Was Kant an Intuitionist about mathematical objects? Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern) Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?How do mathematical objects relate to the real world?Was Kant right about space and time (and wrong about knowledge)?What are the properties of Mathematical Objects?Existence of mathematical objects: how?Was Kant incorrect to assert all maths as 'a priori'?For a mathematical realist, is there a distinction between real mathematical objects and constructed mathematical objects?Modern Mathematical Objects and EmpiricismWas Kant a factor in forming Gauss's abstract view of mathematical objects?Can a physicalist be also realist about mathematical objects?Distinguishing between procedure-like and collection-like mathematical objects

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Was Kant an Intuitionist about mathematical objects?



Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)
Announcing the arrival of Valued Associate #679: Cesar Manara
Unicorn Meta Zoo #1: Why another podcast?How do mathematical objects relate to the real world?Was Kant right about space and time (and wrong about knowledge)?What are the properties of Mathematical Objects?Existence of mathematical objects: how?Was Kant incorrect to assert all maths as 'a priori'?For a mathematical realist, is there a distinction between real mathematical objects and constructed mathematical objects?Modern Mathematical Objects and EmpiricismWas Kant a factor in forming Gauss's abstract view of mathematical objects?Can a physicalist be also realist about mathematical objects?Distinguishing between procedure-like and collection-like mathematical objects










3















In regards to the ontology of mathematics, as far as I can understand, Kant believed that Mathematical objects existed only as features of our perception that influenced how we viewed things-in-themselves, saying that, for example, geometry was the spatial lens by which we viewed phenomena whilst number arithmetic was the temporal lens.



If so, if one were to classify this position into a contemporary position on the ontology of mathematics, would it be fair to say that he was an intuitionist (i.e. he believed mathematical objects to be human constructs)? Or was he still a platonic realist in this regard?










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    3















    In regards to the ontology of mathematics, as far as I can understand, Kant believed that Mathematical objects existed only as features of our perception that influenced how we viewed things-in-themselves, saying that, for example, geometry was the spatial lens by which we viewed phenomena whilst number arithmetic was the temporal lens.



    If so, if one were to classify this position into a contemporary position on the ontology of mathematics, would it be fair to say that he was an intuitionist (i.e. he believed mathematical objects to be human constructs)? Or was he still a platonic realist in this regard?










    share|improve this question









    New contributor




    Aryaman Gupta is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.






















      3












      3








      3








      In regards to the ontology of mathematics, as far as I can understand, Kant believed that Mathematical objects existed only as features of our perception that influenced how we viewed things-in-themselves, saying that, for example, geometry was the spatial lens by which we viewed phenomena whilst number arithmetic was the temporal lens.



      If so, if one were to classify this position into a contemporary position on the ontology of mathematics, would it be fair to say that he was an intuitionist (i.e. he believed mathematical objects to be human constructs)? Or was he still a platonic realist in this regard?










      share|improve this question









      New contributor




      Aryaman Gupta is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.












      In regards to the ontology of mathematics, as far as I can understand, Kant believed that Mathematical objects existed only as features of our perception that influenced how we viewed things-in-themselves, saying that, for example, geometry was the spatial lens by which we viewed phenomena whilst number arithmetic was the temporal lens.



      If so, if one were to classify this position into a contemporary position on the ontology of mathematics, would it be fair to say that he was an intuitionist (i.e. he believed mathematical objects to be human constructs)? Or was he still a platonic realist in this regard?







      philosophy-of-mathematics kant explanation intuitionistic-logic






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      Aryaman Gupta is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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          You're right to see some resonances between intuitionism and Kant. However, there's no uncontroversial sense in which you could blithely categorize Kant as an intuitionist. I'm not sure I've ever seen anyone do it, as a matter of sociological note. More instructive, I think, than trying to get Kant to fit the intuitionist picture is to explore the subtle differences.



          First of all, it would be anachronistic to call Kant an intuitionist, since Brouwer didn't develop intuitionist mathematics until well after Kant's death. As a result, intuitionism as we know it was not an available position in Kant's time. So one would have to make the case that the anachronistic label was really warranted. This by itself, of course, does not mean that it would be inappropriate to retroactively sort Kant into the intuitionist paradigm.



          And there is some initial reason for finding this categorization attractive. Brouwer and Kant certainly agree on one really fundamental issue: namely, that mathematical truths are grounded in the nature of the mind. Certainly Brouwer and other intuitionists drew inspiration from Kant, e.g., from Kant's notion of intuition and the role that time plays in structuring intuition. In particular, Brouwer agrees with Kant that arithmetic truths are synthetic a priori. Brouwer departs strongly from Kant, though, with respect to the view of space. Brouwer denies that geometry is synthetic a priori, therefore denying the possibility of an a priori intuition of space.



          Furthermore, it is not so clear whether Kant was a constructivist about mathematics. By contrast, constructivism is at the heart of intuitionism (i.e., the position that mathematical existence can only be proven by constructing the alleged mathematical object, and not simply showing that its existence is consistent). Intuitionism is a species of constructivism, so that if Kant was not a constructivist, then you could not call him an intuitionist in any sense. It's true that Kant uses the word "construction" and claims that mathematics is in the business of constructing its concepts. But this is a potentially very misleading connection. Kant's definition of construction is:




          to construct a concept means to exhibit a priori the intuition corresponding to it (A713/B741).




          It's not at all clear that this is the notion of construction that intuitionists work with. For example, in constructing proofs the intuitionist will frequently construct an object by demonstrating an algorithm that is sufficient to construct it (this, in fact, is part of the appeal of intuitoinist logic for mathematicians interested in computability). In turn, the logic underlying intuitionism was certainly not Kant's logic; it requires a revisionary definitions of terms such as function and real number that would have certainly been alien to Kant. Let's put it this way: if Kant found himself revived from the dead and walked into a classroom where intuitionist mathematics was being taught, there would be much that he would not recognize.



          Would our revived, 21st-century Kant adopt intuitionism now that he finds it an available option? This is an intriguing question. What you'd have to do is to understand some of the subtle differences (just a couple of which I've highlighted above) and determine which of these doctrines are essential to Kant's philosophy of mathematics and which he might give up while retaining the integrity of his kernel of truth. It's also worth pointing out that several other mathematical philosophies take inspiration from the spirit, if not the letter, of Kant's own philosophy of mathematics (e.g., structuralism). This is important to note because it brings out the sense in which intuitionism certainly has no unique claim to the legacy of Kantian philosophy of mathematics, which points to the sense in which intuitionism is just one option among many that one could find implicit in the spirit of Kant's writings.



          Finally, though it is of course entirely possible to disconnect intuitionist mathematics from the idiosyncracies of its founder (Brouwer), it's also important to note that Brouwer thought of mathematics in a very different way than Kant. Brouwer held a mystical view of the mind's construction of mathematical objects that would be, to say the least, in violation of Kant's transcendental idealism.



          In short: No, Kant was not an intuitionist in any straightforward sense. However, the central resonances of Kant's view with certain general, fundamental intuitonist ideas would make an investigations of the details of the points of departure well worth studying.






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            You're right to see some resonances between intuitionism and Kant. However, there's no uncontroversial sense in which you could blithely categorize Kant as an intuitionist. I'm not sure I've ever seen anyone do it, as a matter of sociological note. More instructive, I think, than trying to get Kant to fit the intuitionist picture is to explore the subtle differences.



            First of all, it would be anachronistic to call Kant an intuitionist, since Brouwer didn't develop intuitionist mathematics until well after Kant's death. As a result, intuitionism as we know it was not an available position in Kant's time. So one would have to make the case that the anachronistic label was really warranted. This by itself, of course, does not mean that it would be inappropriate to retroactively sort Kant into the intuitionist paradigm.



            And there is some initial reason for finding this categorization attractive. Brouwer and Kant certainly agree on one really fundamental issue: namely, that mathematical truths are grounded in the nature of the mind. Certainly Brouwer and other intuitionists drew inspiration from Kant, e.g., from Kant's notion of intuition and the role that time plays in structuring intuition. In particular, Brouwer agrees with Kant that arithmetic truths are synthetic a priori. Brouwer departs strongly from Kant, though, with respect to the view of space. Brouwer denies that geometry is synthetic a priori, therefore denying the possibility of an a priori intuition of space.



            Furthermore, it is not so clear whether Kant was a constructivist about mathematics. By contrast, constructivism is at the heart of intuitionism (i.e., the position that mathematical existence can only be proven by constructing the alleged mathematical object, and not simply showing that its existence is consistent). Intuitionism is a species of constructivism, so that if Kant was not a constructivist, then you could not call him an intuitionist in any sense. It's true that Kant uses the word "construction" and claims that mathematics is in the business of constructing its concepts. But this is a potentially very misleading connection. Kant's definition of construction is:




            to construct a concept means to exhibit a priori the intuition corresponding to it (A713/B741).




            It's not at all clear that this is the notion of construction that intuitionists work with. For example, in constructing proofs the intuitionist will frequently construct an object by demonstrating an algorithm that is sufficient to construct it (this, in fact, is part of the appeal of intuitoinist logic for mathematicians interested in computability). In turn, the logic underlying intuitionism was certainly not Kant's logic; it requires a revisionary definitions of terms such as function and real number that would have certainly been alien to Kant. Let's put it this way: if Kant found himself revived from the dead and walked into a classroom where intuitionist mathematics was being taught, there would be much that he would not recognize.



            Would our revived, 21st-century Kant adopt intuitionism now that he finds it an available option? This is an intriguing question. What you'd have to do is to understand some of the subtle differences (just a couple of which I've highlighted above) and determine which of these doctrines are essential to Kant's philosophy of mathematics and which he might give up while retaining the integrity of his kernel of truth. It's also worth pointing out that several other mathematical philosophies take inspiration from the spirit, if not the letter, of Kant's own philosophy of mathematics (e.g., structuralism). This is important to note because it brings out the sense in which intuitionism certainly has no unique claim to the legacy of Kantian philosophy of mathematics, which points to the sense in which intuitionism is just one option among many that one could find implicit in the spirit of Kant's writings.



            Finally, though it is of course entirely possible to disconnect intuitionist mathematics from the idiosyncracies of its founder (Brouwer), it's also important to note that Brouwer thought of mathematics in a very different way than Kant. Brouwer held a mystical view of the mind's construction of mathematical objects that would be, to say the least, in violation of Kant's transcendental idealism.



            In short: No, Kant was not an intuitionist in any straightforward sense. However, the central resonances of Kant's view with certain general, fundamental intuitonist ideas would make an investigations of the details of the points of departure well worth studying.






            share|improve this answer





























              3














              You're right to see some resonances between intuitionism and Kant. However, there's no uncontroversial sense in which you could blithely categorize Kant as an intuitionist. I'm not sure I've ever seen anyone do it, as a matter of sociological note. More instructive, I think, than trying to get Kant to fit the intuitionist picture is to explore the subtle differences.



              First of all, it would be anachronistic to call Kant an intuitionist, since Brouwer didn't develop intuitionist mathematics until well after Kant's death. As a result, intuitionism as we know it was not an available position in Kant's time. So one would have to make the case that the anachronistic label was really warranted. This by itself, of course, does not mean that it would be inappropriate to retroactively sort Kant into the intuitionist paradigm.



              And there is some initial reason for finding this categorization attractive. Brouwer and Kant certainly agree on one really fundamental issue: namely, that mathematical truths are grounded in the nature of the mind. Certainly Brouwer and other intuitionists drew inspiration from Kant, e.g., from Kant's notion of intuition and the role that time plays in structuring intuition. In particular, Brouwer agrees with Kant that arithmetic truths are synthetic a priori. Brouwer departs strongly from Kant, though, with respect to the view of space. Brouwer denies that geometry is synthetic a priori, therefore denying the possibility of an a priori intuition of space.



              Furthermore, it is not so clear whether Kant was a constructivist about mathematics. By contrast, constructivism is at the heart of intuitionism (i.e., the position that mathematical existence can only be proven by constructing the alleged mathematical object, and not simply showing that its existence is consistent). Intuitionism is a species of constructivism, so that if Kant was not a constructivist, then you could not call him an intuitionist in any sense. It's true that Kant uses the word "construction" and claims that mathematics is in the business of constructing its concepts. But this is a potentially very misleading connection. Kant's definition of construction is:




              to construct a concept means to exhibit a priori the intuition corresponding to it (A713/B741).




              It's not at all clear that this is the notion of construction that intuitionists work with. For example, in constructing proofs the intuitionist will frequently construct an object by demonstrating an algorithm that is sufficient to construct it (this, in fact, is part of the appeal of intuitoinist logic for mathematicians interested in computability). In turn, the logic underlying intuitionism was certainly not Kant's logic; it requires a revisionary definitions of terms such as function and real number that would have certainly been alien to Kant. Let's put it this way: if Kant found himself revived from the dead and walked into a classroom where intuitionist mathematics was being taught, there would be much that he would not recognize.



              Would our revived, 21st-century Kant adopt intuitionism now that he finds it an available option? This is an intriguing question. What you'd have to do is to understand some of the subtle differences (just a couple of which I've highlighted above) and determine which of these doctrines are essential to Kant's philosophy of mathematics and which he might give up while retaining the integrity of his kernel of truth. It's also worth pointing out that several other mathematical philosophies take inspiration from the spirit, if not the letter, of Kant's own philosophy of mathematics (e.g., structuralism). This is important to note because it brings out the sense in which intuitionism certainly has no unique claim to the legacy of Kantian philosophy of mathematics, which points to the sense in which intuitionism is just one option among many that one could find implicit in the spirit of Kant's writings.



              Finally, though it is of course entirely possible to disconnect intuitionist mathematics from the idiosyncracies of its founder (Brouwer), it's also important to note that Brouwer thought of mathematics in a very different way than Kant. Brouwer held a mystical view of the mind's construction of mathematical objects that would be, to say the least, in violation of Kant's transcendental idealism.



              In short: No, Kant was not an intuitionist in any straightforward sense. However, the central resonances of Kant's view with certain general, fundamental intuitonist ideas would make an investigations of the details of the points of departure well worth studying.






              share|improve this answer



























                3












                3








                3







                You're right to see some resonances between intuitionism and Kant. However, there's no uncontroversial sense in which you could blithely categorize Kant as an intuitionist. I'm not sure I've ever seen anyone do it, as a matter of sociological note. More instructive, I think, than trying to get Kant to fit the intuitionist picture is to explore the subtle differences.



                First of all, it would be anachronistic to call Kant an intuitionist, since Brouwer didn't develop intuitionist mathematics until well after Kant's death. As a result, intuitionism as we know it was not an available position in Kant's time. So one would have to make the case that the anachronistic label was really warranted. This by itself, of course, does not mean that it would be inappropriate to retroactively sort Kant into the intuitionist paradigm.



                And there is some initial reason for finding this categorization attractive. Brouwer and Kant certainly agree on one really fundamental issue: namely, that mathematical truths are grounded in the nature of the mind. Certainly Brouwer and other intuitionists drew inspiration from Kant, e.g., from Kant's notion of intuition and the role that time plays in structuring intuition. In particular, Brouwer agrees with Kant that arithmetic truths are synthetic a priori. Brouwer departs strongly from Kant, though, with respect to the view of space. Brouwer denies that geometry is synthetic a priori, therefore denying the possibility of an a priori intuition of space.



                Furthermore, it is not so clear whether Kant was a constructivist about mathematics. By contrast, constructivism is at the heart of intuitionism (i.e., the position that mathematical existence can only be proven by constructing the alleged mathematical object, and not simply showing that its existence is consistent). Intuitionism is a species of constructivism, so that if Kant was not a constructivist, then you could not call him an intuitionist in any sense. It's true that Kant uses the word "construction" and claims that mathematics is in the business of constructing its concepts. But this is a potentially very misleading connection. Kant's definition of construction is:




                to construct a concept means to exhibit a priori the intuition corresponding to it (A713/B741).




                It's not at all clear that this is the notion of construction that intuitionists work with. For example, in constructing proofs the intuitionist will frequently construct an object by demonstrating an algorithm that is sufficient to construct it (this, in fact, is part of the appeal of intuitoinist logic for mathematicians interested in computability). In turn, the logic underlying intuitionism was certainly not Kant's logic; it requires a revisionary definitions of terms such as function and real number that would have certainly been alien to Kant. Let's put it this way: if Kant found himself revived from the dead and walked into a classroom where intuitionist mathematics was being taught, there would be much that he would not recognize.



                Would our revived, 21st-century Kant adopt intuitionism now that he finds it an available option? This is an intriguing question. What you'd have to do is to understand some of the subtle differences (just a couple of which I've highlighted above) and determine which of these doctrines are essential to Kant's philosophy of mathematics and which he might give up while retaining the integrity of his kernel of truth. It's also worth pointing out that several other mathematical philosophies take inspiration from the spirit, if not the letter, of Kant's own philosophy of mathematics (e.g., structuralism). This is important to note because it brings out the sense in which intuitionism certainly has no unique claim to the legacy of Kantian philosophy of mathematics, which points to the sense in which intuitionism is just one option among many that one could find implicit in the spirit of Kant's writings.



                Finally, though it is of course entirely possible to disconnect intuitionist mathematics from the idiosyncracies of its founder (Brouwer), it's also important to note that Brouwer thought of mathematics in a very different way than Kant. Brouwer held a mystical view of the mind's construction of mathematical objects that would be, to say the least, in violation of Kant's transcendental idealism.



                In short: No, Kant was not an intuitionist in any straightforward sense. However, the central resonances of Kant's view with certain general, fundamental intuitonist ideas would make an investigations of the details of the points of departure well worth studying.






                share|improve this answer















                You're right to see some resonances between intuitionism and Kant. However, there's no uncontroversial sense in which you could blithely categorize Kant as an intuitionist. I'm not sure I've ever seen anyone do it, as a matter of sociological note. More instructive, I think, than trying to get Kant to fit the intuitionist picture is to explore the subtle differences.



                First of all, it would be anachronistic to call Kant an intuitionist, since Brouwer didn't develop intuitionist mathematics until well after Kant's death. As a result, intuitionism as we know it was not an available position in Kant's time. So one would have to make the case that the anachronistic label was really warranted. This by itself, of course, does not mean that it would be inappropriate to retroactively sort Kant into the intuitionist paradigm.



                And there is some initial reason for finding this categorization attractive. Brouwer and Kant certainly agree on one really fundamental issue: namely, that mathematical truths are grounded in the nature of the mind. Certainly Brouwer and other intuitionists drew inspiration from Kant, e.g., from Kant's notion of intuition and the role that time plays in structuring intuition. In particular, Brouwer agrees with Kant that arithmetic truths are synthetic a priori. Brouwer departs strongly from Kant, though, with respect to the view of space. Brouwer denies that geometry is synthetic a priori, therefore denying the possibility of an a priori intuition of space.



                Furthermore, it is not so clear whether Kant was a constructivist about mathematics. By contrast, constructivism is at the heart of intuitionism (i.e., the position that mathematical existence can only be proven by constructing the alleged mathematical object, and not simply showing that its existence is consistent). Intuitionism is a species of constructivism, so that if Kant was not a constructivist, then you could not call him an intuitionist in any sense. It's true that Kant uses the word "construction" and claims that mathematics is in the business of constructing its concepts. But this is a potentially very misleading connection. Kant's definition of construction is:




                to construct a concept means to exhibit a priori the intuition corresponding to it (A713/B741).




                It's not at all clear that this is the notion of construction that intuitionists work with. For example, in constructing proofs the intuitionist will frequently construct an object by demonstrating an algorithm that is sufficient to construct it (this, in fact, is part of the appeal of intuitoinist logic for mathematicians interested in computability). In turn, the logic underlying intuitionism was certainly not Kant's logic; it requires a revisionary definitions of terms such as function and real number that would have certainly been alien to Kant. Let's put it this way: if Kant found himself revived from the dead and walked into a classroom where intuitionist mathematics was being taught, there would be much that he would not recognize.



                Would our revived, 21st-century Kant adopt intuitionism now that he finds it an available option? This is an intriguing question. What you'd have to do is to understand some of the subtle differences (just a couple of which I've highlighted above) and determine which of these doctrines are essential to Kant's philosophy of mathematics and which he might give up while retaining the integrity of his kernel of truth. It's also worth pointing out that several other mathematical philosophies take inspiration from the spirit, if not the letter, of Kant's own philosophy of mathematics (e.g., structuralism). This is important to note because it brings out the sense in which intuitionism certainly has no unique claim to the legacy of Kantian philosophy of mathematics, which points to the sense in which intuitionism is just one option among many that one could find implicit in the spirit of Kant's writings.



                Finally, though it is of course entirely possible to disconnect intuitionist mathematics from the idiosyncracies of its founder (Brouwer), it's also important to note that Brouwer thought of mathematics in a very different way than Kant. Brouwer held a mystical view of the mind's construction of mathematical objects that would be, to say the least, in violation of Kant's transcendental idealism.



                In short: No, Kant was not an intuitionist in any straightforward sense. However, the central resonances of Kant's view with certain general, fundamental intuitonist ideas would make an investigations of the details of the points of departure well worth studying.







                share|improve this answer














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                    Tom Holland Mục lục Đầu đời và giáo dục | Sự nghiệp | Cuộc sống cá nhân | Phim tham gia | Giải thưởng và đề cử | Chú thích | Liên kết ngoài | Trình đơn chuyển hướngProfile“Person Details for Thomas Stanley Holland, "England and Wales Birth Registration Index, 1837-2008" — FamilySearch.org”"Meet Tom Holland... the 16-year-old star of The Impossible""Schoolboy actor Tom Holland finds himself in Oscar contention for role in tsunami drama"“Naomi Watts on the Prince William and Harry's reaction to her film about the late Princess Diana”lưu trữ"Holland and Pflueger Are West End's Two New 'Billy Elliots'""I'm so envious of my son, the movie star! British writer Dominic Holland's spent 20 years trying to crack Hollywood - but he's been beaten to it by a very unlikely rival"“Richard and Margaret Povey of Jersey, Channel Islands, UK: Information about Thomas Stanley Holland”"Tom Holland to play Billy Elliot""New Billy Elliot leaving the garage"Billy Elliot the Musical - Tom Holland - Billy"A Tale of four Billys: Tom Holland""The Feel Good Factor""Thames Christian College schoolboys join Myleene Klass for The Feelgood Factor""Government launches £600,000 arts bursaries pilot""BILLY's Chapman, Holland, Gardner & Jackson-Keen Visit Prime Minister""Elton John 'blown away' by Billy Elliot fifth birthday" (video with John's interview and fragments of Holland's performance)"First News interviews Arrietty's Tom Holland"“33rd Critics' Circle Film Awards winners”“National Board of Review Current Awards”Bản gốc"Ron Howard Whaling Tale 'In The Heart Of The Sea' Casts Tom Holland"“'Spider-Man' Finds Tom Holland to Star as New Web-Slinger”lưu trữ“Captain America: Civil War (2016)”“Film Review: ‘Captain America: Civil War’”lưu trữ“‘Captain America: Civil War’ review: Choose your own avenger”lưu trữ“The Lost City of Z reviews”“Sony Pictures and Marvel Studios Find Their 'Spider-Man' Star and Director”“‘Mary Magdalene’, ‘Current War’ & ‘Wind River’ Get 2017 Release Dates From Weinstein”“Lionsgate Unleashing Daisy Ridley & Tom Holland Starrer ‘Chaos Walking’ In Cannes”“PTA's 'Master' Leads Chicago Film Critics Nominations, UPDATED: Houston and Indiana Critics Nominations”“Nominaciones Goya 2013 Telecinco Cinema – ENG”“Jameson Empire Film Awards: Martin Freeman wins best actor for performance in The Hobbit”“34th Annual Young Artist Awards”Bản gốc“Teen Choice Awards 2016—Captain America: Civil War Leads Second Wave of Nominations”“BAFTA Film Award Nominations: ‘La La Land’ Leads Race”“Saturn Awards Nominations 2017: 'Rogue One,' 'Walking Dead' Lead”Tom HollandTom HollandTom HollandTom Hollandmedia.gettyimages.comWorldCat Identities300279794no20130442900000 0004 0355 42791085670554170004732cb16706349t(data)XX5557367