, the subjective from the objective, never to ask for the meaning of a word in isolation, but only in the context of a proposition, never to lose sight of the distinction between concept and object. In compliance with the first principle, I have used the word idea always in the psychological sense, and have distinguished ideas from concepts and from objects. If the second principle is not observed, one is almost forced to take as the meanings of words mental pictures or acts of the individual mind, and so to offend against the first principle as well. As to the third point, it is a mere illusion to suppose that a concept can be made an object without altering it." End quote. Quote, in mathematics, a mere moral conviction, supported by a mass of successful applications, is not good enough. Proof is now demanded of many things that formerly passed as self-evident. Again and again, the limits to the validity of a proposition have been in this way established for the first time. The concepts of function, of continuity, of limit, and of infinity have been shown to stand in need of sharper definition. Negative and irrational numbers, which had long since been admitted into science, have had to submit to a closer scrutiny of their credentials. In all directions, these same ideals can be seen at work. Rigor of proof, precise delimitation of extent of validity, and as a means to this, sharp definitions of concepts. Proceeding along these lines, we are bound eventually to come to the concept of number, and to the simplest propositions holding of positive whole numbers, which form the foundation of the whole of arithmetic. Of course, numerical formulas like 7 plus 5 equals 12, and laws like the associative law of addition, are so amply established by the countless applications made of them every day, that it may seem almost ridiculous to try to bring them into dispute by demanding a proof of them. But it is in the nature of mathematics always to prefer proof, where proof is possible, to any confirmation by induction. End quote. And remember, induction is inferring a general truth from evidence, from an experiment, for example, whereas deduction is more likelogic or a syllogism going from the general to the specific back to the text quote Euclid gives proofs of many things which anyone would Concede him without question and it was when men refused to be satisfied Even with Euclid standards of rigor that they were led to the inquiry set in train by the axiom of parallels End quote and that's something that we got into in looking at Poincaré's writing that in looking at he's talking about Euclid's fifth postulate which is also called the parallel postulate or here for you guys calling it the axiom of parallels and from the inability to prove this postulate came non-euclidean geometries like those of Bobachevsky Bolyai and Riemann so he's saying there that Euclid in the elements gives all kinds of proofs for things that people normally would not think required a proof They seem self-evident, but he did it anyway, and that also it was by Increasing the rigor even beyond what Euclid thought he needed to something higher that in the 19th century Russian and Hungarian and German and other mathematicians made these discoveries about Alternatives to Euclid's geometry which are very relevant today And so he's saying that just by assuming that we understand what a number is or saying this seems Unnecessary to look into it at this level You can't be sure that there isn't something at the bottom of it that may prove Extremely useful for one way or another in opening up new vistas He goes on quote the aim of proof is in fact not merely to place the truth of a proposition Beyond all doubt but also to afford us insight into the dependence of truths upon one another After we have convinced ourselves that a boulder is immovable by trying Unsuccessfully to move it there remains the further question What is it that supports it so securely the further we pursue these inquiries the fewer become the primitive truths to which we reduce? everything and this Simplification is in itself a goal worth pursuing end quote and so he's adding to that in Trying to prove or demonstrate something that seems obvious Either you will find that it can't be proven and then you might end up with something like non Euclidean geometry or something new or You will find that it can be proven and the fact or thing on which that proof is based will be Useful in some other context. It's a Exploration downward probably because when we imagine things being based on other things we think of gravity and they're supporting each other So I'm imagining an investigation Downward and that's just as useful for you guys saying as an investigation upward that is where does this logic lead? It's just as useful to try to go down and see what's there because it might also support some other thing First of all is the boulder movable and if it is we shouldn't be counting on it And if it's not movable, what is it based on what makes it immovable? So it's a useful investigation either way and in this next passage He's talking about a priori and a posteriori Synthetic and analytic and what these terms mean and he specifically says that he's not using them Differently from how other writers like Kant have used them But he wants to be very explicit about what he takes to be meant by them And if you're familiar with these terms, then most of this will be familiar to you also But it's always interesting to hear another person lay them out and he has some other thoughts around the edges of these terms Frege writes quote philosophical motives to have prompted me to inquiries of this kind The answer is to the questions raised about the nature of arithmetical truths. Are they a priori or a posteriori? Synthetic or analytic must lie in this same direction for even though the concepts concerned may themselves belong to philosophy Yet as I believe no decision on these questions can be reached without assistance from mathematics Though this depends of course on the sense in which we understand them skipping ahead now these distinctions between a priori a posteriori Synthetic and analytic concern as I see it not the content of the judgment, but the justification for making the judgment And he has an end note there by this I do not of course mean to assign a new sense to these terms But only to state accurately what earlier writers Kant in particular have meant by them and back to the main text where there is no such justification the possibility of drawing the distinction vanishes an A priori error is thus as complete a nonsense as say a blue concept and quote Let's stop there for a second and look at what he's saying because it's a little bit subtle He's saying that these distinctions refer to not the content of the judgment but the justification for making it and if there's no justification then you can't Make a distinction between any of these because the terms are only there to describe the justification They're characterizing the justification and he says an a priori errors Thus is complete nonsense as say a blue concept and what he means is that if there's an error in the judgment Then you have no justification Itcan't be an a priori error because the presence of the error makes justification impossible and then you can't categorize it one way or another. Back to the text quote when a proposition is called a posteriori or analytic in my sense this is not a judgment about the conditions psychological physiological and physical which have made it possible to form the content of the proposition in our consciousness nor is it a judgment about the way in which some other man has come perhaps erroneously to believe it true rather it is a judgment about the ultimate ground upon which rests the justification for holding it to be true this means that the question is removed from the sphere of psychology and assigned if the truth concerned is a mathematical one to the sphere of mathematics the problem becomes in fact that of finding the proof of the proposition and of following it up right back to the primitive truths if in carrying out this process we come only on general logical laws and on definitions then the truth is an analytic one bearing in mind that we must take account also of all propositions upon which the admissibility of any of the definitions depends if however it is impossible to give the proof without making use of truths which are not of a general logical nature but belong to the sphere of some special science then the proposition is a synthetic one for a truth to be a posteriori it must be impossible to construct a proof of it without including an appeal to facts ie to truths which cannot be proved and are not general since they contain assertions about particular objects but if on the contrary its proof can be derived exclusively from general laws which themselves neither need nor admit of proof then the truth is a priori and he has an end note there if we recognize the existence of general truths at all we must also admit the existence of such primitive laws since from mere individual facts nothing follows unless it be on the strength of a law induction itself depends on the general proposition that the inductive method can establish the truth of a law or at least some probability for it if we deny this induction becomes nothing more than a psychological phenomenon a procedure which induces men to believe in the truth of a proposition without affording the slightest justification for so believing end quote and we talked a little bit about one of his disagreements with Kant already but here's the actual text of it under the heading of our numerical formulas provable Frieger writes quote we must distinguish numerical formulas such as two plus three equals five which deal with particular numbers from general laws which hold good for all whole numbers the former are held by some philosophers to be unprovable and immediately self-evident like axioms Kant declares them to be unprovable and synthetic but hesitates to call them axioms because they are not general because the number of them is infinite Hankel justifiably calls this conception of infinitely numerous unprovable primitive truths incongruous and paradoxical the fact is that it conflicts with one of the requirements of reason which must be able to embrace all first principles in a survey besides is it really self-evident that 135,664 plus 37,863 equals 173,527 it is not and Kant actually urges this is an argument for holding these propositions to be synthetic skipping ahead Kant obviously was thinking only of small numbers so that for large numbers the formulas would be provable though for small numbers they are immediately self-evident through intuition yet it is awkward to make a fundamental distinction between small and large numbers especially as it would scarcely be possible to draw any sharp boundary between them if the numerical formulas were provable from say 10 on we should ask with justice why not from 5 on or from 2 on or from 1 on end quote later he writes quote the basis of arithmetic lies deeper it seems than that of any of the empirical sciences and even than that of geometry the truths of arithmetic govern all that is numerable this is the widest domain of all for to it belongs not only the actual not only the intuitable but everything thinkable should not the laws of number then be connected very intimately with the laws of thought end quote later he writes quote now we have already decided in favor of the view that the individual numbers are best arrived in the way proposed by Leibniz Mill Grossman and others from the number one together with increase by one but that these definitions remain incomplete so long as the number one and increase by one are themselves undefined so he's saying that the numbers besides one and he goes into this in more detail elsewhere are 1 plus 1 plus 1 plus 1 etc and this is similar to the demonstration by recurrence that Poincaré laid out but then Frege says in understanding that we still have to understand the number one and also the increase by one touching on what we now callPsychologism. Frieger writes later, quote, "...no description of this kind of the mental processes which precede the forming of a judgment of number, even if more to the point than this one, can ever take the place of a genuine definition of the concept. It can never be adduced in proof of any proposition of arithmetic. It acquaints us with none of the properties of numbers. For number is no wit more an object of psychology or a product of mental processes than, let us say, the North Sea is. The objectivity of the North Sea is not affected by the fact that it is a matter of our arbitrary choice which part of all the water on the earth's surface we mark off and elect to call the North Sea. This is no reason for deciding to investigate the North Sea by psychological methods." End quote. And on the one hand that might seem a little bit strong, that the notion of number is as distinct from our perception of it as the North Sea is. Because even if you don't take the psychologistic approach and say that number is a manifestation of our psychology in some way, then most people would still say it's somehow more abstract than something observable like the North Sea. But I guess this is one passage that shows how Frege is holding the Platonist position. But later, maybe to expand a little on what he means here, he writes quote, "...astronomy is concerned not with ideas of the planets, but with the planets themselves. And by the same token, the objects of arithmetic are not ideas either. If the number two were an idea, then it would have straightaway to be private to me only. Another man's idea is by definition another idea." End quote. And so maybe he's saying there the way that the number two resembles a planet is that the number two is the same for everybody. Everybody doesn't have their own individual idea of the number two. We have somehow a shared idea of it. And I'm realizing that if I try to go over every argument that he makes in this book, I'll just be reading out the whole thing and I don't want to do that. So I'm gonna jump to the conclusion. Frege writes quote, "...I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts. Calculation becomes deduction. The laws of number will not, as Bauman thinks, need to stand up to practical tests if they are to be applicable to the external world. For in the external world, in the whole of space and all that therein is, there are no concepts, no properties of concepts, no numbers. The laws of number therefore are not really applicable to external things. They are not laws of nature. They are, however, applicable to judgments holding good of things in the external world. They are laws of the laws of nature. They assert not connections between phenomena, but connections between judgments. And among judgments are included the laws of nature. Kant, obviously, as a result, no doubt, of defining them too narrowly underestimated the value of analytic judgments. Though it seems that he did have some inkling of the wider sense in which I have used the term. On the basis of his definition, the division of judgments into analytic and synthetic is not exhaustive. What he is thinking of is the universal affirmative judgment. There, we can speak of a subject concept and ask, as his definition requires, whether the predicate concept is contained in it or not. But how can we do this if the subject is an individual object, or if the judgment is an existential one? In these cases, there can simply be no question of a subject concept in Kant's sense. He seems to think of concepts as defined by giving a simple list of characteristics in no special order. But of all ways of forming concepts, that is one of the least fruitful. If we look through the definitions given in the course of this book, we shall scarcely find one that is of this description. The same is true of the really fruitful definitions in mathematics, such as that of the continuity of a function. What we find in these is not a simple list of characteristics. Every element in the definition is intimately, I might almost say organically, connected with the others. Skipping ahead a little, the conclusions we draw from it extend our knowledge and ought, therefore, on Kant's view, to be regarded as synthetic. And yet, they can be proved by purely logical means and are thus analytic. The truth is that they are contained in the definitions, but as plants are contained in their seeds, not as beams are contained in a house." And then, talking more about Kant, he says, "...I have no wish to incur the reproach of picking petty quarrels with a genius to whom we must all look up with grateful awe. I feel bound, therefore, to call attention also to the extent of my agreement with him, which far exceeds any disagreement. To touch only upon what is immediately relevant, I consider Kant did great service in drawing the distinction between synthetic and analytic judgments. In calling the truths of geometry synthetic and a priori, he revealed their true nature. And this is still worth repeating, since even today it is often not recognized. If Kant was wrong about arithmetic, that's...does not seriously detract, in my opinion, from the value of his work. His point was that there are such things as synthetic judgments a priori. Whether they are to be found in geometry only or in arithmetic as well is of less importance." And that's a little bit ironic because Poincaré in one of his books, I think it's in Science and Hypothesis, he argues also that Kant was wrong about geometry as well. Picking up right where we left off, Frege writes, quote, I do not claim to have made the analytic character of arithmetical propositions more than probable, because it can still always be doubted whether they are deducible solely from purely logical laws or whether some other type of premise is not involved at some point in their proof without our noticing it. This misgiving will not be completely allayed even by the indications I have given of the proof of some of the propositions. It can only be removed by producing a chain of deductions with no link missing, such that no step in it is taken which does not conform to some one of a small number of principles of inference recognized as purely logical. To this day, scarcely one single proof has ever been conducted on these lines. The mathematician rests content if every transition to a fresh judgment is self-evidently correct, without inquiring into the nature of this self-evidence, whether it is logical or intuitive. A single such step is often really a whole compendium, equivalent to several simple inferences and into it there can still creep along with these some element from intuition. In proofs as we know them, progress is by jumps, which is why the variety of types of inference in mathematics appears to be so excessively rich. For the bigger the jump, the more diverse are the combinations it can represent of simple inferences with axioms derived from intuition. Often, nevertheless, the correctness of such a transition is immediately self-evident to us without our ever becoming conscious of the subordinate steps condensed within it. Whereupon, since it does not obviously conform to any of the recognized steps of logical inference, we are prepared to accept its self-evidence forthwith as intuitive, and the conclusion itself as a synthetic truth, and this even when obviously it holds good of much more than merely what can be intuited. On these lines, our synthetic based on intuition cannot possibly be cut cleanly away from our analytic, nor shall we succeed in compiling with certainty a complete set of axioms of intuition, such that from them alone we can derive, by means of the laws of logic, every proof in mathematics." And we can wrap up with a short passage about postulation and rigor, and how we know what we know. Quote, It is common to proceed as if a mere postulation were equivalent to its own fulfillment. We postulate that it shall be possible in all cases to carry out the operation of subtraction, or of division, or of root extraction, and suppose that with that we have done enough. But why do we not postulate that through any three points it shall be possible to draw a straight line? Why do we not postulate that all the laws of addition and multiplication shall continue to hold for a three-dimensional complex number system, just as they do for real numbers? Because this postulate contains a contradiction. Very well then, what we have to do first is to prove that these other postulates of ours do not contain any contradiction. Until we have done that, all rigor, strive for it as we will, is so much moonshine. End quote. And those are some of the passages that I wanted to show you from the Foundations of Arithmetic by Gottlob Frege. This is a short, important book on this topic. If you're interested in the philosophy of math and logic generally, I definitely recommend it. It's satisfying to see in Frege's writing that he had a very high standard for what he considered a valid inference. He was a very serious logician. He tried to look very carefully, almost with a microscope, at the moves that our mind makes when it comes to a conclusion about something. And that is a standard for truth and certainty that would be worth trying to uphold in almost any context. It's not always possible, of course. But when it's not possible, you should notice how far you are from this standard of truth and certainty. And in our own efforts, it's worth noticing when we've been satisfied with a much lower level of demonstration than what Frege requires for himself to be satisfied. And whenever possible, we should raise that standard toward Frege as high as we can. If you enjoyed this podcast, I hope you will send it to a friend who you think will enjoy it as well. And go to vollrathpublishing.com. The link is in the description. And pick up some books for yourself, for your friends, for your family. And do your part to support independent book publishing. Farewell until next time. Take care, and happy reading.