What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable Is there more than one mathematical universe? Joel David Hamkins Professor of Logic Sir Peter Strawson Fellow University of Oxford University College Wijsgerig Festival DRIFT Amsterdam, 11 May 2019 Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What does it mean to make existence assertions in mathematics? Is there an ideal mathematical reality that mathematical assertions are about? Is there more than one such universe? None? What are mathematical objects? What are numbers? Does every mathematical problem have a definite answer? Does every mathematical assertion have a definite truth value? What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable Ontology in mathematics The theme of this conference is ontology. I shall discuss ontology in mathematics. Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Is there an ideal mathematical reality that mathematical assertions are about? Is there more than one such universe? None? What are mathematical objects? What are numbers? Does every mathematical problem have a definite answer? Does every mathematical assertion have a definite truth value? What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable Ontology in mathematics The theme of this conference is ontology. I shall discuss ontology in mathematics. What does it mean to make existence assertions in mathematics? Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Is there more than one such universe? None? What are mathematical objects? What are numbers? Does every mathematical problem have a definite answer? Does every mathematical assertion have a definite truth value? What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable Ontology in mathematics The theme of this conference is ontology. I shall discuss ontology in mathematics. What does it mean to make existence assertions in mathematics? Is there an ideal mathematical reality that mathematical assertions are about? Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What are mathematical objects? What are numbers? Does every mathematical problem have a definite answer? Does every mathematical assertion have a definite truth value? What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable Ontology in mathematics The theme of this conference is ontology. I shall discuss ontology in mathematics. What does it mean to make existence assertions in mathematics? Is there an ideal mathematical reality that mathematical assertions are about? Is there more than one such universe? None? Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Does every mathematical problem have a definite answer? Does every mathematical assertion have a definite truth value? What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable Ontology in mathematics The theme of this conference is ontology. I shall discuss ontology in mathematics. What does it mean to make existence assertions in mathematics? Is there an ideal mathematical reality that mathematical assertions are about? Is there more than one such universe? None? What are mathematical objects? What are numbers? Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable Ontology in mathematics The theme of this conference is ontology. I shall discuss ontology in mathematics. What does it mean to make existence assertions in mathematics? Is there an ideal mathematical reality that mathematical assertions are about? Is there more than one such universe? None? What are mathematical objects? What are numbers? Does every mathematical problem have a definite answer? Does every mathematical assertion have a definite truth value? Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Binary 111001 is another: start with thirty-two, fold in sixteen and eight; garnish with one on top. Chill and serve. Roman numerals LVII: take fifty, add five, then two more. We should distinguish between the number and the numeral, between the mathematical object and the description of that object. What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable What is a number? Consider the number 57. What is it? The notation 57 is a recipe: take five tens and add seven. Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Roman numerals LVII: take fifty, add five, then two more. We should distinguish between the number and the numeral, between the mathematical object and the description of that object. What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable What is a number? Consider the number 57. What is it? The notation 57 is a recipe: take five tens and add seven. Binary 111001 is another: start with thirty-two, fold in sixteen and eight; garnish with one on top. Chill and serve. Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford We should distinguish between the number and the numeral, between the mathematical object and the description of that object. What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable What is a number? Consider the number 57. What is it? The notation 57 is a recipe: take five tens and add seven. Binary 111001 is another: start with thirty-two, fold in sixteen and eight; garnish with one on top. Chill and serve. Roman numerals LVII: take fifty, add five, then two more. Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable What is a number? Consider the number 57. What is it? The notation 57 is a recipe: take five tens and add seven. Binary 111001 is another: start with thirty-two, fold in sixteen and eight; garnish with one on top. Chill and serve. Roman numerals LVII: take fifty, add five, then two more. We should distinguish between the number and the numeral, between the mathematical object and the description of that object. Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford In the mines, they found the largest number! A gigantic number 3. Over four meters tall; made of stone. Broken numbers from the mine were used for fractions. What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable The largest number In the children’s novel, The phantom tollbooth, numbers come from the number mine in Digitopolis. Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford A gigantic number 3. Over four meters tall; made of stone. Broken numbers from the mine were used for fractions. What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable The largest number In the children’s novel, The phantom tollbooth, numbers come from the number mine in Digitopolis. In the mines, they found the largest number! Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Broken numbers from the mine were used for fractions. What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable The largest number In the children’s novel, The phantom tollbooth, numbers come from the number mine in Digitopolis. In the mines, they found the largest number! A gigantic number 3. Over four meters tall; made of stone. Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable The largest number In the children’s novel, The phantom tollbooth, numbers come from the number mine in Digitopolis. In the mines, they found the largest number! A gigantic number 3. Over four meters tall; made of stone. Broken numbers from the mine were used for fractions. Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Triangular numbers can be arranged in a triangle. 1 3 6 10 15 One proceeds to the hexagonal numbers and so forth. What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable Kinds of numbers Square numbers can be arranged in a square. 1 4 9 16 25 Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable Kinds of numbers Square numbers can be arranged in a square. 1 4 9 16 25 Triangular numbers can be arranged in a triangle. 1 3 6 10 15 One proceeds to the hexagonal numbers and so forth. Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford But, this depends on the base we use to represent the number. Thus, palindromicity is not a property of the number, but of the numeral. Every number is a palindrome in any sufficiently large base. It becomes a single digit. What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable Palindromic numbers Palindromic numbers have digits reading the same forwards and backwards. 121 523323325 Like the phrase, “I prefer pi.” Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Every number is a palindrome in any sufficiently large base. It becomes a single digit. What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable Palindromic numbers Palindromic numbers have digits reading the same forwards and backwards. 121 523323325 Like the phrase, “I prefer pi.” But, this depends on the base we use to represent the number. Thus, palindromicity is not a property of the number, but of the numeral. Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable Palindromic numbers Palindromic numbers have digits reading the same forwards and backwards.